1. the difference of two natural numbers is always an in ......math 1250-2 fall 2008 hsiang-ping...

Math 1250-2 Fall 2008

Hsiang-Ping Huang.

WeBWorK assignment number 1.

due 09/09/2008 at 11:59pm MDT.This first home work serves as an introduction to WeBWorK

and as a review of some relevant preCalculus topics, including:The language of the number system,Solution of linear, quadratic, and polynomial equations,Language of PolynomialsCartesian CoordinatesEquations and properties of straight lineslogic, converse, contrapositive, and negation of statementsfunctions, even, odd, graphs, compositionThe questions include a few word problems.Here are some hints on how to use WeBWorK effectively:After first logging into WeBWorK change your pass-

word.Make sure that at all times WeBWorK has your correct email

address.Find out how to print a hard copy on the computer system

that you are going to use. Contact me if you have any problems.Print a hard copy of this assignment. Note, however, that the on-line versions of these problems may have links to the web pagesof this course that are absent on the hard copy.

Get to work on each set right away and answer these ques-tions well before the deadline. Not only will this give you thechance to figure out what’s wrong if an answer is not accepted,you also will avoid the likely rush and congestion prior to thedeadline.

The primary purpose of the WeBWorK assignments in thisclass is to give you the opportunity to learn by having instantfeedback on your active solution of relevant problems. Makethe best of it!

Peter Alfeld, JWB 127, 581-6842.

1. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s1p1.pgRecall that the natural numbers are

1,2,3, . . . ,

the integers are

. . . ,−3,−2,−1,0,1,2,3, . . . ,

rational numbers are ratios of integers (with the denominator be-ing non-zero), and real numbers are all numbers correspondingto points on the number line. You can also think of real numbersas repeating or non-repeating decimals. There are more techni-cal definitions which you will learn in real analysis.

Indicate whether the following statements are True (T) or False(F).

1. The difference of two natural numbers is always an in-teger.

2. The quotient of two natural numbers is always a naturalnumber.

3. The quotient of two natural numbers is always a ratio-nal number

4. The product of two natural numbers is always a naturalnumber.

5. The ratio of two natural numbers is always positive6. The sum of two natural numbers is always a natural

number.7. The difference of two natural numbers is always a nat-

ural number.Correct Answers:

• T• F• T• T• T• T• F

2. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s1p2.pgIndicate whether the following statements are True (T) or False(F).

1. The quotient of two integers is always an integer (pro-vided the denominator is non-zero).

2. The difference of two integers is always a natural num-ber.

3. The quotient of two integers is always a rational num-ber (provided the denominator is non-zero).

4. The sum of two integers is always an integer.5. The ratio of two integers is always positive6. The difference of two integers is always an integer.7. The product of two integers is always an integer.

Correct Answers:

• F• F• T• T• F• T• T

3. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s1p3.pgIndicate whether the following statements are True (T) or False(F).

1. The difference of two rational numbers is always a ra-tional number.

2. The product of two rational numbers is always a ratio-nal number.

3. The quotient of two rational numbers is always a ratio-nal number (provided the denominator is non-zero).

4. The sum of two rational numbers is always a rationalnumber.

1

5. The quotient of two rational numbers is always a realnumber (provided the denominator is non-zero).

6. The ratio of two rational numbers is always positive7. The difference of two rational numbers is always a nat-

ural number.Correct Answers:

• T• T• T• T• T• F• F

4. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s1p4.pgSolve the equation

3(x+2)+4 =−5(x−1)−2

for x = .Hint: This is a linear equation.

Correct Answers:

• -0.875

5. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s1p5.pgSolve the equation

23− t

+1

3+ t+

19− t2 = 0

for t =

Hint: Get rid of the denominators by multiplying with them.Note that the third denominator is a difference of squares.

Correct Answers:

• -10

6. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s1p6.pgThe equation 4x4−2x3−3x2 = 0 has three real solutions A, B,and C where A < B < Cand A is:and B is:and C is:

Correct Answers:

• -0.651387818865997• 0• 1.151387818866

7. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s1p7.pgEvaluate the expression 64−4/3.

(You may enter a fraction as your answer.)

Correct Answers:

• 0.00390625

8. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s1p8.pgThe expression (3a4b5c2)2(2a5b2c4)3 equals narbsct

where n, the leading coefficient, is:and r, the exponent of a, is:and s, the exponent of b, is:and finally t, the exponent of c, is:

Correct Answers:• 72• 23• 16• 16

9. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s1p9.pgConsider the two points (5,−3) and (9,10). The distance be-tween them is:The x co-ordinate of the midpoint of the line segment that joinsthem is:The y co-ordinate of the midpoint of the line segment that joinsthem is:

Correct Answers:• 13.6014705087354• 7• 3.5

10. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s1p10.pgFind the distance between (8, 9) and (-8, -2).

Correct Answers:• 19.4164878389476

11. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s1p11.pgFind the perimeter of the triangle with the vertices at(1, 0), (-2, 4), and (-7, -4).


12. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s1p12.pgThe equation of the line with slope 5 that goes through thepoint (5,5) can be written in the form y = mx + b where m is:

and where b is:

Correct Answers:• 5• -20

13. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s1p13.pgThis problem is like the preceding problem. The equation of theline with slope 4.5 that goes through the point (−8.9,6) can bewritten in the form y = mx+b where m is:and where b is:


2

• 46.05

14. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s1p14.pgThe equation of the line that goes through the point (21,29) andis parallel to the x-axis can be written in the form y = mx + bwhere m is:and where b is:

Correct Answers:

• 0• 29

15. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s1p15.pgThe equation of the line that goes through the point (3,9) andis parallel to the line 2x + 3y = 3 can be written in the formy = mx+b where m is:and where b is:

Correct Answers:

• -0.666666666666667• 11

16. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s1p16.pgThe equation of the line that goes through the point (6,4) and isperpendicular to the line 5x +2y = 4 can be written in the formy = mx+b where m is:and where b is:

Correct Answers:

• 0.4• 1.6

17. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s1p17.pgA line through (-2, -5) with a slope of -9 has a y-intercept at

Correct Answers:

• -23

18. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s1p18.pgAn equation of a line through (-2, 3) which is perpendicular tothe line y = 4x+2 has slope:

and y intercept at:

Correct Answers:

• -0.25• 2.5

19. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s1p19.pgFind the slope of the line through (8, -1) and (6, 6).

Correct Answers:

• -3.5

20. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s1p20.pgThe distance of the point (3,4) from the line y =−3x−1is:

Correct Answers:

• 4.42718872423573

21. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s1p21.pgThis is like the preceding problem. The distance of the point(−1,5) from the line y = 3x−1is:

Correct Answers:

• 2.84604989415154

22. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s1p22.pgYou drop a rock into a deep well. You can’t see the rock’s im-pact at the bottom, but you hear it after 6 seconds. The depthof the well is feet. Ignore air resistance. The time thatpasses after you drop the rock has two components: the time ittakes the rock to reach the bottom of the well, and the time thatit takes the sound of the impact to travel back to you. Assumethe speed of sound is 1100 feet per second.Note: After t seconds the rock has reached a depth of d feetwhere

d = 16t2.

Set up and solve a quadratic equation.Correct Answers:

• 493.140302367796

23. (10 pts) set1/1210s1p23.pgThis is the general version of the preceding problem. Supposethe speed of sound is c, and gravity is g. Suppose you hear theimpact after t seconds. Then the depth of this well isd = .(Your answer will be a mathematical expression involving t, g,and c.)Set up and solve a quadratic equation.

Correct Answers:

• (-c/sqrt(2*g)+sqrt(c**2/(2*g)+c*t))**2

24. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s1p24.pgAssume that the cutoff value for an A is 90 and the weight of allww homeworks is 42%. Suppose you obtain 100 percent crediton all WeBWorK assignments in this class. Then the minimumaverage percentage on the exams that will still get you an A inthis class is . Your answer should be a number be-tween 0 and 100. You may enter a fraction.Hint: Clearly distinguish between the various meanings of per-cent in this question.

Correct Answers:

• 82.7586206896552

3

25. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s1p25.pgJamestown is 10 miles downstream from Aliceville and on theopposite side of a river half a mile wide. Mary will run fromAliceville along the river for 6 miles, and then swim diagonally(i.e., along a straight line) across the river to Jamestown. If sheruns at 8 miles per hour and swims at three miles per hour shewill reach Jamestown after hours. Assume that thecurrent is negligible.

Note that unless otherwise specified WeBWorK expects youranswer to be within ”one tenth of one percent” of the ”true an-swer” which is the answer it has been given by the author of theproblem. Practically speaking this means you should specify atleast 4 digits total (including any before the decimal point). Tobe safe, you want to compute your answer with your calculatormuch more accurately.

Hint: Draw a picture and apply the Pythagorean Theorem;Correct Answers:

• 2.09370962471642

26. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s1p27.pgThe next few problems are simple exercises in logic. Considerthe statement ”A implies B”. The converse of this statementis ”B implies A”, its contrapositive is ”not B implies not A”,and its negation is ”A and not B”. For example, consider thestatement all natural numbers are real numbers. This is a truestatement, but this is actually not important for this discussion.This statement can be put as an implication: if x is a naturalnumber, then x is a real number. The converse of this state-ment is all real numbers are natural numbers, or if x is a realnumber then it is a natural number, (which is a false statement),the contrapositive is if x is not a real number then it isn’t a nat-ural number (which is a true statement), and the negation of thestatement is some natural numbers are not real numbers (whichis a false statement).

The purpose this particular problem is to illustrate the pattern ofthe next couple of problems. You already know the answers, soit is just a matter of entering them. In this problem, WeBWorKwill tell you separately for each answer whether it is correct ornot, but in the next two you will have to enter everything cor-rectly to get credit.

Let S be the statement: All natural numbers are real num-bers.

Complete the sentences below, filling in A-G from the fol-lowing list, and T or F for true or false, as appropriate.

A. Numbers aren’t natural.

B. All real numbers are natural.

C. Some natural numbers aren’t real numbers.

D. Some real numbers aren’t natural numbers.

E. No real number is natural.

F. A number can’t be natural if it isn’t real.

G. A number can’t be real if it isn’t natural.

S is (true or false).

The converse of S is (enter a letter from A-G) and that state-ment is .The contrapositive of S is and that statement is .The negation of S is and that statement is (true or false).

Correct Answers:

• T• B• F• F• T• C• F

27. (10 pts) set1/1210s1p28.pgLet S be the statement: All women are humans.


A. Some women aren’t humans.

B. Some women are humans.

C. Some humans aren’t women.

D. No women are humans.

E. If you aren’t a human you aren’t a woman.

F. Men aren’t human.

G. All humans are women.4


The converse of ”S” is (enter a letter from A-G) and thatstatement is (true or false).The contrapositive of S is and that statement is .The negation of S is and that statement is .

There are two versions of this problem with the roles of ”men”versus ”women” randomly interchanged.

Correct Answers:

• T• G• F• E• T• A• F

28. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s1p29.pgThroughout this problem, suppose that x stands for a real num-ber.

Let S be the statement: if x > 0 then x3 > 0.


A. Real numbers are all zero.

B. If x3 > 0, then x > 0.

C. There is some x > 0 such that x3 ≤ 0

D. If x≥ 0 then x3 ≤ 0.

E. If x > 0 then x3 > x.

F. x3 > x.

G. If x3 ≤ 0 then x≤ 0.


The converse of S is (enter a letter from A-G) and that state-ment is (true or false).

The contrapositive of S is and that statement is .The negation of S is and that statement is .

Correct Answers:

• T• B• T• G• T• C• F

29. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s1p30.pgThis problem addresses some common algebraic errors. For theequalities stated below assume that x and y stand for real num-bers. Assume that any denominators are non-zero. Mark theequalities with T (true) if they are true for all values of x and y,and F (false) otherwise.

(x+ y)2 = x2 + y2.(x+ y)2 = x2 +2xy+ y2.

xx+y = 1

y .x− (x+ y) = y.√

x2 = x.√x2 = |x|.√x2 +4 = x+2.

1x+y = 1

x + 1y .

Correct Answers:

• F• T• F• F• F• T• F• F

30. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s2p1.pgLet

f (x) = mx+b

(where m 6= 0). The graph of f is a straight line with slope .Any line perpendicular to that graph has a slope .The graph of the inverse function of f is a straight line withslope .

Correct Answers:

• m• -1/m• 1/m

31. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s2p2.pgIn this question you will derive a general formula for the dis-tance of a point from a line.

Let P be the point (p,q) and L the line y = mx+b.The slope of L is .

5

The slope of a line perpendicular to L is.

The line through P perpendicular to L can be written asy = sx+ cwhere s is:and c is: .That line intersects L in the point Q = (u,v),

where u is:and v is: .

The distance of P and Q is .(Note, all answers must be in terms of m, b, p and q.)

Hint: If you are bewildered by all the symbols ask yourself whatthey mean in the special cases of the preceding home work, andcompare your calculations for this problem with the numericalcalculations you did earlier.

Correct Answers:• m• -1/m• -1/m• q+p/m• (m*(q+p/m - b))/(1+m*m)• m*(m*(q+p/m - b))/(1+m*m) + b• sqrt((q-m*p-b)**2/(1+m**2))

32. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s2p3.pgThis problem is about polynomial degrees. Recall that the de-gree of a polynomial p is n if p can be written as

p(x) = anxn +an−1xn−1 + . . .+a1x+a0

where the leading coefficient

an 6= 0.

The degree of p(x) = 1 is .The degree of p(x) = x2 +2x+3 is .The degree of p(x) =

((x+2)2(x−1)3

)2 is .Let p and q be polynomials of degree m and n, respectively.The degree of the product of p and q is .and the degree of the composition of p and q is .Hint: The product of two polynomials p and q is

h(x) = p(x)×q(x)

and the composition of p and q is

h(x) = p(q(x))

orh(x) = q(p(x)).

(The two compositions are distinct but they have the same de-gree. If you are still confused look at simple examples.

Correct Answers:• 0• 2• 10• m+n• m*n

33. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s2p4.pgThe next few problems concern even and odd functions. Recallthat a function f is even if

f (x) = f (−x)

for all x in its domain, and it is odd if

f (x) =− f (−x)

for all x in its domain. The graph of an even functions is sym-metric with respect to the y-axis, and an odd function is sym-metric with respect to the origin. This is an example of one ofour major themes: the interplay between algebra and geometry.

For each of the following functions enter ”E” to indicate thatthe function is even, ”O” to indicate it is odd, and ”N” to indi-cate that is neither even nor odd.

1. f (x) = x3 + x9 + x7

2. f (x) = x6−6x10 +3x2

3. f (x) = x−6

4. f (x) = x6 +3x10 +2x7

Correct Answers:

• O• E• E• N

34. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s2p5.pgOddly, there is a function that is both even and odd. It is f (x) =

.Correct Answers:

• 0

35. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s2p6.pgFor each of the following functions enter ”E” to indicate that thefunction is even, ”O” to indicate it is odd, and ”N” to indicatethat is neither even nor odd.

f (x) = sinx.f (x) = cosx.f (x) = tanx.

Correct Answers:

• O• E• O

6

36. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s2p7.pgFor each of the following functions enter ”E” to indicate that thefunction is even, ”O” to indicate it is odd, and ”N” to indicatethat is neither even nor odd.

f (x) = sin2 x.f (x) = sinx2.f (x) = sin(cosx).f (x) = sin(sinx).f (x) = sinx+ cosx.

Correct Answers:

• E• E• E• O• N

37. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s2p8.pgEnter a T or an F in each answer space below to indicate whetherthe corresponding statement is true or false.You must get all of the answers correct to receive credit.

1. The ratio of two odd functions is odd2. The sum of two even functions is even3. The composition of an odd function and an odd func-

tion is even4. The product of two odd function is odd5. The product of two even function is even6. The sum of an even and an odd function is usually nei-

ther even or odd, but it may be even.7. A function cannot be both even and odd.8. The composition of an even and an odd function is even

All of the answers must be correct before you get credit forthe problem.

Correct Answers:

• F• T• F• F• T• T• F• T

38. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s2p9.pgThe next few problems concern the composition of functions,and the fact that this is different from the multiplication of func-tions. If f and g are two functions then their product f g is de-fined by

( f g)(x) = f (x)g(x).

As usual, the absence of an arithmetic operator denotes multi-plication.

The composition f ◦ g of these two functions is quite different.We evaluate first one, and then the other. Thus

( f ◦g)(x) = f(g(x)

).

Let f (x) = 3x+5 and g(x) = 5x2 +3x.( f g)(3) =

Correct Answers:

• 756

39. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s2p10.pgLet f (x) = 4x+6 and g(x) = 4x2 +8x. Then( f g)(x) =

Correct Answers:

• 16*xˆ3+56*xˆ2+48*x

40. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s2p11.pgLet f (x) = 6x+4 and g(x) = 6x2 +6x.( f ◦g)(2) =

Correct Answers:

• 220

41. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s2p12.pgLet f (x) = 3x+3 and g(x) = 3x2 +3x.

Then,( f ◦g)(x) =

Correct Answers:

• 9*x**2 + 9*x + 3

42. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s2p13.pgLet f (x) = x+1 and g(x) = 1

x+1 . Then

( f ◦ f )(x) = ,( f ◦g)(x) = ,(g◦ f )(x) = ,(g◦g)(x) = .

Correct Answers:

• x+2• (x+2)/(x+1)• 1/(x+2)• (x+1)/(x+2)

43. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s2p14.pg

Let g(x) = x+1 andh(x) = ( f ◦g)(x) = x2 +2x+1.Thenf (x) = , and(g◦ f )(x) = .

Correct Answers:

• xˆ2• xˆ2+1

7


Let g(x) = x2 andh(x) = ( f ◦g)(x) = sinx2.Thenf (x) = , and(g◦ f )(x) = .

Correct Answers:

• sin(x)• sin(x)ˆ2

45. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s2p16.pgThe next few questions provide another variation of the inter-play between algebra and geometry. Simple algebraic modifi-cations have simple effects on the graph. Adding to x shifts thegraph left or right, adding to y shifts it up or down, multiplyingx rescales it horizontally, and multiplying y rescales it vertically.These effects can of course be combined.

Relative to the graph of

y = x2

the graphs of the following equations have been changed in whatway?

1. y = x2−102. y = x2 +103. y = (x/10)2

4. y = (10x)2

A. stretched horizontally by the factor 10B. shifted 10 units upC. shifted 10 units downD. compressed horizontally by the factor 10

Correct Answers:

• C• B• A• D

46. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s2p17.pgRelative to the graph of

y = sin(x)


1. y = sin(x)−92. y = sin(x+9)3. y = sin(x)/94. y = sin(9x)

A. shifted 9 units downB. compressed vertically by the factor 9C. compressed horizontally by the factor 9D. shifted 9 units left

Correct Answers:• A• D• B• C


Let f (x) = x2 + sinx and let g(x) = whereg is the function whose graph has been obtained from that of fby shifting it 5 to the right and 8 up.

Correct Answers:• (x-5)ˆ2+sin(x-5) + 8


The function f (x) = x2 + 8x + 23 can be obtained from aneven function g by shifting its graph horizontally and vertically.That even function is g(x) = .

Its graph has been shifted by to the leftand up.

Hint: Complete the Square.Correct Answers:• xˆ2• 4• 7

49. (10 pts) 1250Library/set1 Reviews of Fundamentals/1210s2p20.pgRelative to the graph of

y =1x2


1. y = 1(x−14)2

2. y = 1x2 −14

3. y = 114x2

4. y = 1196x2

A. compressed horizontally by the factor 14B. compressed vertically by the factor 14C. shifted 14 units rightD. shifted 14 units down

Correct Answers:• C• D• B• A

Generated by the WeBWorK system c©WeBWorK Team, Department of Mathematics, University of Rochester

8

Math 1250-2 Fall 2008

Hsiang-Ping Huang.


due 09/16/2008 at 11:59pm MDT.This problem set is on limits and derivatives, and some ap-

plications.Peter Alfeld, JWB 127, 581-6842.

