peter alfeld. webwork assignment number 0. due 8/29/06 at ... · u of u math 1050-1 fall 2006 peter...

U of U Math 1050-1 Fall 2006

Peter Alfeld.

WeBWorK assignment number 0.

due 8/29/06 at 11:59 PM.The main purpose of this first WeBWorK set is to

review some prerequisites for this class and to helpyou familiarize yourself with WeBWorK.

Here are some hints on how to use WeBWorK ef-fectively:

• After first logging into WeBWorK changeyour password.

• Find out how to print a hard copy on the com-puter system that you are going to use. Con-tact me if you have any problems. Print a hardcopy of this assignment.

• Get to work on this set right away and answerthese questions well before the deadline. Thatway you won’t run out of time if you can’ssolve a problem right away, and it will giveyou a chance to to figure out what’s wrong ifan answer is not accepted.

• The primary purpose of the WeBWorK as-signments in this class is to give you the op-portunity to learn, for example by having in-stant feedback on your active solution of rel-evant problems. Make the best of it!

Thought : Math is intrinsically more difficult be-cause it is cumulative. If you fail to learn the his-tory of Greece, it will have minor effect on learningthe history of Rome. But if one fails to understandfractions, the ability to complete more advanced cal-culations, such as algebra or trigonometry, is fatallyimpaired. D. E. Koshland

1.(1 pt) This first question is just an exercise inentering answers into WeBWorK. It also gives youan opportunity to experiment with entering differentarithmetic and algebraic expressions into WeBWorKand seeing what WeBWorK really thinks you are do-ing (as opposed to what you believe it should think).Notice the buttons on this page and try them out be-fore moving to the next problem. Use the ”Back”

Button on your browser to get back here whenneeded.

• ”Prob. List” gets you back to the list of allproblems in this set.

• ”Next” gets you to the next question in thisset.

• ”Submit Answer” submits your answer asyou might expect, but there may be otherways to do so. Specifically, in this problem,there is only one question. In that case youcan submit your answer by typing it into theanswer window and then pressing ”Return”(or ”Enter”) on your keyboard. But even inthis case, you can also type the answer andclick on the ”submit” button. There is noharm in submitting an answer even if you arenot quite sure that it’s correct, since if it isnot you have an unlimited number of addi-tional tries. On the other hand, it is usuallymore efficient to print your own private prob-lem set, work out the answers in a quiet envi-ronment like your home, and then sit down infront of a computer and enter your answers. Ifsome are wrong you can try to fix them rightat the computer, or you may want to go backand work on them quietly elsewhere beforereturning to the computer.

• Pressing on the ”Preview Answer” Buttonmakes WeBWorK display what it thinks youentered in the answer window. After using”Preview” you can modify your answer anduse a ”Preview Again” button.

• ”typeset” denotes the ordinary display modeon your workstation, but ”formatted text” isa little faster. Very occasionally the outputfrom ”typeset” gets corrupted. In that caseyou can send me an email and I will fix it.You can also see the intended output, if notquite as nicely, using ”formated text”. Theoptions of plain ”text” and ”typeset2” are notuseful.

• ”Logout” terminates this WeBWorK sessionfor you. You can of course log back in andcontinue.

1

• ”Feedback” enables you to send a message tome. If you use this way of sending e-mailI receive information about your WeBWorKstate, in addition to your actual message.

• The ”Help” Button transports you to an offi-cial WeBWorK help page that has more infor-mation than this first problem.

• ”Problem Sets” transports you back to thepage where you can select a certain problemset. When you do this particular problem inthis first set, there is only one set, but eventu-ally there will be 13 of them.

• For all problems in this course you will beable to see the Answers to the problems af-ter the due date. Go to a problem, clickon ”show correct answers”, and then click on”submit answer”. You can also download andprint a hard copy with the answers showing.These answers are the precise strings againstwhich WeBWorK compares your answer. Ifthe answer is an algebraic expression your an-swer needs to be equivalent to the WeBWorKanswer, but it may be in a different form. Forexample if WeBWorK thinks the answer is2∗a, it is OK for you to type a+a instead. IfWeBWorK expects a numerical answer thenyou can usually enter it as an arithmetic ex-pression (like 1/7 instead of .142857), andusually WeBWorK will expect your answerto be within one tenth of one percent of whatit thinks the answer is.

• Most of the problems (including this one) inthis course will also have solutions attachedthat you can see after the due date by click-ing on ”show solutions” followed by ”sub-mit answers”. The solutions are text typedby your instructor that gives more informa-tion than the ”answers”, and in particular of-ten explains how the answers can be obtained.

Now for the meat of this problem. Notice that the an-swer window is extra large so you can try the thingssuggested above.Type the number 3 here:

.

Try entering other expressions and use the pre-view button to see what WeBWorK thinks you en-tered. Return to this problem to try out thingswhen you get stuck somewhere else.Here are some good examples to try. Check them allout using the Preview button. (In later questions youwill get to use what you learn here.) Never mind thatyou may have already answered the correct answer 3.Once you get credit for an answer it won’t be takenaway by trying other answers.a/2b versus a/2/b versus a/(2b)a/b+c versus a/(b+c)a+b**2 versus (a+b)**2sqrt a+b versus sqrt(a+b)4/3 pi r**2 versus (4/3) pi r**2 (In other words, ifyou are not sure use parentheses freely.)

Note: WeBWorK will not usually let you enter al-gebraic expressions when the answer is a number, andit will only let you use certain variables when the an-swer is in fact an algebraic expression. So the abovewindow, and the opportunity for experimentationthat it offers is unique. Make good use of it!Presumably this has been your first encounter withWeBWorK. Come back here to try things out andto refresh your memory if you get stuck somewheredown the line.

2.(1 pt) The purpose of this exercise is to illustratefurther the use of the buttons on this page and to showyou the most common way in which WeBWorK pro-cesses partially correct problems. Try entering incor-rect answers in the answer fields below, to see whathappens. (This time WeBWorK will reject algebraicexpressions since I told it to expect a numerical an-swer.)Type the number 4 here: .Type the number 5 here: .

3.(1 pt) WeBWorK will usually consider a numer-ical answer to be correct if it is within one tenth ofpercent of the answer that it has been given. How-ever, you do not usually have to enter a decimal ap-proximation. WeBWorK will do simple arithmeticfor you. The answer in this problem is 2

7 . You canenter this number in various forms, e.g., 2/7, 4/14,1/(7/2), 0.285714. Try it below. Also try to enterrough approximations like 0.3, 0.29, 0.286, 0.2857,0.28571 and see which WeBWorK will accept.

2

Occasionally, WeBWorK will insist on an answer be-ing a decimal expression. If so that expectation willbe stated in the problem.27 = .

4.(1 pt) You will find that your interaction withWeBWorK usually goes pretty smoothly. Suppose,however, you are sure that you have the right answer,but WeBWorK will not accept it. In that case youmay of course actually have the wrong answer. Inrare cases, WeBWorK may have been told the wronganswer. (If that happens send me a message and youwill receive an extra point in our One Point Contest.)A frequently occurring third possibility is that you areexpressing your answer in a way that is inconsistentwith the rule of arithmetic precedence:

• A missing operator means multiplication.• Functions are evaluated first.• Multiplication and Division come before Ad-

dition and Subtraction.• Operations of the same level of precedence

are carried out from left to right.• Expressions in parentheses are evaluated first,

and so parentheses overrule the precedingrules.

• It’s OK to have unnecessary parentheses (forexample, if you are not sure WeBWorK willunderstand what you mean).

In this and the next few problems, you will beasked to enter algebraic expressions. Keep in mindthe above rules, try alternatives, and if WeBWorKwill not accept your answer, use the preview buttonto see what WeBWorK thinks you are saying. It maydiffer from what you think you are saying.a+b

c = .5.(1 pt)

a+bc+d = .

6.(1 pt) Remember that a missing operator meansmultiplication. Thus you can enter the product of cand d as cd or c*d. Try it:cd = .Now enter

a+bcd = .

If you get stuck use the preview button!7.(1 pt) It is useful to understand how WeBWorK

determines whether you entered a correct algebraic

expressions. Algebra is driven by the fact that ex-pressions can be changed to equivalent ones. For ex-ample, the second binomial formula states

(a−b)2 = a2 −2ab+b2

for all real and complex numbers a and b. You canindicate a power with a double asterisk, but how doyou know whether to enter the expression (a-b)**2or a**2 - 2*a*b + b**2?The answer is that it does not matter. The code defin-ing the WeBWorK problem contains a particular ex-pression that’s the “correct” answer. However, WeB-WorK evaluates your expression for some randomvalues of the variables, and it evaluates its expres-sion at the same random numbers. Then it comparesthe two sets of values and if they agree it deems youranswer correct. (I once heard it said that all of math-ematics is driven by the creative confusion of neces-sary and sufficient conditions. This must be an in-stance of this drive.)Try entering both alternatives below. Also try enter-ing something that’s actually wrong, like a**2-b**2.(a−b)2 = .