1. (10 pts) 1250Library/set2 Derivatives and Limits/1210s2p21.pgEvaluate the limitlimx→1

x−1x2 +5x−6

= .Correct Answers:

• 0.142857142857143

2. (10 pts) 1250Library/set2 Derivatives and Limits/1210s2p22.pgEvaluate the limit

limx→1

x3− xx2−1

= .Correct Answers:

• 1

3. (10 pts) 1250Library/set2 Derivatives and Limits/1210s2p23.pgEvaluate the limitlim

b→36

36−b6−

√b

= .

Correct Answers:

• 12

4. (10 pts) 1250Library/set2 Derivatives and Limits/1210s2p24.pg

Evaluate the limit limx→6

1x −

16

x−6= .

Correct Answers:

• -0.0277777777777778


Evaluate the limit limx→−3

7x2−7x+7x−5

=.

Correct Answers:

• -11.375


limh−→0

(x+h)3− x3

h= .

(Your answer will be a mathematical expression in x.)Correct Answers:

• 3xˆ2


Letf (x) = x2−2x+5.

Then

limh−→0

f (x+h)− f (x)h

= .Correct Answers:• 2*x-2


Letf (x) = 8x2−5x+5.

Then

limh−→0

f (x+h)− f (x)h

= .Correct Answers:• 2*8*x-5

9. (10 pts) set2/1210s2p29.pgRecall that

limx−→c

f (x) = Lmeans:

For all ε > 0 there is a δ > 0 such that for all x satisfying0 < |x− c|< δ we have that | f (x)−L|< ε.

What if the limit does not equal L? Think about what that meansin ε,δ language.

Consider the following phrases:

1. ε > 02. δ > 03. 0 < |x− c|< δ

4. | f (x)−L|> ε

5. but6. such that for all7. there is some8. there is some x such that

Order these statements so that they form a rigorous assertionthat

limx−→c

f (x) 6= L

and enter their reference numbers in the appropriate sequence inthese boxes:

Correct Answers:• 7• 1• 6• 2• 8• 3

1

• 5• 4


This exercise will help you review some basic trigonometryon a right triangle. Recall that, by the Pythagorean Theorem,a triangle with sides a. b and c has a right angle opposite the(longest) side c if and only if

a2 +b2 = c2.

This is certainly true for the triangle in this problem. If neces-sary, review the definitions of the trigonometric functions andtheir inverses to solve this problem.

Consider the familiar right triangle with sides of lengths 3, 4,and 5 feet. Let A be the angle opposite the side of length 3, andB the angle opposite the side of length 4 feet.A = radians = degrees, andB = radians = degrees.

Correct Answers:

• 0.643501108793284• 36.869897645844• 0.927295218001612• 53.130102354156

11. (10 pts) 1250Library/set2 Derivatives and Limits/1210s3p1.pgMany times in this class we will solve quadratic equations. Youmay be used to applying the quadratic formula, and that’s fine,but of course the variables a, b, and c may occur in your qua-dratic equations in different ways than they are used in the qua-dratic formula. This problem illustrates the translation from for-mula to problem, that may be necessary.

There are two solutions of the equation

bx2− cx+a2 = 0

(where a, b, and c are constants, and x is the unknown). Theydiffer by the sign of the square root. Enter the one with the plussign here .

Correct Answers:

• (c+sqrt(cˆ2-4*aˆ2*b))/(2*b)


Evaluate limx−→3

x+1x+2

= .Correct Answers:

• 0.8


limx−→5

x2−13x+40x−5

= .Correct Answers:

• -3


limx−→2

x2−5x+6x−2

= .Correct Answers:• -1


limx−→4

x−4√x−2

= .

Correct Answers:• 4


limx−→∞

x−4x+4

= .

limx−→∞

x−4x2 +4

= .Correct Answers:• 1• 0

17. (10 pts) 1250Library/set2 Derivatives and Limits/1210s3p10.pgIf you toss a rock at an initial height H with an initial velocity Vthen its height h(t) after t seconds is given by the formula

h(t) =−12

gt2 +Vt +H

whereg = 32

feetsecond2

on Earth. (On other celestial bodies g would be different. Theminus sign is due to the convention that the positive direction isup and gravity pulls down.)

You have probably seen this formula before, and chances areyou were simply told that this is the way it is. With Calculus,we can make perfect sense of this formula. The underlying ob-servation is the experimentally observed fact that, ignoring airresistance, on Earth a free falling object increases its speed by32 feet per second every second. The velocity of the object isthe derivative of height, and the acceleration is the derivative ofvelocity. So we have to work out that the formula given abovehas the right properties.

If h is given by the above expression, thenh(0) = .The velocity v(t) isv(t) = h′(t) = andv(0) = .Moreover, the acceleration isv′(t) = .

Correct Answers:• H

2

• -g*t+V• V• -g

18. (10 pts) 1250Library/set2 Derivatives and Limits/1210s3p11.pgThe point P(3,19) lies on the curve y = x2 + x + 7. If Q is thepoint (x,x2 + x +7), find the slope of the secant line PQ for thefollowing values of x.If x = 3.1, the slope of PQ is:and if x = 3.01, the slope of PQ is:and if x = 2.9, the slope of PQ is:and if x = 2.99, the slope of PQ is:Based on the above results, guess the slope of the tangent lineto the curve at P(3,19).

Correct Answers:

• 7.1• 7.01• 6.9• 6.99• 7

19. (10 pts) 1250Library/set2 Derivatives and Limits/1210s3p12.pgIf a ball is thrown straight up into the air with an initial ve-locity of 50 ft/s, its height in feet after t second is given byy = 50t − 16t2. Find the average velocity for the time periodbeginning when t = 1 and lasting(i) 0.5 seconds

(ii) 0.1 seconds

(iii) 0.01 seconds

Finally based on the above results, guess what the instanta-neous velocity of the ball is when t = 1.

Correct Answers:

• 10• 16.4• 17.84• 18

20. (10 pts) 1250Library/set2 Derivatives and Limits/1210s3p13.pgThis a simple exercise in computing derivatives of polynomials.The derivative of

p(x) = 8x2 +3x+4

is p′(x) = .The derivative of

q(x) = 7x5−2x4 +8x3−5x2 +7x+8

is q′(x) = .Correct Answers:

• 2*8 x + 3• 5*7*xˆ4 - 4*2 * xˆ3 + 3*8* xˆ2-2*5 x + 7

21. (10 pts) 1250Library/set2 Derivatives and Limits/1210s3p14.pgMore derivatives: The derivative of

p(x) = (2x−1)2

is p′(x) = .The derivative of

q(x) = (2x−1)3

is q′(x) = .Hint: As we will learn soon, there are more advanced ways ofdoing this, but you can compute the derivative by expanding thegiven expressions and writing each of p and q in the standardform of a polynomial.

Correct Answers:• 4*(2x-1)• 3*(2x-1)**2*2

22. (10 pts) 1250Library/set2 Derivatives and Limits/1210s3p15.pgIf f (x) = 12x+25, thenf ′(−8)= .


23. (10 pts) 1250Library/set2 Derivatives and Limits/1210s3p16.pgIf f (x) = 2+4x−2x2 thenf ′(0) = .


24. (10 pts) 1250Library/set2 Derivatives and Limits/1210s3p17.pgIf f (x) = 3x2−2x−35, thenf ′(x) = ,andf ′(1)= .

Correct Answers:• 2*3*x-2• 4

25. (10 pts) 1250Library/set2 Derivatives and Limits/1210s3p18.pgLet

f (x) =1

x+3Use the limit definition of the derivative on page 107 to find(i) f ′(−7) =(ii) f ′(−5) =(iii) f ′(−2) =(iv) f ′(1) =

To avoid calculating four separate limits, I suggest that youevaluate the derivative at the point when x = a. Once you havethe derivative, you can just plug in those four values for ”a” toget the answers.

Correct Answers:• -0.0625• -0.25• -1

3

• -0.0625

26. (10 pts) 1250Library/set2 Derivatives and Limits/1210s3p19.pgThis problem will help you practice computing tangents in thenext problem. Let

f (x) = x2.

Then f ′(x) = .

The tangent to the graph of f through the point (1,1) has theslope and the

y-intercept .It intercepts the x-axis at x = .

Correct Answers:

• 2x• 2• -1• 0.5

27. (10 pts) 1250Library/set2 Derivatives and Limits/1210s3p20.pgThe slope of the tangent line to the parabola y = 2x2 +5x+4 atthe point (4,56) is:The equation of this tangent line can be written in the formy = mx+b where m is:and where b is:

Correct Answers:

• 21• 21• -28

28. (10 pts) 1250Library/set2 Derivatives and Limits/1210s3p21.pgThe slope of the tangent line to the curve y = 4x3 at the point(−2,−32) is:The equation of this tangent line can be written in the formy = mx+b where m is:and where b is:

Correct Answers:

• 48• 48• 64

29. (10 pts) 1250Library/set2 Derivatives and Limits/1210s3p22.pgThis problem is our first introduction to Newton’s Method forthe solution of nonlinear equations.

The positive solution of

f (x) = x2−2 = 0

is obviouslyx =

√2.

How might one calculate a numerical value of√

2? The idea ofNewton’s method is to pick a guess x0, compute the tangent tothe graph of f at

(x0, f (x0)

), and then replace the guess x0 with

x1 which is the x intercept of the tangent. (You should draw apicture of this idea.)

Suppose x0 = 1.5. The tangent to the graph of f at the point(x0, f (x0)) can be written in slope intercept form as

y = x+ .The tangent intercepts the x-axis at x1 = .

Now let’s repeat the process. The tangent to the graph of fat the point (x1, f (x1)) can be written in slope intercept form as

y = x+ .The tangent intercepts the x-axis at x2 = .Note the accuracy of this approximation:√

2− x2 = .Note: To obtain the same result in your last answer as ww youneed to compute all intermediate answers to the full accuracy ofyour calculator. To accomplish this store all intermediate resultsin your calculator. Do not copy them on paper and then reenterthem by hand later. Doing so introduces rounding errors andcompromises the accuracy of your calculations and solutions.In fact, you should make a habit of this every time you use yourcalculator: store intermediate results and avoid having to reen-ter them. Most calculators keep more digits internally than theycan display, so losing accuracy by copying and reentering is in-evitable.

Correct Answers:

• 3• -4.25• 1.41666666666667• 2.83333333333333• -4.00694444444444• 1.41421568627451• -2.12390141474117E-06

30. (10 pts) 1250Library/set2 Derivatives and Limits/1210s3p23.pgA particle moves along a straight line and its position at time tis given by s(t) = 2t3− 24t2 + 90t where s is measured in feetand t in seconds.Find the velocity (in ft/sec) of the particle at time t = 0:

The particle stops moving (i.e. is in a rest) twice, once whent = A and again when t = B where A < B. A isand B isWhat is the position of the particle at time 16?

Finally, what is the TOTAL distance the particle travels betweentime 0 and time 16?

Correct Answers:

• 90• 3• 5• 3488• 3504

4

31. (10 pts) 1250Library/set2 Derivatives and Limits/1210s3p24.pgYou toss a rock up vertically at an initial speed of 26 feet persecond and release it at an initial height of 6 feet. The rock willremain in the air for seconds.It will reach a maximum height of feet after seconds.Note: Ignore air resistance.

Correct Answers:

• 1.82992628725623• 16.5625• 0.8125

32. (10 pts) 1250Library/set2 Derivatives and Limits/1210s3p25.pgThis is the general version of the previous problem. Again, ig-nore air resistance. Assume a free falling object accelerates atg feet per second. (On Earth, of course, g = 32.) Your answerswill be mathematical expressions involving g, V , and H.

You toss a rock up vertically at an initial speed of V feet persecond and release it at an initial height of H feet. The rock willremain in the air for seconds.It will reach a maximum height of feet after

seconds.Correct Answers:

• (V+sqrt(Vˆ2+2*g*H))/g• V*V/2/g+H• V/g

33. (10 pts) 1250Library/set2 Derivatives and Limits/1210s3p26.pgLet’s now combine our rock tossing with a horizontal motion.This kind of problem can be handled by considering the verti-cal and horizontal motions separately. The horizontal velocityis constant. (Again, we ignore air resistance.) Recall from yourstudy of trigonometry that if you release a rock at a speed v in adirection that makes an angle α with the horizontal, then the ini-tial vertical velocity vv and the horizontal velocity vh are givenby

vv = vsinα and vh = vcosα.

You shoot a rifle at an angle of 46 degrees. The bullet leavesyour rifle at a height of 6 feet and a speed of 854 feet per sec-ond. It hits the ground after seconds at a distance of

feet.Note: Again, ignore air resistance. When shooting a rifle this isof course a gross simplification. Even so, that’s some rifle!

Correct Answers:

• 38.4045263171092• 22783.0339219176

34. (10 pts) 1250Library/set2 Derivatives and Limits/1210s3p27.pgThe ideas we are practicing in these exercises can be put to prac-tical use. My son likes to shoot arrows. He was wondering atwhat speed the arrow leaves the bow when it is released. So wewent to the Great Salt Lake and shot the bow at an angle of 45degrees. The arrow hit the ground at a distance of 600 feet. All

of this was measured very roughly, but lets work with these fig-ures, and, again, let’s ignore air resistance. For simplicity let’salso assume that the arrow is released at an initial height of 0feet.

The arrow leaves the bow at a speed of miles perhour.

Note: My son and I did worry about the effect of air resistance.The arrows were sticking out of the sand where they had hit thebeach. So my son shot another arrow downwards at an angle of45 degrees straight into the sand. It penetrated to about the samedepth as the arrows that had flown the distance. So they musthave hit at about the same speed. We concluded that ignoringair resistance in this case was reasonable.

Hint: The speed of the arrow is measured in the direction inwhich it flies. The significance of shooting at an angle of 45 de-grees is that initially the horizontal and the vertical componentsof the speed are equal. There are 3600 seconds in an hour and5280 feet in a mile.

Correct Answers:

• 94.475498594666

35. (10 pts) 1250Library/set2 Derivatives and Limits/1210s3p28.pgActually, once you get the hang of it, you’ll prefer using variablerather than numbers, and putting in the numbers only at the endof the calculation. So let’s do the previous problem in general.However, since we are not likely to leave Earth, let’s continuewith g = 32 feet per second squared. (To write the solution ofthis problem I did very little beyond taking the solution of theprevious problem and replacing 600 with d throughout.)

You shoot an arrow at an angle of 45 degrees. It hits theground at a distance of d feet. The arrow left the bow at a speedof miles per hour.


Correct Answers:

• sqrt(32*d)/5280*3600

36. (10 pts) 1250Library/set2 Derivatives and Limits/1210s3p29.pgIf a ball is thrown vertically upward from the roof of a 64 foottall building with a velocity of 80 ft/sec, its height after t sec-onds is s(t) = 64+80t−16t2. What is the maximum height theball reaches?What is the velocity of the ball when it hits the ground (height0)?

5

Correct Answers:• 164• -102.449987798926

37. (10 pts) 1250Library/set2 Derivatives and Limits/ind.pgThis fun little question was motivated by a discussion I had withstudents after class. Consider the absurd statement:

All numbers are equal.Figure out what is wrong with the proof below and email me

your explanation. I will forward the best explanations to theclass, with your name attached.

Proof: Let n be a natural number. Let Sn be the statementthat in a set of n numbers all numbers are equal. I shall prove byinduction that Sn is true for all natural numbers n = 1,2,3, . . ..

Clearly, S1 is true. If a set contains just one number then allnumbers in that set are equal.

I now need to show that the truth of Sk implies the truth ofSk+1. (Note: In class I used n instead of k. I m hoping that k isclearer. Using k instead of n is not the problem.) Let

M = {a1,a2, . . . ,ak+1}

be a set of k +1 numbers. Then the subset

M1 = {a1,a2, . . . ,ak}

of M contains k numbers, and by the induction hypothesis theseare all equal. Similarly, the subset

M2 = {a2,a3, . . . ,ak+1}

of M contains k numbers all of which are equal.In other words, we have

a1 = a2 = . . . = aka2 = . . . = ak = ak+1

Together these equations imply that

a1 = a2 = . . . = ak = ak+1

Thus all the numbers in M are equal. QEDTo get credit for this ww question, type the sentence ”I read

this”, without the quotation marks, here: .Correct Answers:

• Ireadthis


6

Math 1250-2 Fall 2008

Hsiang-Ping Huang.


due 09/23/2008 at 11:59pm MDT.This set of exercises covers differentiation rules, related

rates, and Newton’s Method.Peter Alfeld, JWB 127, 581-6842.

1. (10 pts) 1250Library/set3 Rates of Change and the Chain Rule-/1210s4p5.pgIf

f (x) = (x2 +5x+2)4

thenf ′(x) =

Correct Answers:

• (4*(x**2+5*x+2)**(4-1))*(2*x+5)


f (x) = (x3 +2x+2)2

thenf ′(x) =

Correct Answers:

• (2*(x**3+2*x+2)**(2-1))*(3*x*x+2)


f (x) = (3x+7)−2

then f ′(x) =

and f ′(3) = .

Correct Answers:

• (-2*(3*x+7)**(-2-1))*(3)• -0.00146484375


f (x) =√

2x+5

then f ′(x) = .

Correct Answers:

• (0.5*(2*x+5)**(-0.5))*2


f (x) =√

4x2 +4x+3

then f ′(x) =and f ′(1) = .

Correct Answers:

• (0.5*(4*x*x+4*x+3)**(-0.5))*(2*4*x+4)• 1.80906806746658


f (x) =8x4

1− x

then f (4)(x) = .Note: There is a way of doing this problem without using thequotient rule 4 times.

Correct Answers:

• 8 * 24 * (1-x)**(-1-4)

7. (10 pts) 1250Library/set3 Rates of Change and the Chain Rule-/1210s4p20.pgA space traveler is moving from left to right along the curve

y = x2.

When she shuts off the engines, she will continue travelingalong the tangent line at the point where she is at that time. Atwhat point

(x,y) = ( , )should she shut off the engines in order to reach the point(4,15)?If she was traveling from right to left she would have to shut offthe engines at the point

(x,y) = ( , ).Correct Answers:

• 3• 9• 5• 25

1

8. (10 pts) 1250Library/set3 Rates of Change and the Chain Rule-/1210s4p22.pgAs discussed in class (and in the textbook in section 11.4), tosolve the equation

f (x) = 0by Newton’s Method we start with a good initial guess x0 andthen run the iteration

xn+1 = xn−f (xn)f ′(xn)

, n = 0,1,2, . . .

until we get an approximation xn+1 that is good enough for ourpurposes.

Suppose you want to compute the cube root of 4 by solvingthe equation

x3−4 = 0.

Since 13 = 1 and 23 = 8 Let’s start with

x0 = 1.5

Thenx1 = ,x2 = ,x3 = , andTo check your answer compute x3

3 = .

Enter x1, x2 and x3 with at least 6 correct digits beyond the dec-imal point.

Correct Answers:• 1.59259259259259• 1.58741795698709• 1.587401052• 4

9. (10 pts) 1250Library/set3 Rates of Change and the Chain Rule-/1210s4p23.pgThe equation

10(x−1)(x−2)(x−3) = 1

has three real solutions

a < b < c

wherea = ,b = , andc = .

Enter your answers with at least six correct digits beyond thedecimal point.Hint: Ask what the solutions are if the right hand side is 0 in-stead of 1, and use Newton’s Method.

Correct Answers:• 1.05435072607641• 1.89896874211899• 3.0466805318046

10. (10 pts) 1250Library/set3 Rates of Change and the Chain Rule-/1210s4p24.pgNewton’s Method will converge to a true solution if you have agood initial approximation. If you don’t it may not converge atall. Consider, for example, the equation

f (x) = x(x−1)(x+1) = x3− x = 0.

Obviously, the solutions are

x = −1, 0, 1.

If we start Newton’s Method with x0 being close to one of thesesolutions we will get convergence to that solution. On the otherhand, note that

f ′(x) = 3x2−1

is zero when

x =± 1√3.

Thus the tangent will be horizontal in those two cases, and New-ton’s can’t even be carried out.

In this problem we’ll investigate what happens in the contrivedcase that

x0 =√

55

.