8.(1 pt) Enter here the ex-pression

11a + 1

bEnter here the expression

a+b+11+ 1

a+b

9.(1 pt)Enter here the expression

ab + c

def + g

h.

10.(1 pt) To enter square roots you can use thefunction sqrt. For example, to enter the square rootof 2 you can type sqrt(2).√

2 = . (Try the exact value and a nu-merical approximation.)

Enter here the expression√

a11.(1.5 pts) Enter here the

expression √a+b

3

Enter here the expressiona√

a+bEnter here the expression

a+b√a+b

12.(1.5 pts)Enter here the expression

√

x2 + y2

Enter here the expression

x√

x2 + y2

Enter here the expressionx+ y

√

x2 + y2

13.(1 pt)Enter here the expression

−b+√

b2 −4ac2a

(This is of course the celebrated quadratic formula.)Use the preview button if you get stuck!

14.(1.5 pts)For each of the WeBWorK phrases below enter a

T (true) if the two given phrases describe the samealgebraic expression and an F (false) otherwise. Oneway you can decide whether the phrases are equiva-lent is to substitute specific values for a, b, etc. If youget two different results the two phrases are certainlynot equivalent. If you get the same values there issmall chance this happened accidentally for just thatchoice of particular values. In any case, pay closeattention to when these phrases are equivalent andwhen they are not, it will help you tremendously withfuture WeBWorK assignments.a+b b+aa+b+ c a+(b+ c)a−b− c a− (b− c)

15.(2 pts)More of the same.

a+b2 (a+b)2

a2 +b2 (a+b)2

a∗b∗ c a∗ (b∗ c)

a/b/c a/(b/c)16.(2.5 pts) The following question reviews the ter-

minology of the number system. See section A.1 fordetails.

Match the statements defined below with the let-ters labeling particular numbers. Use all the letters.Of course a natural number is also a rational num-ber, for example. However, there is only one correctmatching that uses all five letters A through E.

1. x is a rational number2. x is an irrational number3. x is a natural number4. x is neither positive nor negative5. x is an integerA. x = πB. x = −17C. x = 12D. x = 0E. x = 17

12

17.(2.5 pts) This question reviews the terminol-ogy for the four basic arithmetic operations: adding(sum), subtracting (difference), multiplying (prod-uct), and dividing (quotient). Don’t use words like“plussing” or “timesing”, they are juvenile, and youhave moved beyond them.

Match the phrases given below with the letters la-beling the algebraic expression.You must get all of the answers correct to receivecredit.

1. The sum of x and 32. The product of the sum of x and 2 and the sum

of x2 and 23. The quotient of x and the sum of x and 34. The difference of x and x2 divided by the sum

of x and x2

5. The sum of x and 2, all squaredA. x−x2

x+x2

B. xx+3

C. (x+2)2

D. x+3E. (x+2)(x2 +2)

18.(3.5 pts) Indicate whether the following state-ments are True (T) or False (F).

4

1. The product of two natural numbers is alwaysa natural number.

2. The difference of two natural numbers is al-ways a natural number.

3. The sum of two natural numbers is always anatural number.

4. The difference of two natural numbers is al-ways an integer.

5. The quotient of two natural numbers is al-ways a natural number.

6. The ratio of two natural numbers is alwayspositive

7. The quotient of two natural numbers is al-ways a rational number


1. The product of two integers is always an inte-ger.

2. The difference of two integers is always a nat-ural number.

3. The difference of two integers is always aninteger.

4. The quotient of two integers is always an in-teger (provided the denominator is non-zero).

5. The ratio of two integers is always positive6. The sum of two integers is always an integer.7. The quotient of two integers is always a ra-

tional number (provided the denominator isnon-zero).


1. The quotient of two rational numbers is al-ways a rational number (provided the denom-inator is non-zero).

2. The sum of two rational numbers is always arational number.

3. The difference of two rational numbers is al-ways a natural number.

4. The product of two rational numbers is al-ways a rational number.

5. The ratio of two rational numbers is alwayspositive

6. The quotient of two rational numbers is al-ways a real number (provided the denomina-tor is non-zero).

7. The difference of two rational numbers is al-ways a rational number.


1. The ratio of two real numbers is never zero.2. The sum of two real numbers is always a real

number.3. The product of two real numbers is always a

real number.4. The difference of two real numbers is always

a real number.5. The quotient of two real numbers is always a

rational number (provided the denominator isnon-zero).

6. The difference of two real numbers is alwaysan irrational number.

7. The quotient of two real numbers is alwaysa real number (provided the denominator isnon-zero).

22.(1 pt) Remember that we multiply powers withthe same base by adding the exponents, we dividethem by subtracting the exponents, and we take apower to a power by multiplying the exponents.

(

x2y4

x7y7

)2= xayb

where a = and b = .23.(1 pt)

(

x 14 y 2

3

x 43 y 1

8

) 13

= xayb

where a = and b = .

Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c©UR5

Peter AlfeldMath 1050-1, Fall 2006

Assignment 1 due September 5Thought: Thought: If you feel listless, make a list.

1.(1 pt) Evaluate the expression −32.

[NOTE: Your answer cannot be an algebraic expres-sion. ]

2.(1 pt) Evaluate the expression 8−4/3.

[NOTE: Your answer cannot be an algebraic expres-sion. ]

3.(1 pt) The expression 25242−2√

43242−3 equals 2n where nis:

4.(2 pts) The expression (3a3b4c2)2(2a2b4c3)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:[NOTE: Your answers cannot be algebraic expres-sions.]

5.(1 pt) The expression√

x3y5 3√

x3y4 5√x4 equalsxrys

where r, the exponent of x, is:and s, the exponent of y, is:

6.(1 pt) Find x if(3.5)x(3.5)(−5)

(3.5)2 = (3.5)2

x =7.(1.5 pts) The expression (2x + 4)(5x− 4) equals

Ax2 +Bx+Cwhere A equals:and B equals:and C equals:

8.(1.5 pts) The expression (4t−5)(5t +6)+6t −4equals At2 +Bt +Cwhere A equals:and B equals:and C equals:

9.(1.5 pts) The expression (7x + 5)2equals Ax2 +Bx+Cwhere A equals:and B equals:and C equals:

10.(1.5 pts) The expression (x − 7)(x2 + 7x +4)equals Ax3 +Bx2 +Cx+Dwhere A equals:and B equals:and C equals:and D equals:



Assignment 2 due September 11Thought: Thought is already there.

1.(1 pt) Find the slope of the line through (1, 1) and(-7, 11).

2.(1 pt) A line through (-9, -9) with a slope of 10has a y-intercept at

3.(1 pt) An equation of a line through (-1, 1) whichis parallel to the line y = 1x+1 has slope:

and y intercept at:

4.(1 pt) An equation of a line through (2, 3) whichis perpendicular to the line y = 3x+1 has slope:

and y intercept at:

5.(1 pt) The equation of the line with slope 3 thatgoes through the point (6,5) can be written in theform y = mx+b where m is:and where b is:

6.(3 pts)Match each graph to its equation.

(For all graphs on this page, if you are having a hardtime seeing the picture clearly, click on it. It will ex-pand to a larger picture on its own page so that youcan inspect it more closely.)

1.

2.

3.

4.

1

5.

6.

A. (x−1)2 = −2(y−1)B. y2 = −2xC. (y−1)2 = 2(x+1)D. (y−1)2 = −2(x−1)E. y2 = 2xF. (x−1)2 = 2(y+1)

7.(1 pt) The domain of the function f (x) = 4710x−33

is all real numbers x except for x where x equals8.(2 pts) For each of the following functions, de-

cide whether it is even, odd, or neither. Enter E for anEVEN function, O for an ODD function and N for afunction which is NEITHER even nor odd.

NOTE: You will only have four attempts to get thisproblem right!

1. f (x) = −5x2 −3x6 −22. f (x) = x2 −6x6 +3x6

3. f (x) = x−4

4. f (x) = x3 + x7 + x7

9.(1 pt)Let f (x) = 5x+2 and g(x) = 2x2 +4x.( f +g)(5) =

10.(1 pt)Let f (x) = 4x+2 and g(x) = 4x2 +4x.( f g)(3) =

11.(2 pts) Almost any kind of quantitative data can be represented by a graph and most of these graphsrepresent functions. This is why functions and graphs are the objects analyzed by calculus. The next twoproblems illustrate data which can be represented by a graph. Match the following descriptions with theirgraphs below:

1. The graph of the distance traveled by a car as it drives along a city street vs. time.2. The graph of the velocity of a car entering a superhighway vs. time.3. The graph of the distance traveled by a car as it enters a superhighway vs. time.4. The graph of the velocity of a car as it drives along a city street vs. time.