You can enter x0 into WeBWorK as sqrt(5)/5. Try it:

x0 = ,Now do your computations using exact arithmetic, and you’llrecognize a pattern:

x1 = ,x2 = ,x3 = ,x4 = ,Draw a picture to see what’s going on. Note, however, that New-ton’s method may fail in many different ways. A detailed anal-ysis of Newton’s method and related methods is a huge subjectand well beyond the scope of our class.

Correct Answers:

• 0.447213595499958• -0.447213595499958• 0.447213595499958• -0.447213595499958• 0.447213595499958

2

11. (10 pts) 1250Library/set3 Rates of Change and the Chain Rule-/1210s4p25.pgAs discussed in class, there may be more variables floatingaround than the one with respect to which we differentiate. Inthat situation it is important to be aware with respect to whichvariable we differentiate.

For example, the volume of a square box width s and heighth is given

V = s2h.

ThusDsV = andDhV = .

Correct Answers:• 2*s*h• s*s

12. (10 pts) 1250Library/set3 Rates of Change and the Chain Rule-/1210s4p26.pgThe total resistance R of two parallel resistors S and T is givenby

R =ST

S +T.

ThusDSR = andDT R = .

Correct Answers:• T**2/(S+T)**2• S**2/(S+T)**2

13. (10 pts) 1250Library/set3 Rates of Change and the Chain Rule-/1210s4p27.pgThis problem is in preparation for the next two problems. Sup-pose you have a cube of length s. The volume of that cube is

V = s3.

Now let’s suppose the dimensions of that cube (and hence itsvolume) depend on time. We are wondering about the relation-ship between the growth of the length versus the growth of thevolume.

Supposes(t) = t.

Then s′(t) = and V ′(t) = .Next, suppose

V (t) = t.Then s′(t) = and V ′(t) = .

Correct Answers:• 1• 3*t*t• t**(-2/3)/3

• 1

14. (10 pts) 1250Library/set3 Rates of Change and the Chain Rule-/1210s4p28.pgThe radius of a spherical watermelon is growing at a constantrate of 2 centimeters per week. The thickness of the rind is al-ways one tenth of the radius. The volume of the rind is growingat the rate cubic centimeters per week at the end ofthe fifth week. Assume that the radius is initially zero.


15. (10 pts) 1250Library/set3 Rates of Change and the Chain Rule-/1210s4p29.pgYour neighbor is growing a slightly different watermelon. Italso has a rind whose thickness is one tenth of the radius ofthat watermelon. However, the rind of your neighbor’s wa-ter melons grows at a constant rate of 20 cubic centimeters aweek. The radius of your neighbor’s watermelon after 5 weeksis and at that time it is growing at cen-timeters per week.

Correct Answers:• 4.44952891419823• 0.296635260946549

16. (10 pts) 1250Library/set3 Rates of Change and the Chain Rule-/1210s5p1.pgLet

f (x) = x(x2 +4)15.

f ′(x) = .f ′′(x) = .

Correct Answers:• (31*x**2 + 4)*(x**2 + 4)**14• 30*(31*x**2 + 12)*(x**2 + 4)**13*x

17. (10 pts) 1250Library/set3 Rates of Change and the Chain Rule-/1210s5p6.pgComputeDxx = ,D2

xx2 = ,D3

xx3 = ,D4

xx4 = ,D5

xx5 = ,D6

xx6 = .Do you recognize a pattern?Compute D6

x(2x6 +3x5 +4x4 +5x3 +17x2−15x+π) = andD5

x(2x+1)5 = .Correct Answers:• 1• 2• 6• 24• 120• 720• 1440• 3840

3

18. (10 pts) 1250Library/set3 Rates of Change and the Chain Rule-/1210s5p7.pgConsider these statements written in ordinary language:

A The speed of the car is proportional to the distance it has trav-eled.

B The car is speeding up.

C The car is slowing down.

D The car always travels the same distance in the same timeinterval.

E We are driving backwards.

F Our acceleration is decreasing.Denoting by s(t) the distance covered by the car at time t,

and letting k denote a constant, match these statements withthe following mathematical statements by entering the lettersA through E on the appropriate boxes:

s′′ < 0.

s′ is constant

s′ < 0

s′′′ < 0

s′′ > 0

s′ = ksCorrect Answers:

• C• D• E• F• B• A

19. (10 pts) 1250Library/set3 Rates of Change and the Chain Rule-/1210s5p8.pgFind the slope of the tangent line to the curve

−2x2−4xy−3y3 =−1440

at the point (−4,8).Your answer:

Hint: Differentiate implicitly.Correct Answers:

• -0.0285714285714286

20. (10 pts) 1250Library/set3 Rates of Change and the Chain Rule-/1210s5p9.pgSuppose 3x2 + 5x + xy = 5 and y(5) = −19. Find y′(5) by im-plicit differentiation.Your answer:

Hint: You’ll also have to solve for y.Correct Answers:

• -3.2

21. (10 pts) 1250Library/set3 Rates of Change and the Chain Rule-/1210s5p11.pgLet A be the area of a circle with radius r. If dr

dt = 2, find dAdt

when r = 3.Your answer:

Correct Answers:

• 37.6991118

22. (10 pts) 1250Library/set3 Rates of Change and the Chain Rule-/1210s5p12.pgUse implicit differentiation to find the equation of the tangentline to the curve xy3 + xy = 14 at the point (7,1). The equationof this tangent line can be written in the form y = mx+b wherem is:and where b is:

Correct Answers:

• -0.0714285714285714• 1.5

23. (10 pts) 1250Library/set3 Rates of Change and the Chain Rule-/1210s5p13.pg

Suppose x2

36 + y2

36 = 1 and y(2) = 5.65685. Find y′(2) by implicitdifferentiation.

Correct Answers:

• -0.353553390593274

24. (10 pts) 1250Library/set3 Rates of Change and the Chain Rule-/1210s5p14.pgFind the slope of the tangent line to the curve (a lemniscate)

2(x2 + y2)2 = 25(x2− y2)

at the point (3,1).m =Correct Answers:

• -0.692307692307692

4

25. (10 pts) 1250Library/set3 Rates of Change and the Chain Rule-/1210s5p15.pgSuppose

√x +

√y = 11 and y(9) = 64. Find y′(9) by implicit

differentiation.

Correct Answers:

• -2.66666666666667

26. (10 pts) 1250Library/set3 Rates of Change and the Chain Rule-/1210s5p16.pgLet

xy = 4

and let

dydt

= 1

Find dxdt when x = 1.

Correct Answers:

• -0.25

27. (10 pts) 1250Library/set3 Rates of Change and the Chain Rule-/1210s5p17.pgThe graph of the equation

x2 + xy+ y2 = 9

is a slanted ellipse. Think of y as a function of x. Differentiatingimplicitly and solving for y′ gives:y′ = . (Your answer will depend on x and y.)

The ellipse has two horizontal tangents. The upper one hasthe equation

y = .The right most vertical tangent has the equationx = .That tangent touches the ellipse wherey = .Hint: The horizontal tangent is of course characterized byy′ = 0. To find the vertical tangent use symmetry, or think ofx as a function of y, differentiate implicitly, solve for x′ and thenset x′ = 0.

Correct Answers:

• -(2*x+y)/(x+2*y)• 3.46410161513775• 3.46410161513775• -1.73205080756888

28. (10 pts) 1250Library/set3 Rates of Change and the Chain Rule-/1210s5p18.pgA street light is at the top of a 12.0 ft. tall pole. A man 5.6 fttall walks away from the pole with a speed of 3.5 feet/sec alonga straight path. How fast is the tip of his shadow moving whenhe is 32 feet from the pole?Your answer:Hint: Draw a picture and use similar triangles.

Correct Answers:

• 6.5625

29. (10 pts) 1250Library/set3 Rates of Change and the Chain Rule-/1210s5p19.pgThe altitude of a triangle is increasing at a rate of 2.5 centime-ters/minute while the area of the triangle is increasing at a rateof 5.0 square centimeters/minute. At what rate is the base of thetriangle changing when the altitude is 8.5 centimeters and thearea is 98.0 square centimeters?Your answer:Hint: The area A of a triangle with base b and height h is givenby

A =12

bh.

Differentiate implicitly.Correct Answers:

• -5.60553633217993

30. (10 pts) 1250Library/set3 Rates of Change and the Chain Rule-/1210s5p20.pgGravel is being dumped from a conveyor belt at a rate of 30 cu-bic feet per minute. It forms a pile in the shape of a right circularcone whose base diameter and height are always equal to eachother. How fast is the height of the pile increasing when the pileis 20 feet high? Recall that the volume of a right circular conewith height h and radius of the base r is given by V = π

3 r2h.Your answer: feet per minute.

Correct Answers:

• 0.0954929658551372

31. (10 pts) 1250Library/set3 Rates of Change and the Chain Rule-/1210s5p21.pgWater is leaking out of an inverted conical tank at a rate of 10900cubic centimeters per minute at the same time that water is be-ing pumped into the tank at a constant rate. The tank has height7 meters and the diameter at the top is 6.5 meters. If the waterlevel is rising at a rate of 18 centimeters per minute when theheight of the water is 1.0 meters, find the rate at which water isbeing pumped into the tank in cubic centimeters per minute.Your answer: cubic centimeters per minute.

Correct Answers:

• 132796.998586735

5

32. (10 pts) 1250Library/set3 Rates of Change and the Chain Rule-/1210s5p23.pgA spherical snowball is melting in such a way that its diameteris decreasing at rate of 0.2 cm/min. At what rate is the volumeof the snowball decreasing when the diameter is 18 cm.Your answer (cubic centimeters per minute) shouldbe a positive number.Hint: The volume of a sphere of radius r is

V =4πr3

3.

The diameter is twice the radius.Correct Answers:• 101.78760186

33. (10 pts) 1250Library/set3 Rates of Change and the Chain Rule-/1210s5p24.pgYou are blowing air into a spherical balloon at a rate of 4π

3 cubicinches per second. (The reason for this strange looking rate isthat it will simplify your algebra a little.) Assume the radius ofyour balloon is zero at time zero. Let r(t), A(t), and V (t) denotethe radius, the surface area, and the volume of your balloon attime t, respectively. (Assume the thickness of the skin is zero.)All of your answers below are expressions in t:r′(t) = inches per second,A′(t) = square inches per second, andV ′(t) = cubic inches per second.Hint: The surface area A and the volume V of a sphere of radiusr are given by

A = 4πr2 and V =4πr3

3.

Correct Answers:• tˆ(-2/3)/3• 8*3.14159265358979/3*tˆ(-1/3)• 4.18879020478639

34. (10 pts) 1250Library/set3 Rates of Change and the Chain Rule-/1210s5p25.pgYou may want to solve this problem only after you solve the nextone, which is the general version of this one. On the other hand,you also may want to solve this one first, and when solving thegeneral one compare your evolving solution against your resultsin this problem.A child is flying a kite. If the kite is 150 feet above the child’shand level and the wind is blowing it on a horizontal course at4 feet per second, the child is paying out cord at feetper second when 210 feet of cord are out. Assume that the cord

remains straight from hand to kite. (If you have ever flown akite you know that this is an unrealistic assumption.)For a solution of this problem (after the set closes) see the nextproblem.


35. (10 pts) 1250Library/set3 Rates of Change and the Chain Rule-/1210s5p26.pgA child is flying a kite. If the kite is h feet above thechild’s hand level and the wind is blowing it on a horizon-tal course at v feet per second, the child is paying out cord at

feet per second when s feet of cordare out. Assume that the cord remains straight from hand tokite.

Correct Answers:• sqrt(s*s-h*h)*v/s

36. (10 pts) 1250Library/set3 Rates of Change and the Chain Rule-/1210s5p29.pgYou say goodbye to your friend at the intersection of two per-pendicular roads. At time t = 0 you drive off North at a (con-stant) speed v and your friend drives West at a (constant) speedw. You badly want to know: how fast is the distance betweenyou and your friend increasing at time t? Enter here the deriva-tive of the distance from your friend with respect to t:

Being scientifically minded you ask yourself how does thespeed of separation change with time. In other words, whatis the second derivative of the distance between you and yourfriend?

Suppose that after your friend takes off (at time t = 0) youlinger for an hour to contemplate the spot on which he or shewas standing. After that hour you drive off too (to the North).How fast is the distance between you and your friend increasingat time t (greater than one hour)?

Again, you ask what is the second derivative of your separa-tion:

If you wish e-mail me your comments on how lingering makes things harder,mathematically speaking.

Correct Answers:• sqrt(vˆ2+wˆ2)• 0• (vˆ2*(t-1)+wˆ2*t)/sqrt(vˆ2*(t-1)ˆ2+wˆ2*tˆ2)• (vˆ2*wˆ2)/(sqrt((t-1)ˆ2*vˆ2+wˆ2*tˆ2)*((t-1)ˆ2*vˆ2+wˆ2*tˆ2))


6

Math 1250-2 Fall 2008

Hsiang-Ping Huang.


due 09/30/2008 at 11:59pm MDT.This problem set covers our class work during the week of

September 15.

1. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210s6p3.pg

ddx (ax2 +bx+ c) = .

Correct Answers:

• 2*a*x + b

2. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210s6p4.pgSuppose

f (t) =√

t2 +1.

Then

f ′(t) = .

f ′′(t) = .

f ′′′(t) = .Correct Answers:

• t/sqrt(t**2 + 1)• 1/(sqrt(t**2 + 1)*(t**2 + 1))• ( - 3*t)/(sqrt(t**2 + 1)*(t**4 + 2*t**2 + 1))

3. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210s6p5.pgSuppose you want to compute the fifth root of 6 by solving theequation

f (x) = x5−6 = 0 (∗)using Newton’s method. Newton’s method starts with an initialapproximation x0 and then computes a sequence of approxima-tions x1, x2, x3, . . . via the formula

xk+1 = g(xk), k = 0,1,2, . . .

where

g(x) = x− f (x)f ′(x)

.

For the function defined above in (∗), g(x) = .

Lettingx0 = 1

you obtainx1 = ,

x2 = , andx3 = .

Correct Answers:

• x-(xˆ5-6)/(5xˆ4)• 2• 1.675• 1.49244809457499

4. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210s6p7.pgFor what values of x does the graph of

f (x) = 2x3−9x2 +0x+0

have a horizontal tangent? Enter the x values in order, smallestfirst, to 4 places of accuracy:x1 = < x2 =

Correct Answers:

• 0• 3

5. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210s6p8.pgThe function

f (x) =−4x3−30x2 +72x−2

is increasing on the interval ( , ).It is decreasing on the interval ( −∞, ) and the interval (, ∞ ).The function has a local maximum at .Correct Answers:

• -6• 1• -6• 1• 1

6. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210s6p9.pgFind the point on the line 4x+7y−7 = 0 which is closest to thepoint (−4,−2).

( , )Correct Answers:

• -1.72307692307692• 1.98461538461538

7. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210s6p10.pgA rectangle is inscribed with its base on the x-axis and its uppercorners on the parabola y = 7− x2. What are the dimensions ofsuch a rectangle with the greatest possible area?

Width = Height =Correct Answers:

• 3.05505046330389• 4.66666666666667

1

8. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210s6p11.pgA cylinder is inscribed in a right circular cone of height 7 andradius (at the base) equal to 6. What are the dimensions of sucha cylinder which has maximum volume?

Radius = Height =Correct Answers:

• 4• 2.33333333333333

9. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210s6p12.pgA fence 6 feet tall runs parallel to a tall building at a distanceof 2 feet from the building. What is the length of the shortestladder that will reach from the ground over the fence to the wallof the building?

Correct Answers:

• 10.811197454847

10. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210s6p13.pgIf 1600 square centimeters of material is available to make abox with a square base and an open top, find the largest possiblevolume of the box.Volume = cubic centimeters.

Correct Answers:

• 6158.40287135601

11. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210s6p14.pgThe function f (x) = 4x+9x−1 has one local minimum and onelocal maximum.It is helpful to make a rough sketch of the graph to see what ishappening.This function has a local minimum at x equals withvalueand a local maximum at x equals with value

Correct Answers:

• 1.5• 12• -1.5• -12

12. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210s6p16.pgA Norman window has the shape of a semicircle atop a rectan-gle so that the diameter of the semicircle is equal to the width ofthe rectangle. What is the area of the largest possible Normanwindow with a perimeter of 26 feet?

Correct Answers:

• 47.3283784955167

13. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210s6p17.pgA rancher wants to fence in an area of 2000000 square feet in arectangular field and then divide it in half with a fence down themiddle parallel to one side. What is the shortest length of fencethat the rancher can use?

Correct Answers:

• 6928.20323027551

14. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210s6p18.pgA University of Rochester student decided to depart from Earthafter his graduation to find work on Mars. Before building ashuttle, he conducted careful calculations. A model for the ve-locity of the shuttle, from liftoff at t = 0 s until the solid rocketboosters were jettisoned at t = 52.7 s, is given by

v(t) = 0.001341833t3−0.081295t2 +34.76t +0.25

(in feet per second). Using this model, estimatethe absolute maximum valueand absolute minimum valueof the acceleration of the shuttle between liftoff and the jettison-ing of the boosters.

Correct Answers:

• 37.371487895• 33.1182469196373

15. (10 pts) set4/1210s6p19.pgYou are going to make many cylindrical cans. The cans willhold different volumes. But you’d like them all to be such thatthe amount of sheet metal used for the cans is as small as possi-ble, subject to the can holding the specific volume. How do youchoose the ratio of diameter to height of the can? Assume thatthe thickness of the wall, top, and bottom of the can is every-where the same, and that you can ignore the material needed forexample to join the top to the wall.

Put differently, you ask what ratio of diameter to height willminimize the area of a cylinder with a given volume?

That ratio equals .Correct Answers:

• 1

16. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210s6p20.pgIt takes a certain power P to keep a plane moving along at aspeed v. The power needs to overcome air drag which increasesas the speed increases, and it needs to keep the plane in the airwhich gets harder as the speed decreases. So assume the powerrequired is given by

P = cv2 +dv2

2

where c and d are positive constants. (They depend on yourplane, your altitude, and the weather, among other things.) En-ter here the choice of v that will minimize thepower required to keep moving at speed v.

Suppose you have a certain amount of fuel and the fuel flowrequired to deliver a certain power is proportional to to thatpower. What is the speed v that will maximize your range (i.e.,the distance you can travel at that speed before your fuel runsout)? Enter your speed here

Finally, enter here the ratio of the speedthat maximizes the distance and the speed that minimizes therequired power.

Correct Answers:

• (d/c)ˆ(1/4)• (3d/c)ˆ(1/4)• 1.31607401295249


This is a related rates problem with a twist.

Suppose you have a street light at a height H. You drop a rockvertically so that it hits the ground at a distance d from the streetlight. Denote the height of the rock by h. The shadow of therock moves along the ground. Let s denote the distance of theshadow from the point where the rock impacts the ground. Ofcourse, s and h are both functions of time. To enter your answerinto WeBWorK use the notation v to denote h′:

v = h′.

Then the speed of the shadow at any time while the rock is inthe air is given by s′ = (where s′ is an expressiondepending on h, s, H, and v (You will find that d drops out ofyour calculation.)

Now consider the time at which the rock hits the ground. Atthat time

h = s = 0.

The speed of the shadow at that time is s′ = whereyour answer is an expression depending on H, v, and d.Hint: Use similar triangles and implicit differentiation. For thesecond part of the problem you will need to compute a limit.

Correct Answers:• v*s*H/h/(H-h)• v*d/H

18. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210s7p1.pgThis differentiation problem is a little sneaky. Let

f (x) =x2−1x+1

.

Thenf ′(x) = and

f ′′(x) = .Hint: Begin by canceling common factors in numerator and de-nominator.

Correct Answers:• 1• 0

19. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210s7p4.pgConsider the function

f (x) =−2x3−4x2 +3x−2

Find the average slope of this function on the interval (1,2).

By the Mean Value Theorem, we know there exists a c in theopen interval (1,2) such that f ′(c) is equal to this mean slope.Find the value of c in the interval which works:

Correct Answers:• -23• 1.51914617476733

20. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210s7p5.pgAnswer the following questions for the function

f (x) = x√

x2 +4

defined on the interval [−5,7].A. f (x) is concave down on the interval toB. f (x) is concave up on the interval toC. The inflection point for this function is at x =D. The minimum for this function occurs at x =E. The maximum for this function occurs at x =

Correct Answers:• -5• 0• 0• 7• 0• -5• 7

3

21. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210s7p6.pgThe function f (x) = 2x3−36x2 + 120x + 1 has one local mini-mum and one local maximum.It is helpful to make a rough sketch of the graph to see what ishappening.This function has a local minimum at x = with valuef (x) = ,and a local maximum at x = with value f (x) =

.Correct Answers:• 10• -399• 2• 113

22. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210s7p7.pgAt what point does the normal to y = −5− 4x + 3x2 at (1,−6)intersect the parabola a second time?( , )

The normal line is perpendicular to the tangent line. If twolines are perpendicular their slopes are negative reciprocals – i.e.if the slope of the first line is m then the slope of the second lineis −1/m

Correct Answers:• 0.166666666666667• -5.58333333333333

23. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210s7p8.pgThe function f (x) =−2x3 +33x2−60x+8 has one local mini-mum and one local maximum.It is helpful to make a rough sketch of the graph to see what ishappening.This function has a local minimum at x = with value f (x) =

,and a local maximum at x = with value f (x) = .

Correct Answers:• 1• -21• 10• 708

24. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210s7p9.pgConsider the function f (x) = 3x3 − 3x on the interval [−2,2].Find the average or mean slope of the function on this interval.

By the Mean Value Theorem, we know there exists at least onec in the open interval (−2,2) such that f ′(c) is equal to thismean slope.For this problem, there are two values of c that work. Thesmaller one is and the larger one is .


• -1.15470053837925• 1.15470053837925

25. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210s7p10.pgConsider the function f (x) = 2x3−6x2−90x+8 on the interval[−5,10]. Find the average or mean slope of the function on thisinterval.

By the Mean Value Theorem, we know there exists a c in theopen interval (−5,10) such that f ′(c) is equal to this meanslope. For this problem, there are two values of c that work. Thesmaller one is , and the larger one is .

Correct Answers:

• 30• -3.58257569495584• 5.58257569495584

26. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210s7p13.pgConsider the function f (x) = 12x5 +75x4−120x3 +3.f (x) has inflection points at (reading from left to right) x = D,E, and Fwhere D isand E isand F isFor each of the following intervals, tell whether f (x) is concaveup (type in CU) or concave down (type in CD).(−∞,D]:[D,E]:[E,F ]:[F,∞):

Correct Answers:

• -4.42757223208277• 0• 0.677572232082767• CD• CU• CD• CU

27. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210s7p14.pgA right circular cone is to be inscribed in another right circularcone of given volume, with the same axis and with the vertex ofthe inner cone touching the base of the outer cone.

Draw a picture of the cones.What must be the ratio of their altitudes for the inscribed

cone to have maximum volume?

Correct Answers:

• 0.333333333333333

4

28. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210s7p15.pgI have enough pure silver to coat 4 square meter of surface area.I plan to coat a sphere and a cube. Allowing for the possibilityof all the silver going onto one of the solids, what dimensionsshould they be if the total volume of the silvered solids is to bea maximum?

The radius of the sphere is ,and the length of the sides of the cube is .

Again, allowing for the possibility of all the silver going ontoone of the solids, what dimensions should they be if the totalvolume of the silvered solids is to be a minimum?

The radius of the sphere is ,and the length of the sides of the cube is .Correct Answers:

• 0.564189583510922• 0• 0.330741786042061• 0.661483572084122


One end of a ladder of length L rests on the ground and theother end rests on the top of a wall of height h, as illustrated inthe Figure on this page. As the bottom end is pushed along theground towards the wall, the top end extends beyond the wall.The value of x that maximizes the horizontal overhang s is x =

. (Your answer will depend on L and h.)In the particular case that L = 18 and h = 7 this value is x =

.The corresponding numerical value of s = .

Correct Answers:

• (-h*h+(L*h*h)**(2/3))**(1/2)• 6.55514316683058

• 5.74844610338649

30. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210s7p17.pgThe illumination at a point is inversely proportional to the squareof the distance of the point from the light source and directlyproportional to the intensity of the light source. Suppose twolight sources are s feet apart and their intensities are I and J,respectively. Let P be the point between them where the sumof their illuminations is a minimum. The distance of P fromlight source I will be feet. (Your answer willdepend on I, J, and s.)

Correct Answers:

• s*Iˆ(1/3)/(Iˆ(1/3)+Jˆ(1/3))

32. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210set8p3.pgA car traveling at 49 ft/sec decelerates at a constant 8 feet persecond squared. How many feet does the car travel before com-ing to a complete stop?

Correct Answers:

• 150.0625

33. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210set8p4.pgA ball is shot straight up into the air with initial velocity of 50ft/sec. Assuming that the air resistance can be ignored, howhigh does it go? (Assume that the acceleration due to gravity is32 ft per second squared.)

Correct Answers:

• 39.0625

34. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210set8p6.pgConsider the function f (x) = 6x3−7x2 +5x−1.An antiderivative of f (x) is F(x) = Ax4 +Bx3 +Cx2 +Dxwhere A is and B is and C is and D is

Correct Answers:

• 1.5• -2.33333333333333• 2.5• -1

35. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210set8p7.pgConsider the function f (x) = 36x3− 18x2 + 12x− 5. Enter anantiderivative of f (x)

Correct Answers:

• 9*xˆ4-6*xˆ3+6*xˆ2-5*x

5

36. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210set8p10.pgGiven that the graph of f (x) passes through the point (10,8)and that the slope of its tangent line at (x, f (x)) is 6x + 6, whatis f (3)?

Correct Answers:• -307

37. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210set8p11.pgConsider the function f (x) = 4x9 +9x5−6x2−4.Enter an antiderivative of f (x)

Correct Answers:• 0.4*xˆ10+1.5*xˆ6-2*xˆ3-4*x

38. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210set8p12.pgConsider the function f (x) = 9

x2 − 7x5 .

Let F(x) be the antiderivative of f (x) with F(1) = 0.Then F(4) equals


39. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210set8p13.pgConsider the function f (x) = 7

x2 − 10x7 .

Let F(x) be the antiderivative of f (x) with F(1) = 0.Then F(x) =

Correct Answers:• -7*xˆ(-1)+(1.66666666666667)*xˆ(-6)+5.33333333333333

40. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210set8p15.pgSuppose

f (x) =1

(x−1)2

and F is an antiderivative of f that satisfies

F(0) = 1.

ThenF(x) = .

Correct Answers:• 1/(1-x)

41. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210set8p17.pgThis problem revisits our familiar formulas for vertically mov-ing objects.

Suppose we know that

h′′(t) =−g, h′(0) = V, and h(0) = H.

Thenh(t) = .

Correct Answers:

• -g/2*t*t+V*t+H

42. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210set8p18.pg

This problem is a very simple example of a differential equa-tion : an equation that relates a function to one or more of itsderivatives. You can solve this problem by doing some edu-cated guessing. (”educated” means ”remember what we did inthe past.”)

Suppose f is the function that satisfies

f ′(x) =− f 2(x)

for all x in its domain, and

f (1) = 1.

Thenf (x) = .

Correct Answers:

• 1/x

43. (10 pts) 1250Library/set4 Graphing and Maximum-Minimum Problems/1210set8p23.pg

The rectangle with the largest area that can be enclosed in acircle of radius r is of course a square. Its sides have lengths = and its area is A = .Consider now the ellipse defined by

x2

a2 +y2

b2 = 1.

The rectangle with the largest area that can be inscribed in thatellipse has a horizontal side of length , a vertical side oflength , and an area .Hint: You can solve the ellipse problem by cranking the handleand proceeding as we did for several similar geometric prob-lems. Or you can look at the results for the circle and make aninspired guess.

Correct Answers:

• sqrt(2)*r• 2*r*r• sqrt(2)*a• sqrt(2)*b• 2*a*b


6

Math 1250-2 Fall 2008

Hsiang-Ping Huang.


due 10/07/2008 at 11:59pm MDT.

This problem set deals mostly with sums and simple inte-grals. However, it starts out with an intriguing minimizationproblem.

1. (10 pts) set5/cone.pgYou are going to make tanks in the shape of a right circular cone.They will hold various volumes, but the amount of sheet metalrequired to build the tank will be as small as possible subjectto the requirement of holding the required volume. These tankswill forever after be known all over the world as the famousHsiang-Ping Huang tanks.

The quotient radius/height of your tanks equals.

Correct Answers:

• 0.353553390593274

2. (10 pts) 1250Library/set5 The Integral/1210set8p20.pg

Compute the sum98

∑i=1

i = .

Correct Answers:

• 4851


Compute the sum59

∑i=1

(2i−1) = .

Correct Answers:

• 3481


Compute the sumn

∑i=1

(2i−1) = .

Correct Answers:

• n*n

5. (10 pts) 1250Library/set5 The Integral/c4s4p1.pgIf f (x) =

R x0 t4dt

thenf ′(x) =f ′(6) =Hint: You don’t need to compute f to answer this question.

Correct Answers:

• xˆ4• 1296

6. (10 pts) 1250Library/set5 The Integral/c4s4p5.pgThe value of

R 40 (x+1)2dx is

Correct Answers:

• 41.3333333333333

7. (10 pts) 1250Library/set5 The Integral/c4s4p6.pgThe value of

R 21

1x2 dx is

Correct Answers:

• 0.5

8. (10 pts) 1250Library/set5 The Integral/q1.pg

Enter a formula for∑

ni=m i =

where 0 < m < n are positive integers.Correct Answers:

• (nˆ2+n - mˆ2+m)/2

9. (10 pts) 1250Library/set5 The Integral/q2.pg

Enter a formula for(∑n

i=1 i)2−∑ni=1 i3 =

where n is a positive integer.Correct Answers:

• 0

10. (10 pts) 1250Library/set5 The Integral/s4 4 20.pgThe value of

R 72 (18x2−8x+4)dx is .

Correct Answers:

• 1850

11. (10 pts) 1250Library/set5 The Integral/sc5 2 24.pgEvaluate the integral below by interpreting it in terms of areas.In other words, draw a picture of the region the integral repre-sents, and find the area using high school geometry.R 4−4

√16− x2dx =

Correct Answers:

• 25.132741232

1

12. (10 pts) 1250Library/set5 The Integral/sc5 2 28.pgEvaluate the integral by interpreting it in terms of areas. In otherwords, draw a picture of the region the integral represents, andfind the area using high school geometry.R 9

0 |9x−5|dx =Correct Answers:

• 322.277777777778

13. (10 pts) 1250Library/set5 The Integral/sc5 3 17.pgEvaluate the integralR 2

3 sin(t)dt = .Correct Answers:

• -0.573845660053303

14. (10 pts) 1250Library/set5 The Integral/sc5 4 19a.pgFind the derivative of

g(x) =Z 3x

5x

u+5u−3

du

BR g′(x) =HINT: Z 3x

5x

u+5u−3

du =Z 3x

0

u+5u−3

du+Z 0

5x

u+5u−3

du

Correct Answers:

• 3*(3*x+5)/(3*x-3)-5*(5*x+5)/(5*x-3)

15. (10 pts) 1250Library/set5 The Integral/ur in 0 12.pg

Consider the function f (x) = x2

4 +7.

In this problem you will calculateR 4

0 ( x2

4 +7)dx by using thedefinition Z b

af (x)dx = lim

n→∞

[n

∑i=1

f (xi)∆x

]The summation inside the brackets is Rn which is the Rie-

mann sum where the sample points are chosen to be the right-hand endpoints of each sub-interval.

Calculate Rn for f (x) = x2

4 +7 on the interval [0,4] and writeyour answer as a function of n without any summation signs.

Rn =limn→∞ Rn =

Correct Answers:• 7*4 + 4**3*(n+1)*(2*n+1)/(6*(4)*n**2)• 33.3333333333333

16. (10 pts) 1250Library/set5 The Integral/ur in 3 11.pgThe velocity function is v(t) =−t2 + 3t−2 for a particle mov-ing along a line. Find the displacement and the distance traveledby the particle during the time interval [-2,6].

displacement =distance traveled =Correct Answers:• -42.6666666666667• 43


2

Math 1250-2 Fall 2008

Hsiang-Ping Huang.


due 10/21/2008 at 11:59pm MDT.This set has more on integrals and derivatives. To stay in

sync, and have home works of roughly consistent size trigono-metric functions will start only on the next home work.

1. (10 pts) 1250Library/set6 The Integral/1210s3p7.pgFor what value of the constant c is the function f continuous on(−∞,∞) where

f (x) =

{c if x = 0xsin 1

x otherwise

Hint: Ask what is limx−→0 f (x)?Correct Answers:• 0

2. (10 pts) 1250Library/set6 The Integral/1210s3p8.pgFor what value of the constant c is the function f continuous on(−∞,∞) where

f (b) =

{cb+7 if b ∈ (−∞,8]cb2−7 if b ∈ (8,∞)


3. (10 pts) 1250Library/set6 The Integral/1210s3p9.pgIn this problem we consider three functions f . Each of them iscontinuous at x = 0, i.e.,

limx−→0

f (x) = f (0).

In order to show by the ε/δ definition that this is true one has togive a definition of δ in terms of ε such that

|x−0|< δ =⇒ | f (x)− f (0)|< ε.

Match these choices of δ

1. δ = ε2

2. δ = ε

3. δ =√

ε

with the functions so that that choice of δ establishes conti-nuity of the function (at x = 0. You can use each choice only

once. Enter the reference numbers of the given functions in theappropriate answer boxes.

f (x) = x:f (x) = x2:f (x) =

√x:

Correct Answers:

• 2• 3• 1

4. (10 pts) 1250Library/set6 The Integral/1210set8p2.pgGiven

f ′′(x) = 5x+3

and f ′(0) =−3 and f (0) =−5.Find f ′(x) =

and find f (4) =Correct Answers:

• 5/2*xˆ2+ 3*x + -3• 60.3333333333333

5. (10 pts) 1250Library/set6 The Integral/1210set8p16.pgSuppose

f (x) = x+1


F(0) = 1.

ThenF(x) = .

Correct Answers:

• xˆ2/2+x+1

6. (10 pts) 1250Library/set6 The Integral/1220s1p11.pgA word problem. What number exceeds its square by the max-imum amount?

Answer: .

Correct Answers:

• 0.5

7. (10 pts) 1250Library/set6 The Integral/1220s1p12.pgUps and Downs. Consider the function f (x) = x3−3x2 +2x.

This function is increasing on the interval (−∞,A) where A=

It has an inflection point at x = Call that point B. For eachof the following intervals, tell whether f (x) is concave up (typein CU) or concave down (type in CD).(−∞,B):(B,∞):

Correct Answers:

• 0.422649730810374• 1• CD• CU

1

8. (10 pts) 1250Library/set6 The Integral/1220s1p16.pgDefinite Integrals. EvaluateR 5

1(5x2−8x+4

)dx = .


9. (10 pts) 1250Library/set6 The Integral/1220s1p17.pgThe Fundamental Theorem of Calculus. Use the Fundamen-tal Theorem of Calculus to find the derivative of

f (x) =Z x2

2

(12

t2−1)8

dt

f ′(x) =

Correct Answers:• 2*(1/2*xˆ4-1)ˆ8*x

10. (10 pts) 1250Library/set6 The Integral/1220s1p18.pgAreas. Sketch the region enclosed by the given curves. Decidewhether to integrate with respect to x or y. Then find the area ofthe region.y = 7x,y = 6x2


11. (10 pts) 1250Library/set6 The Integral/1220s1p22.pgMore on Areas. Farmer Jones, and his wife, Dr. Jones, bothmathematicians, decide to build a fence in their field, to keep thesheep safe. Being mathematicians they decide that the fencesare to be in the shape of the parabolas y = 4x2 and y = x2 + 6.What is the area of the enclosed region?


12. (10 pts) 1250Library/set6 The Integral/1220s1p23.pgYet More on Areas. There is a line through the origin that di-vides the region bounded by the parabola y = 3x− 8x2 and thex-axis into two regions with equal area. What is the slope of thatline?

Hint: Draw a picture, write the area between the parabola andthe line in terms of the slope of the line, and solve for the slope.


13. (10 pts) 1250Library/set6 The Integral/1220s1p27.pgHow Far is That Point? Let P be the point (p,q) and L the liney = mx+b.

The distance of P from L isd = .(Of course, your answer will be in terms of m, b, p and q.)Correct Answers:• sqrt((q-m*p-b)**2/(1+m**2))

14. (10 pts) 1250Library/set6 The Integral/1220s1p30.pgShooting Arrows. The ideas we are practicing in these exer-cises can be put to practical use. My son likes to shoot arrows.He was wondering at what speed the arrow leaves the bow whenit is released. So we went to a large deserted beach at the GreatSalt Lake and shot the bow at an angle of 45 degrees. The arrowhit the ground at a distance of 600 feet. All of this was mea-sured very roughly, but lets work with these figures, and let’signore air resistance. For simplicity let’s also assume that thearrow is released at an initial height of 0 feet. As usual, assumethat gravity causes an object to accelerate at 32 feet per secondsquared.

The arrow leaves the bow at a speed of miles perhour.

Note: My son and I did worry about the effect of air resistance.The arrows were sticking out of the sand where they had hit thebeach. So my son shot another arrow downwards at an angle of45 degrees straight into the sand. It penetrated to about the samedepth as the arrows that had flown the distance. So they musthave hit at about the same speed. We concluded that ignoringair resistance in this case was reasonable.


Correct Answers:

• 94.475498594666

15. (10 pts) 1250Library/set6 The Integral/1220s1p31.pgThe Columbia Accident. On January 16, 2003, 81.7 secondsinto the ascent, a piece of foam, weighing about 1.7 pounds,and being roughly the size, shape and weight of a large loafof bread, separated from the bipod ramp of the space shuttleColumbia’s external fuel tank and impacted the leading edge ofthe shuttle’s left wing. At that time the shuttle was traveling atapproximately 1,568 mph (Mach 2.46) and was at an altitudeof 66,000 feet. Based on film evidence, the foam traveled the58-foot distance from the ramp to the wing in 0.16 seconds. As-suming a constant acceleration relative to the body of the spaceshuttle of feet per second squared, the foam hit thewing at a speed of feet per second, ormiles per hour. Unbeknownst to the astronauts and observerson earth, it struck the wing hard enough to cause the destructionof the Columbia two weeks later during re-entry on February 1,2003.

Note: A much more involved calculation estimated the speedof the impact to be about 530 mph. You can find very detailedinformation about all aspects of the Columbia accident in thisofficial report.

2

Remember that ww expects your answer to be within onetenth of one percent of what it considers the true answer. Calcu-late your answers based on the given data and assumptions to atleast four digits.Hint: Figure everything relative to the moving space shuttle,i.e., the origin is traveling with the shuttle. Acceleration is thederivative of velocity, and velocity is the derivative of displace-ment. In this case, the acceleration is constant (by assumption),and the initial velocity and displacement are 0. After 0.16 sec-onds the displacement is 58 feet. There are 3,600 seconds in anhour, and 5280 feet in a mile. As in many real problems, theproblem statement contains extraneous information. You haveto decide what parts of the information are pertinent.

Correct Answers:• 4531.25• 725• 494.318181818182

16. (10 pts) 1250Library/set6 The Integral/p12.pgAreas

Sketch the region enclosed by the given curves. Decidewhether to integrate with respect to x or y. Then find the area ofthe region.y = 4x,y = 6x2


17. (10 pts) 1250Library/set6 The Integral/q3.pgYou may think you have seen this problem before, and indeedyou have. This time, however, you need to simplify your answerbefore entering it.

Enter a formula for(∑n

i=1 i)2−∑ni=1 i3 = where n is a positive

integer.Correct Answers:• 0

18. (10 pts) 1250Library/set6 The Integral/q4.pgThis question summarizes a few simple facts about polynomials.Recall that a polynomial p can be written in the form

p(x) =n

∑i=0

αixi

where we assume that αn 6= 0. The degree of a polynomial isthe largest exponent present, and so the degree of p is n. Fill inthe blanks in the following questions:

The degree of p′(x) is .The degree of

Rp(x)dx is .

The degree of p2(x) (i.e., the square of p) is .The (n+1)-th derivative of p is .Correct Answers:• n-1• n+1• 2n• 0

19. (10 pts) 1250Library/set6 The Integral/q5.pgUse the Fundamental Theorem of Calculus to carry out the fol-lowing differentiation:

ddx

Z √x

1ttdt = .

Correct Answers:

• sqrt(x)**sqrt(x)/2/sqrt(x)

20. (10 pts) 1250Library/set6 The Integral/s4 4 21.pgThe value of

R 2−2(4− x2)dx is .

Correct Answers:

• 10.6666666666667

21. (10 pts) 1250Library/set6 The Integral/s4 4 27.pg

The value ofR 9

36x2+6√

x dx is .Correct Answers:

• 561.003092865686

22. (10 pts) 1250Library/set6 The Integral/sc5 2 3.pgConsider the integral Z 7

1(4x2 +3x+6)dx

(a) Find the Riemann sum for this integral using right end-points and n = 3.

(b) Find the Riemann sum for this same integral, using left end-points and n = 3

Correct Answers:

• 790• 370

23. (10 pts) 1250Library/set6 The Integral/z1.pgIn the next few problems we will explore the mathematics ofmotion with constant acceleration, and apply it to falling ob-jects and accelerating and decelerating vehicles. Unless other-wise stated assume that we use feet and seconds as our units forlength and time. Some of the problems may not require Cal-culus (i.e., differentiation or integration) for their solution, butthat’s OK, you want to be able to solve all problems that comealong.

As a warmup, let’s practice some conversions. A speed of mmiles per hour equals a speed of feet per second,and a speed f feet per second equals a speed ofmiles per hour. Recall that there are 5280 feet in a mile, and3600 seconds in an hour.

Recall that velocity is the derivative of distance, and acceler-ation is the derivative of velocity. Suppose that acceleration isconstant, and equal to A. Suppose at time 0 your velocity is V .(To accommodate WeBWorK idiosyncrasies, I’m using uppercase to denote constants.)

Let v(t) denote your velocity at time t. Thenv(t) = .

3

Suppose at time t = 0 your distance (in whatever directionyou are moving) is D.

Let d(t) denote your distance at time t. Thend(t) = .Let’s translate that into a familiar context. Suppose A =−32

feet per second squared, h(t) is you height at time t, H is yourinitial height, and V is your initial velocity. Then your heighth(t) at time t is h(t) = , your velocity at time t isv(t) = h′(t) = , and your acceleration at time(t) is,of course, a(t) = v′(t) = h′′(t) = feet per secondsquared.

Correct Answers:• m*5280/3600• f*3600/5280• A*t + V• A*t*t/2 + V*t +D• -16*t*t + V*t +H• -32*t + V• -32

24. (10 pts) 1250Library/set6 The Integral/z2.pgSuppose now you are driving along the highway at a speed vfeet per second. You apply the brakes which cause a constantdeceleration A feet per second squared. You come to a stop af-ter seconds. (Enter an expression that assumes apositive value for A.

A typical safe deceleration of a moving car is about onehalf g, half the acceleration of gravity, i.e., 16 feet per secondsquared.

Suppose you are driving at 60 miles per hour. You press onthe brakes and decelerate at 16 feet per second squared. It willtake you seconds to stop. During that time youwill travel feet.

Assuming that it takes you 1 second to react to an emergencybefore you start braking, at the same initial speed, and the sameconstant deceleration, you will travel a total offeet, before coming to a stop.

Correct Answers:• v/A• 5.5• 242• 330

25. (10 pts) 1250Library/set6 The Integral/z3.pgLet’s now replay that scenario at 80 miles per hour.

Again decelerating at 16 feet per second squared it will takeyou seconds to stop.

During that time you will travel feet.Assuming that it takes you 1 second to react to an emergency

before you start braking, at the same initial speed, and the sameconstant deceleration, you will travel a total offeet, before coming to a stop.

Correct Answers:

• 7.33333333333333• 430.222222222222• 547.555555555556

26. (10 pts) 1250Library/set6 The Integral/z4.pgYou are testing your brandnew Ferrari Testarossa. To see howwell the brakes work you accelerate to 100 miles per hour, slamon the brakes, and determine that you brought the car to a stopover a distance of 481 feet. Assuming a constant decelerationyou figure that that deceleration is feet per secondsquared. (Enter a positive number.)

I trust that you don’t have the courage to try this, but thatnight you wonder how long it would take you to stop (with thesame constant deceleration) if your were moving at 200 milesper hour. Remembering your Calculus you figure out that yourstopping distance would be feet. (Enter a number,not an arithmetic expression.)

Correct Answers:

• 22.3608223608224• 1924

27. (10 pts) 1250Library/set6 The Integral/z5.pgAccording to Richard Graham in ”SR-71 Revealed: The InsideStory”, the Lockheed SR-71 Reconnaissance plane acceleratesfrom a standing start to a speed of Mach 3 (3 times the speed ofsound) ”in about 14 minutes”. (During that time it also climbsto an altitude of about 70,000 feet, so the acceleration is not asimpressive as you might otherwise expect.) Assuming that thetime is exactly 14 minutes, and the speed of sound is 1100 feetper second, this is an average acceleration of feetper second squared. Assuming that the acceleration is constant,during that time your SR-71 covers a distance ofmiles.

By the way, you can see an SR-71 on display in the Hill AirForce Museum in Roy. It looks a little dusty, and it’s clear thatits best days are over, but the plane is fiercely elegant and pow-erful. It still presents an impressive sight.

Correct Answers:

• 3.92857142857143• 262.5


4

Math 1250-2 Fall 2008

Hsiang-Ping Huang.


due 10/28/2008 at 11:59pm MDT.This set is focused mostly on trigonometric functions.

1. (10 pts) 1250Library/set7 Trigonometric Functions/1210s4p2.pgLet

f (x) =−7cosx+9tanx.

Thenf ′(x) = .

Correct Answers:

• (-1)*-7*sin(x)+9/(cos(x))ˆ2

2. (10 pts) 1250Library/set7 Trigonometric Functions/1210s4p4.pgIf

f (x) =2sinx

1+ cosxthenf ′(x) =

Correct Answers:

• (2*cos(x)*1 +2)/(1+cos(x))**2


f (x) = sin(x2)then f ′(x) = .

Correct Answers:

• (cos(x**2))*(2*x**(2-1))


f (x) = sin3 x

then f ′(x) =and f ′(1) =

Correct Answers:

• (3*sin(x)**(3-1))*(cos(x))• 1.14772110185144


f (x) = cos(sin(x2)),then f ′(x) =

Correct Answers:

• -sin(sin(x**2))*cos(x**2)*2*x

6. (10 pts) 1250Library/set7 Trigonometric Functions/1210s4p17.pgThe purpose of this problem is to show pretty much all of ourrules at work at once.

If

f (x) =xsin2 x2√

1+ x2

then f ′(x) =

Correct Answers:• (sin(x**2)*(4*cos(x**2)*x**4 + 4*cos(x**2)*x**2

+ sin(x**2)))/(sqrt(x**2 + 1)*(x**2 + 1))


f (x) = tanxThenf ′(x)=

f ′′(x)=f ′′′(x)=f ′′′′(x)=

Correct Answers:• 1/cos(x)ˆ2• 2*sin(x)/cos(x)ˆ3• (2+4*sin(x)**2)/cos(x)**4• (16*cos(x)**2*sin(x)+24*sin(x)**3)/cos(x)**5

8. (10 pts) 1250Library/set7 Trigonometric Functions/1210s4p21.pgDirections (or bearings ) on Earth are measured in degrees, run-ning from zero to 360 degrees, clockwise, starting with 0 de-grees being due North. So due East for example, is 90 degrees,due South 180 degrees, and Northwest is 315 degrees.

You are swinging a rock clockwise (looking from above)around your head and you are trying to hit a broom stick 15feet due east of you. The rock moves in a circle of a radius of3 feet around your head. When you release your sling the rockwill continue to move along the tangent to the circle though itsposition at the time of the release. When you release the rockthe sling is pointing in a direction of degrees. Ignore thevertical motion of the rock.

It’s unrealistic, but remember that unless otherwise stated WeB-WorK expects your answer to be within one tenth of one percentof the true answer.Hint: You can solve this problem using calculus and comput-ing the tangent to a circle. However, you can also solve it usingplain trigonometry. The moral is that you want to use whateverrequires the least amount of fuss for the problem at hand.


1


f (x) = sin1x.

f ′(x) = .Let

g(x) =1

sinx.

g′(x) = .Correct Answers:

• -cos(1/x)/(x*x)• -cos(x)/sin(x)**2


f (x) = tanx2.

f ′(x) = .Let

g(x) = tan2 x.

g′(x) = .Correct Answers:

• 2*x/cos(x*x)**2• 2tan(x)/cos(x)/cos(x)

11. (10 pts) 1250Library/set7 Trigonometric Functions/1210s5p10.pgFind y′ by implicit differentiation. Match the expressions defin-ing y implicitly with the letters labeling the expressions for y′.

1. 3cos(x− y) = 6ycosx2. 3cos(x− y) = 6ysinx3. 3sin(x− y) = 6ysinx4. 3sin(x− y) = 6ycosx

A. 3cos(x−y)+6ysinx3cos(x−y)+6cosx

B. 3cos(x−y)−6ycosx3cos(x−y)+6sinx

C. −3sin(x−y)+6ysinx6cosx−3sin(x−y)

D. −3sin(x−y)−6ycosx6sinx−3sin(x−y)

Correct Answers:

• C• D• B• A

12. (10 pts) 1250Library/set7 Trigonometric Functions/1210s5p22.pgA plane flying with a constant speed of 27 km/min passes over aground radar station at an altitude of 10 km and climbs at an an-gle of 45 degrees. At what rate, in km/min is the distance fromthe plane to the radar station increasing 3 minutes later?Your answer: kilometers per minute.Hint: The law of cosines for a triangle is

c2 = a2 +b2−2abcos(θ)

where θ is the angle between the sides of length a and b.Correct Answers:

• 26.9133948328552

13. (10 pts) 1250Library/set7 Trigonometric Functions-/1210set8p14.pgSuppose

f (x) = x2 + sinx


F(π) = 0.

ThenF(x) = .

Correct Answers:

• x**3/3-cos(x)-1 - 3.14159265358979ˆ3/3

14. (10 pts) 1250Library/set7 Trigonometric Functions-/1210set8p19.pg

Solve this problem also by educated guessing.

Suppose f is the function that satisfies

f ′′(x) =− f (x)

for all x in its domain, f (0) = 0, and f ′(0) = 1. Thenf (x) = .

Correct Answers:

• sin(x)

15. (10 pts) 1250Library/set7 Trigonometric Functions/1220s1p17.pgThe Fundamental Theorem of Calculus. Use the Fundamen-tal Theorem of Calculus to find the derivative of

f (x) =Z x2

5

(15

t2−1)11

dt

f ′(x) =

Correct Answers:

• 2*(1/5*xˆ4-1)ˆ11*x

2

16. (10 pts) 1250Library/set7 Trigonometric Functions/1220s3p1.pgThe next few problems in this set ask you to find inverse func-tions and their derivatives. You can find the derivative of theinverse function by differentiating the inverse explicitly, or byusing the formula we obtained in class:

( f−1)′(x) =1

f ′( f−1(x)).

Suit yourself!Find a formula for f−1(x) if :

f (x) =− x3 +1

f−1(x) = .(f−1

)′ (x) = .

Correct Answers:

• -3x + 3• -3

17. (10 pts) 1250Library/set7 Trigonometric Functions/1220s3p2.pg

Find a formula for f−1(x) if :f (x) =−

√1− x

f−1(x) = .

(f−1

)′ (x) = .Correct Answers:

• 1 - xˆ2• -2x

18. (10 pts) 1250Library/set7 Trigonometric Functions/1220s3p3.pgFind a formula for f−1(x) if :

f (x) =√

1x−2

f−1(x) = .

(f−1

)′ (x) = .Correct Answers:

• 1/x**2 + 2• -2/x**3

19. (10 pts) 1250Library/set7 Trigonometric Functions/1220s3p4.pgFind a formula for f−1(x) if :f (x) = ( x−1

x+1 )3

f−1(x) = .

(f−1

)′ (x) = .

Hint: Undo the operations of f (x) from the outside in. Undothe cubed operation first.

Correct Answers:• -(x**(1/3) + 1)/(x**(1/3) - 1)• 2/(3(x**(1/3)-1)**2*x**(2/3))

20. (10 pts) 1250Library/set7 Trigonometric Functions/1220s3p5.pgApplying your knowledge of Inverse Functions. A ball is thrownvertically upward from the ground with velocity s. Find themaximum height H of the ball as a function of s. Then findthe velocity s required to achieve a height of H. As usual, ig-nore air resistance, and assume that gravity causes a downwardacceleration of 32 feet per second squared.

H = .s = .Hint: The velocity of the ball is v(t) a function of time, t. It isgiven by v(t) = s−32t. Integrate to get a function describing theheight of the ball as a function of time.This also determines theheight of the ball as a function of s. Use techniques you remem-ber from Calculus I to maximize this function. For the secondpart, let H be some given height and set this equal to H(s). Thenuse techniques of this chapter to find a value of s which achievesthe height H. Remember that Mathematics and WeBWorK arecase sensitive. So you need to use an upper case H and a lowercase s.

Correct Answers:• (s**2)/64• 8*Hˆ.5

21. (10 pts) 1250Library/set7 Trigonometric Functions/1220s5p4.pgEvaluate Z

π/3

0tanxdx.

Answer:

Hint: Recall that tanx =sinxcosx

and use logarithms.Correct Answers:• 0.693147180559945

22. (10 pts) 1250Library/set7 Trigonometric Functions/s7p23.pgThis entertaining problem does not require Calculus for its so-lution. It is included here to help set the stage for our next topic:the Calculus of exponential and logarithmic functions.

Your village depends for its food on a nearby fishpond. Oneday the wind blows a water lily seed into the pond, and the lilybegins to grow. It doubles its size every day, and its growth issuch that it will cover the entire pond (and smother the fish pop-ulation) in 45 days. You are away on vacation, and the villagersdepend on your vigilance and intelligence to handle their affairs.Without you present they will take action about that lily only onregular work days, and only when the lily covers half the pondor more. The lily will cover half the pond days after itfirst starts growing. The day it reaches that size happens to be aholiday....

3

Hint: You don’t need algebra for this problem, just commonsense.


23. (10 pts) 1250Library/set7 Trigonometric Functions/sc5 4 18.pgFind the derivative

h′(x) =of

h(x) =Z sin(x)

−1(cos(t3)+ t) dt

Correct Answers:

• cos(x)*(cos((sin(x))ˆ3)+sin(x))


4

Math 1250-2 Fall 2008

Hsiang-Ping Huang.


due 11/04/2008 at 11:59pm MST.

This set is on exponentials and logarithms.Note that in WeBWorK ln(x) and log(x) both mean the nat-

ural logarithm. As we discussed in class, to compute the loga-rithm of x to any base a use the formula

loga(x) =lnxlna

.

You can enter powers like ax as a**x . WeBWorK does knowthe base e of the natrual exponential, and you can also enter thenatural exponential as exp(x) .

1. (10 pts) 1250Library/set8 Exponentials and Logarithms-/1220s2p1.pg(Section 7.1, No 4). Find the indicated derivative.Dx ln(3x3 +2x) = .

Hint: Recall the definition of the derivative of the natural loga-rithm function.

Correct Answers:

• (9*x**2+2)/(3x**3 + 2*x)

2. (10 pts) 1250Library/set8 Exponentials and Logarithms-/1220s2p2.pg(Section 7.1, No 6). Find the indicated derivative.Dx ln

√3x−2 = .

Hint: This is similar to the previous problem. Use the chainRule.

Correct Answers:

• 3/(6*x-4)

3. (10 pts) 1250Library/set8 Exponentials and Logarithms-/1220s2p3.pgSection 7.1, No 8. Understand basic derivatives. Find y′ if

y = x2 lnx

Hint: Use the Product and Chain Rules.Correct Answers:

• 2*x*ln(x) + x

4. (10 pts) 1250Library/set8 Exponentials and Logarithms-/1220s2p4.pgSection 7.1, No 10. This is a more complicated differentiationexercise which will involve all three of our main rules: the prod-uct, quotient, and chain rules. You’ll find the right derivative byproceeding carefully and deliberately. You may want to edit theresulting expression in a separate file and then cut and paste itinto the answer window. Find

drdx

ifr =

lnxx2 lnx2 +(ln

1x)3

Correct Answers:• (x*ln(xˆ2)-(2*x*ln(xˆ2)+2*x)*ln(x))/xˆ4/(ln(xˆ2))ˆ2 - 3*(ln(1/x))ˆ2/x

5. (10 pts) 1250Library/set8 Exponentials and Logarithms-/1220s2p5.pgSection 7.1, No 16. Logarithms as anti-derivatives. As youknow, indefinite integrals have an undetermined constant. Intheses exercises, use any numerical integration constant youlike, such as 0. Do not use a symbol, such as C.R 1

1−2x dx = .

Hint: Think about the natural log function.Correct Answers:• -.5*ln(1-2*x)

6. (10 pts) 1250Library/set8 Exponentials and Logarithms-/1220s2p6.pgSection 7.1, No 18. Logarithms as anti-derivatives. Use anynumerical integration constant you like, such as 0. Do not use asymbol, such as C. Z z

2z2 +8dz

= .

Hint: Use the natural log function, and notice that the numera-tor is closely related to the derivative of the denominator.

Correct Answers:• .25*ln(2zˆ2+8)

7. (10 pts) 1250Library/set8 Exponentials and Logarithms-/1220s2p7.pgSection 7.1, No 20. Logarithms as anti-derivatives.Z −1

x(lnx)2 dx

=

Hint: Use the natural log function and substitution.Correct Answers:• 1/ln(x)

1

8. (10 pts) 1250Library/set8 Exponentials and Logarithms-/1220s2p8.pgSection 7.1, No 38 A Word Problem. The rate of transmissionin a telegraph cable is observed to be proportional to

x2 ln(1/x)

where x is the ratio of the radius of the core to the thickness ofthe insulation (0 < x < 1). What value of x gives the maximumrate of transmission?

=

Hint: Think about what does ’proportional’ means in this case.Essentially, this means that the rate of transmission is a constantmultiple of the expression given. Write the rate, R, this way,differentiate and apply your knowledge of rates of change andmaxima and minima to solve for x.

Correct Answers:

• 1/(2.718281828)ˆ.5

9. (10 pts) 1250Library/set8 Exponentials and Logarithms-/1220s2p10.pgSuppose

f (x) = xxx

In expressions like this one, conventionally the upper power iscomputed first, i.e.,

f (x) = x(xx).

Thenf ′(x) =

Hint: Recall how we differentiated xx in class.Correct Answers:

• xˆx*xˆ(xˆx)*(1/x+log(x)ˆ2+log(x))

10. (10 pts) 1250Library/set8 Exponentials and Logarithms-/1220s4p1.pg(Section 7.3, No 12). Find Dxy if y = e2x2−x

Dxy =

Hint: Remember the derivative of the natural exponential func-tion and use the chain rule..

Correct Answers:

• (4*x - 1)*e**(2x**2 - x)

11. (10 pts) 1250Library/set8 Exponentials and Logarithms-/1220s4p2.pg(Section 7.3, No 14). Find Dxy if

y = e−1/x2

Dxy =

Hint: Recall the definition of the derivative of the natural expo-nential function and use the chain rule.

Correct Answers:

• (2/x**3)*e**(-1/x**2)

12. (10 pts) 1250Library/set8 Exponentials and Logarithms-/1220s4p3.pg(Section 7.3, No 20). Find Dxy if y = e1/x2

+ 1

ex2

Dxy =Hint: Apply the chain rule and watch for the right sign.

Correct Answers:

• (-2/x**3)*e**(1/(x**2)) + (-2*x)/(e**(x**2))

13. (10 pts) 1250Library/set8 Exponentials and Logarithms-/1220s4p4.pg(Section 7.3, No 30). Compute the integral. You can use anyintegration constant you like, including 0. Do not use a variablelike C.R

xex2−3d x =Hint: Use substitution and recall the example in the textbookof how to integrate the exponential function.

Correct Answers:

• .5*e**(x**2 - 3)

14. (10 pts) 1250Library/set8 Exponentials and Logarithms-/1220s4p5.pg(Section 7.3, No 32). Compute the integralR ex

ex−1dx = . Again, use any specific

numerical integration constant.Hint: Use substitution and remember what you know about thenatural logarithm.

Correct Answers:

• ln(e**x - 1)

15. (10 pts) 1250Library/set8 Exponentials and Logarithms-/1220s5p1.pgYou should be able to do the first two problems without using acalculator. Use the basic definitions of logarithms and exponen-tials. If you see a base 3 logarithm, for example, compute thefirst few powers of 3, and see how they relate to the number ofwhich you are taking the logarithm.

Evaluate the following expressions.(a) log3

( 1243

)=

(b) log4 1 =(c) log4

√64 =

(d) 8log8 14 =Correct Answers:

• -5• 0• 1.5• 14

2

16. (10 pts) 1250Library/set8 Exponentials and Logarithms-/1220s5p2.pgEvaluate the following expressions.

(a) lne9 =(b) eln2 =(c) eln

√3 =

(d) ln(1/e5) =Correct Answers:

• 9• 2• 1.73205080756888• -5

17. (10 pts) 1250Library/set8 Exponentials and Logarithms-/1220s5p5.pgSuppose y = e1/x2

+1/ex2. Find Dxy.

Dxy =

Hint: Apply the chain rule and watch for the right sign.Correct Answers:

• -2*exp(1/(x**2))/(x**3) - 2*x/exp(x**2)

18. (10 pts) 1250Library/set8 Exponentials and Logarithms-/1220s5p6.pg(This is an improved version of No 38 in Section 7.4). Theloudness of sound is measured in decibels in honor of Alexan-der Graham Bell, inventor of the telephone. A decibel is adimensionless measure of the ratio of two pressures. If thesound pressure to be measured is P, then the sound loudness Lin decibels is defined by

L = 20log10(121.3PP0

)

where P0 is a reference pressure at the limits of audibility. Findthe ratio P/P0 if the pressure P is caused by a rock band at 115decibels.

The pressure caused by the rock concert is times higherthan the barely audible reference pressure.Hint: Use the basic properties of logs to isolate P.

Correct Answers:

• 4635.95486554286

19. (10 pts) 1250Library/set8 Exponentials and Logarithms-/1220s6p1.pgThe half-life of Radium-226 is 1590 years. If a sample contains500 mg, how many mg will remain after 1000 years?

Hint: Recall P = P0ekt where P0 is the initial amount of sucha substance, P is the amount at a given time t andk is the expo-nential rate of decay. What would it mean if P = .5P0?

Correct Answers:

• 323.327737322167

20. (10 pts) 1250Library/set8 Exponentials and Logarithms-/1220s6p2.pgThe magnitude M on the Richter scale of an earthquake as afunction of its intensity I is given by

M = log10

(II0

),

where I0 is some fixed reference level of intensity.The 1906 San Francisco earthquake had a magnitude of

8.3 on the Richter scale. Suppose that at the same time inSouth America there was an earthquake with magnitude 4.9 thatcaused only minor damage. How many times more intense wasthe San Francisco earthquake than the South American one?

Answer:

Hint: You don’t need to know what I0 is. Plug in the respectivemagnitudes and use the log properties you know to compare theresulting values for I in terms of this I0.

Correct Answers:

• 2511.88643150958

21. (10 pts) 1250Library/set8 Exponentials and Logarithms-/1220s6p3.pgHuman hair from a grave in Africa proved to have only 66% ofthe carbon 14 of living tissue. When was the body buried? Thehalf life of carbon 14 is 5730 years. See Problem 13 of Section7.5 of the course text.

The body was buried about years ago.

Hint: From the text we learn the half-life of Carbon-14 is 5730years. Use this and the the information about 66 % to help youfind r as in exercise 1 of this same set. Then find t.

Correct Answers:

• 3434.91766348523

22. (10 pts) 1250Library/set8 Exponentials and Logarithms-/1220s6p4.pgNewton’s Law of Cooling states that the rate at which an objectcools is proportional to the difference in temperature betweenthe object and the surrounding medium. Thus, if an objectis taken from an oven at 308◦F and left to cool in a room at74◦F, its temperature T after t hours will satisfy the differentialequation

dTdt

= k(T −74).

If the temperature fell to 209◦F in 0.9 hour(s), what will it beafter 3 hour(s)?

After 3 hour(s), the temperature will be degree F.

Hint: Separate variables. Work through example 2, p. 342 as itis a similar problem.

Correct Answers:

• 111.406082527677

3

23. (10 pts) 1250Library/set8 Exponentials and Logarithms-/1220s6p5.pgAssume the world population will continue to grow exponen-tially with a growth constant k = 0.0132 (corresponding to adoubling time of about 52 years),it takes 1

2 acre of land to supply food for one person, andthere are 13,500,000 square miles of arable land in in the world.

How long will it be before the world reaches the maximumpopulation? Note: There were 6.06 billion people in the year2000 and 1 square mile is 640 acres.

Answer: The maximum population will be reached sometime in the year .

Hint: Convert .5 acres of land per person (for food) to the num-ber of square miles needed per person. Use this and the numberof arable square miles to get the maximum number of peoplewhich could exist on Earth. Proceed as you have in previousproblems involving exponential growth.


24. (10 pts) 1250Library/set8 Exponentials and Logarithms-/1220s8p1.pgLet

f (x) = x2 tan−1(2x)f ′(x) =

NOTE: The WeBWorK system will accept arctan(x) but nottan−1(x) as the inverse of tan(x).Hint: use the product and chain rules and recall the derivativeof tan−1(x).

Correct Answers:• 2*x**(2-1)*arctan(2*x) +

x**2*2/(1+2*2*x**2)

25. (10 pts) 1250Library/set8 Exponentials and Logarithms-/1220s8p2.pgLet

f (x) = 5sin(x)sin−1(x)f ′(x) =

NOTE: The WeBWorK system will accept arcsin(x) and notsin−1(x) as the inverse of sin(x).Hint: Use the product rule and recall the derivative of sin−1.

Correct Answers:• 5*(cos(x)*asin(x)+sin(x)/sqrt(1-x**2))

26. (10 pts) 1250Library/set8 Exponentials and Logarithms-/1220s8p3.pgEvaluate the integral:Z 2

√2

dx

x√

x2−1=

Hint: Recall the derivative of sec−1(x).

Correct Answers:

• 0.261799387833333

27. (10 pts) 1250Library/set8 Exponentials and Logarithms-/1220s8p4.pgEvaluate the integral:

Zπ/2

0

sinθ

1+ cos2 θdθ

Answer: .

Hint: Use substitution and recall the derivative of tan−1(x).

Correct Answers:

• 0.7853981635

28. (10 pts) 1250Library/set8 Exponentials and Logarithms-/1220s8p5.pgSection 7.7 No 48: Evaluate the indefinite integralZ

x2 sin(x3)dx = .

Hint: Use substitution.

Correct Answers:

• -0.333333333333333 * cos(x**(3))

29. (10 pts) 1250Library/set8 Exponentials and Logarithms-/1220s8p6.pgSection 7.7 No 50: Evaluate the indefinite integral. As usual,do not use a letter for the integration constant.Z

tan(x)dx = .

Hint: write tan(x) =sin(x)cos(x)

Correct Answers:

• -ln(cos(x))

30. (10 pts) 1250Library/set8 Exponentials and Logarithms-/1220s8p7.pgSection 7.7 No 58: Evaluate the indefinite integral. As usual,do not use a letter for the integration constant.Z ex

1+ e2x dx =.

Hint: Think inverse trig functions.

Correct Answers:

• arctan(e**x)

4

31. (10 pts) 1250Library/set8 Exponentials and Logarithms-/1220s8p8.pgSection 7.7 No 76: An airplane in Australia is flying at a con-stant altitude of 2 miles and a constant speed of 600 miles perhour on a straight course that will take it directly over a kan-garoo on the ground. How fast is the angle of elevation of thekangaroo’s line of sight increasing when the distance from thekangaroo to the plane is 3 miles? Give your answer in radiansper minute.

Answer: .

Hint: The angle of elevation is that angle that the line of sightmakes with the horizontal. Construct a triangle representing thesituation. Use this to get a equation first involving x and y, then

differentiate to involvedxdt

anddydt

. Think about what it meansthat the plane is getting closer to the kangaroo.


32. (10 pts) 1250Library/set8 Exponentials and Logarithms/intro.pgThis first problem is just an exercise in entering logarithmic andexponential functions into WeBWorK. Remember that the onlylogarithm that WeBWorK knows is the natural logarithm, andyou write it as log(x). Any other logarithm needs to be ex-pressed in terms of the natural logarithm. You can use a doubleasterisk or a caret ˆ for exponentiation. The number e in WeB-WorK is e and the natural exponential can also be written asexp(x).

Try these notations by entering the following functions:

f (x) = ex .

f (x) = ln(x) .

f (x) = 2x .

f (x) = bx .

f (x) = log2(x) .

f (x) = logb(x) .

Correct Answers:• exp(x)• log(x)• 2ˆx• bˆx• log(x)/log(2)• log(x)/log(b)

33. (10 pts) 1250Library/set8 Exponentials and Logarithms/rope.pgA free hanging cylindrical rope will break under its own weightif it exceeds a certain critical length. Suppose you have a wellthat’s deeper than the critical length of your rope, and you needa rope that will reach the bottom of that well.

A rope will break at any point if the stress at that point ex-ceeds a certain critical value. The stress is the ratio of the weightbelow that point and the area of the cross section at that point.Thus it does not help to increase the radius of the rope by a cer-tain factor. You’d increase the weight and the area of the crosssection by the square of that factor. When computing the stressthose squares would cancel. The critical length of a cylindricalrope is independent of its radius.

However, you can increase the depth a rope can reach by in-creasing its radius towards the top. This problem explores thatidea. The rope will break when the stress s at a certain pointexceeds a specific value c. That critical value c depends on thematerial of which the rope is made. So we want to construct arope which has a constant stress everywhere along the rope, andthat carries a weight w, say.

Putting this information into mathematical terms we obtainthe equation

uR x

0 πr2(t)dt +wπr2(x)

= c

or

uZ x

0πr2(t)dt +w = cπr2(x)

for all x. Here

u is the specific weight of the rope

r(x) is the radius of the rope at a distance x measured upwardsfrom its bottom

w is the weight carried by the rope.

c is the critical stress (incorporating any safety factors).

Differentiate with respect to x in the above equation, obtaina differential equation for r, determine r(0) by the fact that w isattached at x = 0 and enter the radius of your special rope:

r(x) = .Of course, your answer will depend on x, w, u, and c.

Correct Answers:

• (w/c/3.14159265358979)**(1/2)*exp(u/c/2*x)

5

34. (10 pts) 1250Library/set8 Exponentials and Logarithms/s7p38.pgIn a well known story the inventor of the game of chess wasasked by his well pleased King what reward he desired. ”Oh,not much, your majesty”, the inventor responded, ”just place agrain of rice on the first square of the board, 2 on the next, 4on the next, and so on, twice as many on each square as on thepreceding one. I will give this rice to the poor.” (For the unini-tiated, a chess board has 64 squares.) The king thought this amodest request indeed and ordered the rice to be delivered.Let f (n) denote the number of rice grains placed on the first nsquares of the board. So clearly, f (1) = 1, f (2) = 1 + 2 = 3,f (3) = 1 + 2 + 4 = 7, and so on. How does it go on? Computethe next two values of f (n): f (4) = f (5) =Ponder the structure of this summation and then enter an alge-braic expression that definesf (n) = as a function of n.

Supposing that there are 25,000 grains of rice in a pound, 2000pounds in a ton, and 6 billion people on earth, the inventor’sreward would work out to approximately tonsof rice for every person on the planet. Clearly, all the rice inthe kingdom would not be enough to begin to fill that request.The story has a sad ending: feeling duped, the king caused theinventor of chess to be beheaded.

Hint: Note the relationship between the number of grains oneach square, and the number of grains on the preceding squarescombined.

Correct Answers:

• 15• 31• 2ˆn-1• 61.4891469123652


6

Math 1250-2 Fall 2008

Hsiang-Ping Huang.


due 11/11/2008 at 11:59pm MST.

This set just has a few differentiation and integration exer-cises.

1. (1 pt) 1250Library/set9 Basic Methods of Integration/1220s9p1.pgSuppose y = sinh(x2 + x). Find Dxy.

Answer: Dxy = .

Hint: Recall the derivative of sinh(x) and use the chain rule.Correct Answers:

• (2*x+1)*cosh(x**2+x)

2. (1 pt) set9/1220s9p2.pgSuppose y = ln(cothx). Find Dxy.

Answer: Dxy = .

Hint: compute the derivative of coth(x) = cosh(x)sinh(x) and don’t

ever forget the derivative of lnuCorrect Answers:

• -sech(x)/sinh(x)

3. (1 pt) 1250Library/set9 Basic Methods of Integration/1220s9p3.pgEvaluate the integral Z cosh

√z√

zdz

Answer: .

Hint: Use Substitution.Correct Answers:

• 2*sinh(sqrt(z))

4. (1 pt) 1250Library/set9 Basic Methods of Integration/1220s10p1.pgPerform the indicated integration.Z ex

2+ ex dx = .

Hint: Use substitution with u = 2+ ex

Correct Answers:

• ln(2 + eˆx)

5. (1 pt) 1250Library/set9 Basic Methods of Integration/1220s10p2.pgPerform the indicated integrations.Z ex

ex +1dx = .Z ex

ex+1 dx = .Z ex+1

ex +1dx = .

Correct Answers:

• ln(1 + eˆx)• x/exp(1)• exp(1)*log(eˆx+1)

6. (1 pt) 1250Library/set9 Basic Methods of Integration/1220s10p3.pgR 2x3√1−x4

dx = .

Hint: Use substitution with u = 1− x4

Correct Answers:

• -(1-x**4)**.5

7. (1 pt) 1250Library/set9 Basic Methods of Integration/1220s10p4.pgZecosz sinzdz = .

Hint: Use substitution with u = coszCorrect Answers:

• -e**cos(z)

8. (1 pt) 1250Library/set9 Basic Methods of Integration/1220s10p5.pgZ sin(4t−1)1− sin2(4t−1)

dt = .

Hint: In the denominator, express the sin in terms of the cos,and use substitution.

Correct Answers:

• sec(4t-1)/4

9. (1 pt) 1250Library/set9 Basic Methods of Integration/1220s10p6.pgZ 1

0

e2x− e−2x

e2x + e−2x dx = .

Hint: Try substitution with u = e2x + e−2x

Correct Answers:

• 0.662501373678932

10. (1 pt) 1250Library/set9 Basic Methods of Integration-/1220s11p1.pgR

sin2(3x)dx = .Correct Answers:

• -1/6 cos(3 x) sin(3 x) + x/2

1


xcos3(x2)dx = .Correct Answers:

• (sin(x**2)*( - sin(x**2)**2 + 3))/6


ln(x2)dx = .Correct Answers:

• 2*(x*ln(x)-x)

13. (1 pt) 1250Library/set9 Basic Methods of Integration-/1220s13p1.pgUse integration by parts to find the following:Z

xe3xdx = .

Hint: Use the substitution

u = x and dv = e3xdx

Correct Answers:

• e**(3x) * (x/3 - 1/9)


(t +7)e2t+3dt = .

Hint: Use the substitution

u = t +7 and dv = e2t+3dx

Correct Answers:

• e**(2t+3) * ((t+7)/2 - 1/4)


xsin2xdx = .Correct Answers:

• -x/2*cos(2x) +sin(2x)/4

16. (1 pt) 1250Library/set9 Basic Methods of Integration-/1220s13p4.pgUse integration by parts to find the following:Z x

0eat sin tdt = .

Hint: Use Integration by Parts twice.Correct Answers:• (-exp(a*x) cos(x) + a*exp(a*x) sin(x))/(1+a**2)

+ 1/(1+a**2)

17. (1 pt) set9/q1.pgEvaluate the integral below. Your answer will of course involvea and b (and x). Don’t worry about the integration constant.R

eax cosbxdx= .

Correct Answers:• exp(a*x)*(b*sin(b*x)+a*cos(b*x))/(a*a+b*b)

18. (1 pt) 1220Library/set5 Techniques of Integration/set5 pr7.pgFind the following indefinite integrals:

(a)R

x( 3√

x−1)dx =+ C.

(b)R

x3(√

x−1)dx =+ C.

(c)R x3√

x−1dx =

+ C.Correct Answers:• 3/7*(x-1)**(7/3) + 3/4*(x-1)**(4/3)• 2/9*(x-1)**(9/2) + 6/7*(x-1)**(7/2)

+ 6/5*(x-1)**(5/2) + 2/3*(x-1)**(3/2)• 2/7*(x-1)**(7/2) + 6/5*(x-1)**(5/2)

+ 2*(x-1)**(3/2) + 2*(x-1)**(1/2)

19. (1 pt) 1220Library/set5 Techniques of Integration/set5 pr8.pgEvaluate the following definite integrals:

(a)R 1

0 x2exdx = .(b)

R 10 x22xdx = .

(c)R 2

1 (ln(x))2dx = .

(d)R π

20 cos6(x)dx = .

Correct Answers:• 0.718• 0.565475572069307• 0.188317305596622• 0.4908734375


2

Math 1250-2 Fall 2008

Hsiang-Ping Huang.


due 11/18/2008 at 11:59pm MST.In this set you are asked to solve a few differential equations.

1. (10 pts) 1250Library/set10 Differential Equations/1220s7p1.pgSolve the following differential equation. As you know, indefi-nite integrals are used to solve these equations and have an un-determined constant. In this exercise use C = 0.dydx

+2y = xy =

Use the formula:Zxe2xdx = e2x

(x2− 1

4

).

Of course, you can check that formula easily by differentiation.(Also, check your solution of the differential equation by differ-entiation.)

Hint: Recognize this as a first-order linear differential equa-tion and follow the general method for solving these.

Correct Answers:

• .5*x-.25

2. (10 pts) 1250Library/set10 Differential Equations/1220s7p2.pg

Solve the following differential equation.dydx

= e2x − 3y andy = 1 when x = 0y =

Hint: Recognize this as a first-order linear differential equa-tion and follow the general method for solving these and use theinitial conditions to find the integration constant.

Correct Answers:

• .2*e**(2*x) + .8*e**(-3x)

3. (10 pts) 1250Library/set10 Differential Equations/1220s7p3.pgA tank initially contains 200 gallons of brine, with 50 poundsof salt in solution. Brine containing 2 pounds of salt per gallonis entering the tank at the rate of 4 gallons per minute and is isflowing out at the same rate. If the mixture in the tank is keptuniform by constant stirring, find the amount of salt in the tankat the end of 40 minutes.The amount of salt in the tank at the end of 40 minutes is

pounds.Correct Answers:

• 242.734862537726

4. (10 pts) 1250Library/set10 Differential Equations/1220s7p4.pgA tank initially contains 50 gallons of brine, with 30 pounds ofsalt in solution. Water runs into the tank at 6 gallons per minuteand the well-stirred solution runs out at 5 gallons per minute.How long will it be until there are 25 pounds of salt in the tank?

The amount of time until 25 pounds of salt remain in the tank isminutes.

Hint: This problem is a bit complicated. Set up a differentialequation and separate variables.


5. (10 pts) 1250Library/set10 Differential Equations/q0.pgHere are some initial value problems with obvious solutions, asdiscussed in class. In all cases the solutions are functions of x.All letters other than y and x denote constants.

The solution of

y′ = ky, y(0) = A

isy(x) = .The solution of

y′′ = k2y, y(1) = y(−1) = A


y′′ = k2y, y(1) =−y(−1) = A


y′′ =−k2y, y(0) = 1, y′(0) = 0


y′′ =−k2y, y(0) = 0, y′(0) = 1


y′′ =−k2y, y(0) = A, y′(0) = B

isy(x) = .Correct Answers:• A*exp(k*x)• A*(exp(k*x)+exp(-k*x))/(exp(k)+exp(-k))• A*(exp(k*x)-exp(-k*x))/(exp(k)-exp(-k))• cos(k*x)• sin(k*x)/k• A*cos(k*x)+B*sin(k*x)/k

1

6. (10 pts) 1250Library/set10 Differential Equations/q1.pgThis is a warmup for the next question.

The solution of the differential equation

y′ = y(1− y), y(0) =12

isy(t) = .

Hint: Separate variables. Note that1

y(1− y)=

1y

+1

1− y.

Correct Answers:• exp(t)/(1+exp(t))

7. (10 pts) 1250Library/set10 Differential Equations/q2.pgConsider the growth of a population p(t). It starts out withp(0) = A. Suppose the growth is unchecked, and hence

p′ = kp

for some constant k.Then p(t) = .Of course populations don’t grow forever. Let’s say there is

a stable population size Q that p(t) approaches as time passes.Thus the speed at which the population is growing will approachzero as the population size approaches Q. One way to model thisis via the differential equation

p′ = kp(Q− p), p(0) = A.

The solution of this initial value problem is p(t) =.

Hint: Proceed as in the last problem.Correct Answers:• A*exp(k*t)• Q*A/(Q-A)*exp(k*Q*t)/(1+A/(Q-A)*exp(k*Q*t))

8. (10 pts) set10/ur de 2 7.pgSuppose you have just poured a cup of freshly brewed coffeewith temperature 90◦C in a room where the temperature is 25◦C.Newton’s Law of Cooling states that the rate of cooling of anobject is proportional to the temperature difference between theobject and its surroundings. Therefore, the temperature of thecoffee, T (t), satisfies the differential equation

dTdt

= k(T −Troom)

where Troom = 25 is the room temperature, and k is some con-stant.Suppose it is known that the coffee cools at a rate of 2◦C perminute when its temperature is 75◦C.

A. What is the limiting value of the temperature of the cof-fee?limt→∞

T (t) =

B. What is the limiting value of the rate of cooling?

limt→∞

dTdt

=

C. Find the constant k in the differential equation.k = .

D. Use Euler’s method with step size h = 3 minutes to esti-mate the temperature of the coffee after 15 minutes.T (15) = .

Correct Answers:• 25• 0• -0.04• 59.302574592

9. (10 pts) Library/Rochester/setDiffEQ4Linear1stOrder-/osu de 4 15.pgFind the particular solution of the differential equation

dydx

+ ycos(x) = 7cos(x)

satisfying the initial condition y(0) = 9.Answer: y(x)= .

Correct Answers:• 7 + 2*eˆ(-sin(x))

10. (10 pts) Library/Rochester/setDiffEQ4Linear1stOrder-/ur de 4 16.pgFind the function satisfying the differential equation

f ′(t)− f (t) =−5t

and the condition f (2) =−5.f (t) = .

Correct Answers:• -2.70670566473225*2.71828182845905**t - -5*t - -5

11. (10 pts) Library/Rochester/setDiffEQ4Linear1stOrder-/ur de 4 15.pgSolve the initial value problem

dxdt

+2x = cos(2t)

with x(0) =−1.x(t) = .

Correct Answers:• -1.25*2.71828182845905ˆ(- 2*t) + 2/8*cos(2*t) + 2/8*sin(2*t)

12. (10 pts) Library/Rochester/setDiffEQ3Separable/ur de 3 15.pgSolve the separable differential equation

dydx

=−0.2cos(y)

,

and find the particular solution satisfying the initial condition

y(0) =π

3.

y(x) = .2

Correct Answers:

• arcsin(-0.2*x + 0.866025403784439)

13. (10 pts) Library/Rochester/setDiffEQ3Separable/jas7 4 5.pgFind u from the differential equation and initial condition.

dudt

= e1.1t−1.7u, u(0) = 2.

u = .Correct Answers:

• (1/1.7)*ln(28.4186455019425 + ((1.7/1.1)*2.71828182845905ˆ(1.1*t)))

14. (10 pts) Library/Rochester/setDiffEQ3Separable/ns7 4 8a.pgSolve the seperable differential equation for.

dydx

=1+ xxy17 ; x > 0

Use the following initial condition: y(1) = 2.y18 = .

Correct Answers:• 18*ln(x) + 18*x + 262126


3

Math 1250-2 Fall 2008

Hsiang-Ping Huang.


due 11/25/2008 at 11:59pm MST.

This set has applications of integration, and a few partial frac-tions problems.

1. (10 pts) 1250Library/set11 Applications of Integration-/1220s1p19.pgA Solid of Revolution. Find the volume of the solid obtainedby rotating the region bounded by the given curves about thespecified axis.

y = 4x2,x = 1,y = 0, about the x-axis

Correct Answers:

• 10.0530964914873

2. (10 pts) 1250Library/set11 Applications of Integration-/1220s1p21.pgWork. A force of 6 pounds is required to hold a spring stretched0.5 feet beyond its natural length. How much work (in foot-pounds) is done in stretching the spring from its natural lengthto 0.8 feet beyond its natural length?

Correct Answers:

• 3.84

3. (10 pts) 1250Library/set11 Applications of Integration/p12-1.pgFind the volume of the solid generated by revolving about thex-axis the region bounded by the upper half of the ellipse

x2

a2 +y2

b2 = 1

and the x-axis, and thus find the volume of a prolate spheroid.Here a and b are positive constants, with a > b.

Volume of the solid of revolution: .Correct Answers:

• (4/3)*3.141592654*a*(b**2)

4. (10 pts) 1250Library/set11 Applications of Integration/p12-2.pgThe region bounded by y = 2 + sinx, y = 0, x = 0 and 2π isrevolved about the y-axis. Find the volume that results.

Hint: Zxsinx dx = sinx− xcosx+C.

Volume of the solid of revolution: .Correct Answers:

• 208.571795924897

5. (10 pts) 1250Library/set11 Applications of Integration/p12-8.pgFind the volume of the solid obtained by rotating the regionbounded by the given curves about the specified axis.

y = x2,y = 1; about y = 9

Correct Answers:

• 70.3716754404114

6. (10 pts) 1250Library/set11 Applications of Integration/p13-1.pgFind the volume of the solid formed by rotating the region in-side the first quadrant enclosed byy = x4

y = 8xabout the x-axis.

Correct Answers:

• 357.443430808439

7. (10 pts) 1250Library/set11 Applications of Integration/p13-2.pgA ball of radius 11 has a round hole of radius 8 drilled throughits center. Find the volume of the resulting solid.

Correct Answers:

• 1802.6063342226


y = 3x2,x = 1,y = 0, about the x-axis

Correct Answers:

• 5.65486677646163

1


y = x2,y = 0,x = 0,x = 3, about the y-axis

Correct Answers:

• 127.234502470387


y = 0,y = x(1− x) about the axis x = 0

Correct Answers:

• 0.523598775598299

11. (10 pts) 1250Library/set11 Applications of Integration/p14.pgarc length

Find the length of the curve defined by

y = 4x3/2 +1

from x = 2 to x = 6.

Correct Answers:

• 47.6462469775597

12. (10 pts) 1250Library/set11 Applications of Integration/p15.pgWork

A force of 4 pounds is required to hold a spring stretched 0.2feet beyond its natural length. How much work (in foot-pounds)is done in stretching the spring from its natural length to 0.7 feetbeyond its natural length?


13. (10 pts) 1250Library/set12 Further Techniques and Applications of Integration-/1220s14p1.pgUse the method of partial fraction decomposition to perform thefollowing integration.Z 2

x2 +3xdx = .

Hint: See example 5 in section 8.5 of the text.Correct Answers:• 2/3*(ln(x) - ln(x+3))

14. (10 pts) 1250Library/set12 Further Techniques and Applications of Integration-/1220s14p2.pgUse the method of partial fraction decomposition to perform thefollowing integration.Z 5x

2x3 +6x2 dx = .

Hint: See example 5 in section 8.5 of the text.Correct Answers:• 5/6*(ln(x) - ln(x+3))

15. (10 pts) 1250Library/set12 Further Techniques and Applications of Integration-/1220s14p3.pgUse the method of partial fraction decomposition to perform thefollowing integration.Z 2x2− x−20

x2 + x−6dx = .

Hint: 2x2− x−20 = 2(x2 + x−6)−3x−8Correct Answers:• 2x - 1/5*(ln(x+3) + 14*ln(x-2))


2

Math 1250-2 Fall 2008

Hsiang-Ping Huang.


due 12/02/2008 at 11:59pm MST.

This set has more applications of integration.

1. (1 pt) 1250Library/set12 Further Techniques and Applications of Integration-/1220s14p4.pgIn many population growth problems, there is an upper limit be-yond which the population cannot grow. Many scientists agreethat the earth will not support a population of more than 16 bil-lion. There were 2 billion people on earth in 1925 and 4 billionin 1975. If y is the population t years after 1925, an appropriatemodel is the differential equation

dydt

= ky(16− y).

Note that the growth rate approaches zero as the population ap-proaches its maximum size. When the population is zero thenwe have the ordinary exponential growth described by y′ = 16ky.As the population grows it transits from exponential growth tostability.

(a) Solve this differential equation. y =(b) The population in 2015 will be y = billion.(c) The population will be 9 billion some time in the year

Note that the data in this problem are out of date, so the nu-merical answers you’ll obtain will not be consistent with currentpopulation figures.

Hint: (a) Separate variables and use the given information tosolve for y. (b) Evaluate y. (c) Solve for the appropriate time.

Correct Answers:

• 16/(1+7e**(-(1/50*ln(7/3))t))• 6.34• 2054

2. (1 pt) 1250Library/set12 Further Techniques and Applications of Integration-/1220s14p5.pgComputeR √

x+1√x−1 dx = .

Hint: Use a substitution and partial fractions.Correct Answers:

• x+4*sqrt(x)+4*log(sqrt(x)-1)

3. (1 pt) 1250Library/set12 Further Techniques and Applications of Integration-/1220s17p2.pgIf a 1000-pound capsule weighs only 165 pounds on the moon,how much work is done in propelling this capsule out of themoon’s gravitational field? (Note: The radius of the moon isroughly 1080 miles)

The amount of work is mile-pounds.Correct Answers:

• 1.782*10**5

4. (1 pt) 1250Library/set12 Further Techniques and Applications of Integration-/q1.pgThe solution of the initial value problem

dydx

= ysinx−2sinx, y(0) = 0

isy = .

Hint: Find an integrating factor.Correct Answers:

• 2-2*exp(1)exp(-cos(x))

5. (1 pt) 1250Library/set12 Further Techniques and Applications of Integration-/q2.pgA lake contains an amount V liters of water. Initially, the watercontains an amount P liters of a pollutant. Water is flowing intothe lake at a rate of F liters per second, and well mixed lakefluid is flowing out at the same rate. The ratio of pollutant tototal fluid in the inflow is r. The amount of p(t) of pollutant inthe lake is

p(t) = .Hint: Set up a linear differential equation and find an integrat-ing factor.

Correct Answers:

• r*V+(P-r*V)*exp(-F*t/V)

6. (1 pt) 1250Library/set12 Further Techniques and Applications of Integration-/q3.pgThe average value of sinx in the interval [0,π] is

.Hint: Apply the Mean Value Theorem for integrals.

Correct Answers:

• 0.636619772367581

7. (1 pt) set12/q4.pgThe center of mass of a quarter circle given by

y =√

r2− x2, x ∈ [0,r]

is the pointP = ( , ).

Hint: Use symmetry.Correct Answers:

• 4*r/3/3.14159265358979• 4*r/3/3.14159265358979

1

8. (1 pt) 1250Library/set12 Further Techniques and Applications of Integration-/q5.pgLet T be the triangle with vertices (a,r), (b,s), and (c, t) (andassume constant density). The center of mass of T is

C = ( , ).Hint: You can do this the easy or the hard way.

Correct Answers:

• (a+b+c)/3• (r+s+t)/3

9. (1 pt) 1250Library/set12 Further Techniques and Applications of Integration-/q6.pgI got the following data from this web page: The diameter ofthe Sun is 1.392 million kilometers. The mean distance of theSun from the Earth (measured from center to center) is 149 mil-lion kilometers. The surface gravity on the Sun is 274 meters

per second squared. Using those numbers, the escape velocityon the surface of the Sun is kilometers per second.

For the next part, let’s first solve a general problem. Sup-pose you have a planet or star of radius R and surface gravity G.Then the escape velocity on the surface of that object is .(We answered that question in class.). Now suppose you are al-ready at a distance S > R from the center of that object. Thenthe escape velocity is only .

Let’s apply this to our actual world. Suppose you are launch-ing a space craft from a point in Earth’s orbit. Ignoring earth’sgravity the minimum speed required to cause your craft to leavethe solar system is kilometers per second.

Correct Answers:

• 617.582383168432• sqrt(2*G*R)• sqrt(2*G*R*R/S)• 42.2091284379331


2

Math 1250-2 Fall 2008

Hsiang-Ping Huang.


due 12/09/2008 at 11:59pm MST.

This set is on limits, indeterminate expressions, improper in-tegrals, and Newton’s Method. There is a couple of problemson bisection, a simple alternative rootfinding technique. It’s de-scribed in the textbook on page 142.

1. (10 pts) 1250Library/set13 Limits LHopitals Rule and Numerical Methods-/1220s15p1.pgCompute

limx−→12x− sinx

x=

Compute

limx−→1lnx2

x2−1=

Correct Answers:

• 2-sin(1)• 1


limx−→0ex− e−x

2sinx=

Correct Answers:

• 1


limx−→0sinx− tanx

x2 sinx=

Hint: Apply l’Hopital’s Rule repeatedly.Correct Answers:

• -0.5


limx−→0coshx−1

x2 =Hint: Apply l’Hopital’s Rule repeatedly.

Correct Answers:

• 0.5

5. (10 pts) 1250Library/set13 Limits LHopitals Rule and Numerical Methods-/1220s15p5.pgFind the indicated limit. Begin by deciding why l’Hopital’s ruleis not applicable. Then find the limit by other means.

limx−→0x2 sin 1

xtanx

=

Hint: |sinx| ≤ 1.Correct Answers:

• 0


limx−→∞

(lnx)2

2x =Correct Answers:

• 0


limx−→∞

3xln(100x+ ex)

Correct Answers:

• 3

8. (10 pts) set13/1220s16p3.pgCompute

limx−→0

(cosx)1/x2=

Correct Answers:

• 1/sqrt(e)


limx−→∞

(1+

1x

)x

=

Correct Answers:

• e


limx−→0+ xx =Correct Answers:

• 1

11. (10 pts) 1250Library/set13 Limits LHopitals Rule and Numerical Methods-/1220s17p1.pg

Find the area of the region under the curve y =1

x2 + xto the

right of x = 1.The area isHint: Use Partial FractionsCorrect Answers:

• ln(2)

1

12. (10 pts) set13/1220s17p3.pgSuppose that a company expects its annual profits t years fromnow to be f (t) dollars and that interest is considered to be com-pounded continuously at an annual rate r. Then the presentvalue of all future profits can be shown to be

FP =Z

∞

0e−rt f (t)dt

Find FP if r = 0.08 and f (t) = 100,000+1000t.The present value is dollars.Correct Answers:

• 1406250

13. (10 pts) 1250Library/set13 Limits LHopitals Rule and Numerical Methods-/1220s17p4.pgIn electromagnetic theory, the magnetic potential u at a point onthe axis of a circular coil is given by

u = ArZ

∞

a

dx(r2 + x2)(3/2)

where A,r,a are constants. Computeu = .Hint: The integration is a little tricky. Substitute x = r tanθ.Correct Answers:

• A/r(1-(a)/sqrt(r**2+a**2))

14. (10 pts) 1250Library/set13 Limits LHopitals Rule and Numerical Methods-/1220s18p1.pgEvaluate the improper integral if it converges. In this and allother problems of this set, enter the capital letter ”D” if it di-verges. Z 3

1

dx(x−1)4/3

Correct Answers:

• D

15. (10 pts) 1250Library/set13 Limits LHopitals Rule and Numerical Methods-/1220s18p2.pgEvaluate the improper integral.Z 9

0

dx√9− x

Correct Answers:

• 6

16. (10 pts) 1250Library/set13 Limits LHopitals Rule and Numerical Methods-/1220s18p3.pgEvaluate the improper integral.Z 1

0

x3√1− x2

dx

Correct Answers:

• .75

17. (10 pts) 1250Library/set13 Limits LHopitals Rule and Numerical Methods-/1220s18p4.pgEvaluate the improper integral.Z

π/2

0

cosx3√sinx

dx

Correct Answers:

• 1.5

18. (10 pts) set13/1220s18p5.pgEvaluate the improper integral.Z 4

2

dx√4x− x2

Hint: Use the technique of completing the square.Correct Answers:

• 1.5707963267949

19. (10 pts) 1250Library/set13 Limits LHopitals Rule and Numerical Methods-/1220s28p1.pgUse the Bisection Method to approximate the real root of thegiven equation on the given interval. Your answer should beaccurate to two decimal places.

x4 +5x3 +1 = 0; [−1,0]The root r ≈Correct Answers:

• -.61

20. (10 pts) 1250Library/set13 Limits LHopitals Rule and Numerical Methods-/1220s28p2.pgUse the Bisection Method to approximate the real root of thegiven equation on the given interval. Your answer should beaccurate to two decimal places.

x−2+2lnx = 0; [1,2]The root r ≈Correct Answers:

• 1.37

21. (10 pts) 1250Library/set13 Limits LHopitals Rule and Numerical Methods-/1220s28p3.pgUse Newton’s Method to approximate the root of x lnx = 2accurate to five decimal places. Begin by sketching a graph.

The root r ≈Correct Answers:

• 2.345751

22. (10 pts) 1250Library/set13 Limits LHopitals Rule and Numerical Methods-/1220s28p4.pgSection 11.3, Problem 6. Use Newton’s Method to approxi-mate the real root of 7x3 + x− 5 = 0 accurate to five decimalplaces. Begin by sketching a graph.

The root r ≈Correct Answers:

• .84070

2

23. (10 pts) 1250Library/set13 Limits LHopitals Rule and Numerical Methods-/s16p1.pgCompute

limx−→12x− sinx

x=

Compute

limx−→1lnx2

x2−1=

Correct Answers:

• 2-sin(1)• 1

24. (10 pts) 1250Library/set13 Limits LHopitals Rule and Numerical Methods-/s16p2.pgCompute


2sinx=

Correct Answers:

• 1

25. (10 pts) 1250Library/set13 Limits LHopitals Rule and Numerical Methods-/s16p3.pgSection 9.1, Problem 10. Compute


2sinx=

Correct Answers:

• 1

26. (10 pts) 1250Library/set13 Limits LHopitals Rule and Numerical Methods-/s16p4.pgSection 9.1, Problem 10. Compute


2sinx=

Correct Answers:

• 1

27. (10 pts) 1250Library/set13 Limits LHopitals Rule and Numerical Methods-/s16p5.pgFind the indicated limit. Begin by deciding why l’Hopital’s ruleis not applicable. Then find the limit by other means.

limx−→0x2 sin 1

xtanx

=

Hint: |sinx| ≤ 1.Correct Answers:

• 0


3

Math 1250-2 Fall 2008

WeBWorK assignment number 14

Hsiang-Ping Huang

due 12/18/2008 at 11:59pm MSTThis is the final home work set for this semester. Its purpose

is to review the entire semester and help you prepare for the finalexam.

You should be able to answer questions like these (but not justthese particular questions) on the final exam. So make sure youunderstand everything surrounding each of the questions on thisset. Use them as a guide in reviewing our work this semester.

Remember that the final exam will take place

Friday, December 19, 10:30am-12:30pm,

in our regular classroom. It will be closed books and notes, noelectronics, as usual.

The original schedule class states that the deadline for thisset will be the last day of classes. The reason for that choice isthat I don’t think we should have assignments due during final’sweek. On the other hand, our final is on the very last day of thatweek, and that deadline would be a full week before the final. Itoccurred to me that you might prefer to work on this set duringfinal’s week. So I set the deadline at midnight on the day beforethe final. Of course, to be really on top of things you shouldfinish work on this set by Friday, December 12.

Peter Alfeld, JWB 127, 581-6842.

1. (10 pts) 1250Library/set14 Review/1250s14q1.pg

Limits: Compute limx−→3x2−7x+12

x−3 = .

Correct Answers:• -1

2. (10 pts) 1250Library/set14 Review/1250s14q2.pgThe Product Rule: Compute Dx(sinx)(x2 − 1) =

Correct Answers:• cos(x)*x**2 - cos(x) + 2*sin(x)*x

3. (10 pts) 1250Library/set14 Review/1250s14q3.pgThe Quotient Rule:

Compute Dxsinxx2−1 =

Correct Answers:• (cos(x)*x**2 - cos(x) - 2*sin(x)*x)/

(x**4 - 2*x**2 + 1)

4. (10 pts) 1250Library/set14 Review/1250s14q4.pgThe Chain Rule:

Compute Dx cos2(sinx)Correct Answers:• - 2*cos(sin(x))*cos(x)*sin(sin(x))

5. (10 pts) 1250Library/set14 Review/1250s14q5.pgA word problem:

What number exceeds its square by the maximum amount?Answer: .

Correct Answers:

• 0.5

6. (0 pts) 1250Library/set14 Review/1250s14q6.pgGraphing: This question does not actually require an answersince it’s hard in WeBWorK to ask for a graph. But you shouldbe able to draw the graphs of simple functions using symme-try and derivatives. Your graphs should clearly show importantfeatures like stationary points, minima, maxima, points of in-flection, and asymptotes. Practice until you get it down. Forexample, draw the graph of

f (x) =x2

x+1.

7. (10 pts) 1250Library/set14 Review/1250s14q7.pgIndefinite Integrals: Evaluate the indefinite integral.Z

x3√

1+ x4dx

Correct Answers:

• 2/3 * 0.25 * (xˆ4 + 1)ˆ(3/2)

8. (10 pts) 1250Library/set14 Review/1250s14q8.pgDefinite Integrals:Evaluate

R 51

(6x2−10x+7

)dx = .

Correct Answers:

• 156

9. (10 pts) 1250Library/set14 Review/1250s14q9.pgAreas: There is a line through the origin that divides the regionbounded by the parabola y = 6x− 2x2 and the x-axis into tworegions with equal area. What is the slope of that line?

Correct Answers:

• 1.2377968440954

10. (10 pts) 1250Library/set14 Review/1250s14q10.pgTrigonometric Integrals: Some trigonometric integrals can beevaluated by repeated integration by parts, by using trigonomet-ric identities or by using suitable trigonometric identities.R 2π

0 sin2 xdx = .Rsin2 xdx = ,R 1tanx dx = .Correct Answers:

1

• 3.14159265358979• -1/2*cos(x)*sin(x)+x/2• log(sin(x))

11. (10 pts) 1250Library/set14 Review/1250s14q11.pgIntegration by Substitution: Some integrals are obvious, oth-ers impossible, and yet others can be handled only with a bitof cleverness. In class we have been referring to the use of theformula Z

f (u(x))u′(x)dx

as the ”obvious” substitution. Look at the integrals below. Re-member that in WeBWorK we ignore the integration constant,and for definite integrals you may have to change the limits ofintegration when doing the integration.

Resin(x) cosxdx = ,R 2

1 xsin(x2)dx = .R 1x lnx dx = .Correct Answers:

• exp(sin(x))• 0.596972963365876• log(log(x))

12. (10 pts) 1250Library/set14 Review/1250s14q12.pgMore Integration by Substitution:

Compute these integrals:R x√3x+4

dx = .Rπ

0πx−1√

x2+πdx = .

Correct Answers:

• (2*sqrt(3*x+4)*(3*x-8))/27• 4.42673805745442

13. (10 pts) 1250Library/set14 Review/1250s14q13.pgInverse Trig Functions: The trigonometric functions are notinvertible because by the nature of their definition in terms of apoint moving around the unit circle there are many angles thatgive rise to the same function value. However, there are prob-lems where you need to know an angle having a certain prop-erty, and to facilitate finding that angle a notion of ”inverses”of the trig functions are created by suitably restricting the do-main of the trigonometric functions. You should understand theconventions and definitions involved, and you should be able todifferentiate the inverses of the trigonometric functions.

ddx

arcsin(x2) = .

ddx

arctan(ex) = .

You should also be able to use the inverse trigonometric func-tions to solve trigonometric equations.

For example, if the angle x is known to be in the interval[−π

4 , π

4

]and 3sin(2x) = 2, thenx = .

The negative angle θ closest to zero that satisfies cosθ = 3/7 is.

Correct Answers:

• 2x/sqrt(1-xˆ4)• exp(x)/(1+exp(2*x))• 0.364863828113483• -1.12788528272126

14. (10 pts) 1250Library/set14 Review/1250s14q14.pgInverse Functions and their derivatives: This section buildsup to defining the exponential as the inverse function of the log-arithm.

Suppose f (x) = 3x+4. Then the inverse of f is given byf−1(x) = .Moreover,f ′(x) = , and(f−1

)′ (x) = .In general, if f ′(x) = A, then(

f−1)′ ( f (x)) = (Note: Your answer must be in

terms of A.)Now let f (x) = x+ x5. Thenf (1) = ,f ′(1) = ,f−1(2) = , and(f−1

)′ (2) = .Notice that you can obtain the last two results without know-

ing a general expression for the inverse function of f . Indeed,if you feel enterprising try to come up with such an expression.Let me know what you find.

Correct Answers:

• (x-4)/3• 3• 0.333333333333333• 1/A• 2• 6• 1• 0.166666666666667

15. (10 pts) 1250Library/set14 Review/1250s14q15.pgLogarithms: Computeddx

ln(x2 +1) =andR cosx

2+sinx dx = .Correct Answers:

• 2x/(xˆ2+1)• ln(2 + sin(x))

2

16. (10 pts) 1250Library/set14 Review/1250s14q16.pgExponentials and Logarithms: The (natural) exponentialequals its own derivative and its own antiderivative. Once youaccept that the derivative of the natural logarithm is 1/x thesefacts flow from the fact that the exponential is the inverse ofthe natural logarithm. Both exponentials and logarithms appearfrequently in applications.

Here are some exercises reinforcing the unique status of theexponential.

ddx

2x = .

ddx

esinx = .Rxex2

dx = . (As usual ignore the integration con-stant for indefinite integrals in this home work.)

ddx

x√

x = .Correct Answers:

• 2ˆx*log(2)• exp(sin(x))*cos(x)• exp(xˆ2)/2• (x**sqrt(x)*(log(x) + 2))/(2*sqrt(x))

17. (10 pts) 1250Library/set14 Review/1250s14q17.pgExponential Growth and Decay: Here are a couple of exam-ples for applications of exponentials and logarithms.

You find out that in the year 1800 an ancestor of yours in-vested 100 dollars at 6 percent annual interest, compoundedyearly. You happen to be her sole known descendant and in theyear 2005 you collect the accumulated tidy sum ofdollars. You retire and devote the next 10 years of your life towriting a detailed biography of your remarkable ancestor.

Strontium-90 is a biologically important radioactive isotopethat is created in nuclear explosions. It has a half life of 28years. To reduce the amount created in a particular explosion bya factor 1,000 you would have to wait years. Roundyour answer to the nearest integer.

Seeds found in a grave in Egypt proved to have only 66%of the Carbon-14 of living tissue. Those seeds were harvested

years ago. The half life of Carbon-14 is 5,730 years.Correct Answers:

• 15406443• 279• 3434.91766348523

18. (10 pts) 1250Library/set14 Review/1250s14q18.pgThe Hyperbolic Functions and their Inverses: For our pur-poses, the hyperbolic functions, such as

sinhx =ex− e−x

2and coshx =

ex + e−x

2are simply extensions of the exponential, and any questions con-cerning them can be answered by using what we know aboutexponentials. They do have a host of properties that can become

useful if you do extensive work in an area that involves hyper-bolic functions, but their importance and significance is muchmore limited than that of exponential functions and logarithms.

Let

f (x) = sinhxcoshx.

ddx

f (x) = .Rf (x) = .

f−1(x) = .

( f−1)′(x) = .Correct Answers:

• (exp(2*x)+exp(-2*x))/2• (exp(2*x)+exp(-2*x))/8• ln(sqrt(2*x+sqrt(4*x**2+1)))• 1/sqrt(4*x*x+1)

19. (10 pts) 1250Library/set14 Review/1250s14q19.pgIntegration by Parts: This is the most important integrationtechnique we’ve discussed in this class. It has a wide range ofapplications beyond increasing our list of integration rules.R

z3 lnzdz = .Ret cos tdt = .

R 2π

0 sin(x)sin(x+1)dx = .Correct Answers:

• zˆ4*(4*log(z)-1)/16• 1/2 exp(t) (cos(t) + sin(t))• 1.69740975483297

20. (10 pts) 1250Library/set14 Review/1250s14q20.pgIntegration by Partial Fractions: This technique involves anunusual decomposition of rational expressions: instead of fac-toring them you write them as sums of simpler terms.R 3

x2−1 dx = .R x−11x2+3x−4 dx = .

R 1x4−16 dx = .

Correct Answers:

• (3*(log(x - 1) - log(x + 1)))/2• -2*log(x - 1) + 3*log(x + 4)• (- 2*atan(x/2) + log(x - 2) - log(x + 2))/32

3

21. (10 pts) 1250Library/set14 Review/1250s14q21.pgThe Fundamental Theorem of Calculus: Use the Fundamen-tal Theorem of Calculus to find the derivative of

f (x) =Z x2

4

(14

t2−1)5

dt

f ′(x) =

Correct Answers:

• 2*(1/4*xˆ4-1)ˆ5*x

22. (10 pts) 1250Library/set14 Review/1250s14q22.pgarc length: Find the length of the curve defined by

y = 4x3/2−1

from x = 4 to x = 7.

Correct Answers:

• 42.1885210626592

23. (10 pts) 1250Library/set14 Review/1250s14q23.pgVolumes: A ball of radius R has a round hole of radius r drilledthrough its center. Find the volume of the resulting solid.

Correct Answers:

• 4/3*pi*((Rˆ2)-(rˆ2))ˆ(3/2)

24. (10 pts) 1250Library/set14 Review/1250s14q24.pgMore Volumes: Find the volume of the solid obtained by rotat-ing the region bounded by the given curves about the specifiedaxis.

y = axn,x = 1,y = 0, , where n≥ 1 and a > 0, about the x-axis

Correct Answers:

• aˆ2*pi/(2*n+1)

25. (10 pts) 1250Library/set14 Review/1250s14q25.pgImproper Integrals, Infinite Limits of Integration: Integralsmay have infinite limits of integration, or integrands that havesingularities. Such integrals are called ”improper” even thoughthere is nothing wrong with such integrals. They may have welldefined values, in which case we say they converge, or they maynot, in which case we say they diverge. In this and the next prob-lem, give the value of the integral if it converges, and enter theletter ”D” if it diverges.R

∞

11x2 dx = .R

∞

11x dx = .R

∞

1 e−xdx = .

R∞

1 exdx = .R∞

−∞

11+x2 dx = .R

π

−∞exdx = .

Correct Answers:

• 1• D• 0.367879441171442• D• 3.14159265358979• 23.1406926327793

26. (10 pts) 1250Library/set14 Review/1250s14q26.pgImproper Integrals, Infinite Integrands: Compute the im-proper integrals below. Enter the letter ”D” if they diverge.R 1

01x2 dx = .R 1

01x dx = .R 1

01√x dx = .R 1

0 lnxdx = .R 1−1

1√1−x2

dx = .

Correct Answers:

• D• D• 2• -1• 3.14159265358979

27. (10 pts) 1250Library/set14 Review/1250s14q27.pgIndeterminate Expressions of the Form 0/0: Find the indi-cated limits. Make sure you have an Indeterminate Expressionbefore you apply the Rule of L’Hopital. Enter the letter ”D” ifthe limit does not exist.

limx−→0

sinxx

= .

limx−→0

ex−1− xx2 = .

limx−→0

ex− e−x

sin(x)= .

limx−→−2

x2 +5x+6x−2

= .

limx−→2

x2 +5x+6x−2

= .Correct Answers:

• 1• 0.5• 2• 0

4

• D

28. (10 pts) 1250Library/set14 Review/1250s14q28.pgOther Indeterminate Expressions: Find the indicated limits.You may have to manipulate your expression before the apply-ing the Rule of L’Hopital. Enter the letter ”D” if the limit doesnot exist.

Suppose p(x) is a polynomial of degree greater than 0. Then

limx−→∞

p(x)ex = and

limx−→∞

p(x)lnx

= .

limx−→0

(1+ x)1/x = .

limx−→∞

(1+

1x

)x

= .

limx−→∞

(1+

1x

)2x

= .

limx−→0+

xx = .

limx−→0+

xxx= .

Correct Answers:

• 0• D• 2.71828182845905• 2.71828182845905• 7.38905609893065• 1• 0

29. (10 pts) 1250Library/set14 Review/1250s14q29.pgSequences: A sequence is of the form

a1,a2,a3,a4, . . .

where the an are real numbers. Technically, a sequence is afunction whose domain is the set of natural numbers, and whoserange is a subset of the real numbers. Sequences may be definedin various ways:

By listing, and appealing (via the three dots) to your intuition.Suppose the sequence is

12,

25,

310

,417

,5

26,

637

, . . .

Then the n-th term isan = .

Explicitly. For example, suppose an = nn. Then a1 = , a2 =, and a3 = .

Recursively. For example, the Fibonacci Sequence is definedby

a1 = a2 = 1, an+1 = an +an−1, n = 2,3,4, . . . .

Thus a3 = , a4 = , and a5 = .

A sequence may or may not have a limit. For the following se-quences, enter the limit, or enter the letter ”D” if the sequencediverges.

an = n .

an = 1n .

an = n2+4n−5(2n−1)(3n−1) .

Correct Answers:• n/(nˆ2+1)• 1• 4• 27• 2• 3• 5• D• 0• 0.166666666666667

30. (10 pts) set14/1250s14q30.pgSeries: A Series (Or Infinite Series ) is obtained from a sequenceby adding the terms of the sequence. Another sequence associ-ated with the series is the sequence of partial sums. A seriesconverges if its sequence of partial sums converges. The sum ofthe series is the limit of the sequence of partial sums.

For example, consider the geometric series defined by the se-quence

an =1rn , n = 0,1,2, . . .

Then the n-th partial sum Sn is given by

Sn =n

∑k=0

1rk = and, for −1 < 1/r < 1,

∞

∑k=0

1rk = lim

n−→∞Sn = .

Thus∑

∞k=0

( 2π

)k = and

∑∞k=2

( 2π

)k = .

Another situation in which we we can actually compute the par-tial sums occurs if those sums are collapsing . It may not beobvious that that is the case, but look for it in this example:

∑nk=0

(1

k2+9k+20

)= and thus

∑∞k=0

(1

k2+9k+20

)= .

Correct Answers:• (1-(1/r)**(n+1))/(1-(1/r))• 1/(1-(1/r))• 2.75193839388411• 1.11531862151653• 1/4-1/(n+5)

5

• 0.25

31. (10 pts) 1250Library/set14 Review/1250s14q31.pgTaylor and MacLaurin Series: Consider the approximation ofthe exponential by its third degree Taylor Polynomial:ex ≈ P3(x) = 1+ x+ x2

2 + x3

6 .

Compute the error ex−P3(x) for various values of x:e0−P3(0) = .e0.1−P3(0.1) = .e0.5−P3(0.5) = .e1−P3(1) = .e2−P3(2) = .e−1−P3(−1) = .

Correct Answers:

• 0• 4.2514089810392E-06• 0.00288793736679486• 0.0516151617923784• 1.05572276559732• 0.034546107838109

32. (10 pts) 1250Library/set14 Review/1250s14q32.pgPower Series: Compute the convergence set for the followingpower series. Enter ”¡” or ”¡=” (without the quotation marks) asappropriate. If the convergence radius is infinite enter ”-I” and”I” for minus infinity and plus infinity, respectively.

For example (see Example 2 on page 459), the series∞

∑n=0

xn

(n+1)2n

converges on the interval [−2,2), so you’d enter -2, ¡=, ¡ 2. Trythis now:

∞

∑n=0

xn

(n+1)2n converges for

x .Similarly answer the following questions:

∞

∑n=1

(x−1)n

nconverges for

x .

∞

∑n=1

(x−1)n

n!converges for

x .

∞

∑n=17

(x+1)n

3n converges for

x .Correct Answers:

• -2• <=• <• 2• 0• <=

• <• 2• -I• <• <• I• -4• <=• <• 2

33. (10 pts) 1250Library/set14 Review/1250s14q33.pgTaylor and MacLaurin Series: Compute the Taylor Series be-low. WeBWorK does not understand factorials, so you will needto enter numerical values. For example, enter ”1/6” instead of”1/3!”. You may also need to enter factors 0 or 1.

ex = + x+ x2+ x3+ x4 + . . ..

cos2 x = + x+ x2+ x3+ x4 + . . ..

xx = + (x− 1)+ (x− 1)2+ (x− 1)3+ (x−1)4 + . . ..

4x4 +3x3 +2x2 + x +1 = + (x−1)+ (x−1)2+(x−1)3+ (x−1)4.

Correct Answers:• 1• 1• 0.5• 0.166666666666667• 0.0416666666666667• 1• 0• -1• 0• 0.333333333333333• 1• 1• 1• 0.5• 0.333333333333333• 11• 30• 35• 19• 4

34. (10 pts) 1250Library/set14 Review/1250s14q34.pgDifferential Equations: A differential equation is an equationthat involves and unknown function and some of its derivatives.Differential Equations are a large subject on which we teach sev-eral courses. In Calculus you only get a first exposure to themand you see some very simple examples. Set up and solve adifferential equation to solve the following problem:

You have a pond with decorative fish in your back yard. Thepond holds 800 gallons of water. Once a week you pour freshwater into the pond at the rate of 100 gallons per hour. The pond

6

is filled to the brim, and so as you pour water into the tank wa-ter flows out at the same rate. There is a pump in the pond thatkeeps the water perfectly mixed. Your goal is to add water un-til any pollutants in the pond are reduced by a factor 1/2. Youkeep the fresh water flowing for hours. (Rememberthat ww expects your answer to be accurate within one tenth ofone percent. You may want to enter a mathematical expressionrather than a numerical value.)


35. (10 pts) 1250Library/set14 Review/1250s14q35.pgNewton’s Method: As discussed in class, to solve the equation

f (x) = 0

by Newton’s Method we start with a good initial guess x0 andthen run the iteration

xn+1 = xn−f (xn)f ′(xn)

, n = 0,1,2, . . .

until we get an approximation xn+1 that is good enough for ourpurposes.

Suppose you want to compute the cube root of 4 by solvingthe equation

x3−4 = 0.

Since 13 = 1 and 23 = 8 Let’s start with

x0 = 1.5

Thenx1 = ,

x2 = ,x3 = , andTo check your answer compute x3

3 = .

Enter x1, x2 and x3 with at least 6 correct digits beyond the dec-imal point.

Correct Answers:• 1.59259259259259• 1.58741795698709• 1.587401052• 4

36. (10 pts) 1250Library/set14 Review/1250s14q36.pgMore on Newton’s Method: The equation

10(x−1)(x−2)(x−3) = 1

has three real solutions

a < b < c

wherea = ,b = , andc = .

Enter your answers with at least six correct digits beyond thedecimal point.Hint: Ask what the solutions are if the right hand side is 0 in-stead of 1, and use Newton’s Method.

Correct Answers:• 1.05435072607641• 1.89896874211899• 3.0466805318046


7

1. the difference of two natural numbers is always an in ......math 1250-2 fall 2008 hsiang-ping...

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