A B C D

2

12.(2.5 pts)The following questions concern the profits of firmN. The graph of the profits vs. time is given above.For each of the intervals enter the letters correspond-ing to the descriptions which describe the behavior ofthe graph on that interval. (The letters in each answermust be in alphabetical order with no spaces betweenthe letters.)

1. The interval from a to b2. The interval from b to c3. The interval from c to d4. The interval from d to e5. The interval from e to fA. The firm makes a profit on this interval.B. The firm registers a loss on this interval.C. The profit of the firm increases on this inter-

val.D. The profit of the firm decreases on this inter-

val.E. Assuming the profits are reinvested in the

firm the networth of the company is increas-ing on this interval.

F. Assuming the profits are reinvested in thefirm the networth of the company is decreas-ing on this interval.

13.(1 pt) Let f (x) = 15x , g(x) = 9x2 +6, and h(x) =

7x3.Then f ◦g◦h(2) =

14.(2 pts) Relative to the graph of

y = x3

the graphs of the following equations have beenchanged in what way?

1. y = (x+11)3

2. y = (x3)/103. y = x3 −114. y = (x)3/1000A. compressed vertically by the factor 10B. shifted 11 units leftC. stretched horizontally by the factor 10D. shifted 11 units down

15.(2 pts) Let g be the function below.For all graphs on this page, if you are having a hard

time seeing the picture clearly, click on it. It will ex-pand to a larger picture on its own page so that youcan inspect it more closely.

The domain of g(x) is of the form [a,b], where a isand b is .

The range of g(x) is of the form [c,d], where c isand d is .

Enter the letter of the graph which corresponds toeach new function defined below:

1. g(x−2)+2 is .2. g(2x) is .3. 2+g(−x) is .4. g(x+2)−2 is .

A B C D

E F G H

16.(2 pts) Enter a T or an F in each answer spacebelow to indicate whether the corresponding equationis true or false. An equation is true ony if it is true forall values of the variables. Disregard values that makedenominators 0.

3

You must get all of the answers correct to receivecredit.

1. x+41y+41 = x

y2. 41+a

41 = 1+ a41

3. 4148+x = 41

48 + 41x

4. 4848−c = 1− 48

c

17.(1.5 pts) The equation 2x4 − 8x3 − 3x2 = 0 hasthree real solutions A, B, and C where A < B < Cand A is:

and B is:and C is:


Peter Alfeld

Math 1050-1, Fall 2006

Assignment 3 due September 25Thought: Math is FUNctional!

1.(1 pt) Let f (x) = 12x , g(x) = 4x2 + 6, and h(x) =

3x3.Then f ◦g◦h(4) =

2.(2 pts) Relative to the graph of

y = x3

the graphs of the following equations have beenchanged in what way?

1. y = (x−11)3

2. y = x3 +113. y = (x)3/80004. y = x3 −11A. shifted 11 units rightB. stretched horizontally by the factor 20C. shifted 11 units upD. shifted 11 units down

3.(2 pts) Let g be the function below.For all graphs on this page, if you are having a hard

time seeing the picture clearly, click on it. It will ex-pand to a larger picture on its own page so that youcan inspect it more closely.

The domain of g(x) is of the form [a,b], where a isand b is .

The range of g(x) is of the form [c,d], where c isand d is .

Enter the letter of the graph which corresponds toeach new function defined below:

1. g(x−2)+2 is .2. g(2x) is .3. 2+g(−x) is .4. g(x+2)−2 is .

A B C D

E F G H

4.(1 pt) Let

f (x) =x+2

x+10f−1(−4) =

5.(2 pts) Find the inverse for each of the followingfunctions.

f (x) = 1x+1f−1(x) =

g(x) = 10x3 −10g−1(x) =

h(x) = 1x+10

h−1(x) =j(x) = 3√x+10

j−1(x) =

6.(2 pts) This problem gives you some practiceidentifying how more complicated functions can bebuilt from simpler functions.

Let f (x) = x3 + 1and let g(x) = x + 1. Match thefunctions defined below with the letters labeling theirequivalent expressions.

1. (g(x))2

2. g(x2)3. f (x2)4. f (x)/g(x)A. 1− x+ x2

B. 1+2x+ x2

C. 1+ x2

D. 1+ x6

7.(2 pts)Let f (x) = 2x+5 and g(x) = 3x2 +4x.After simplifying,

( f ◦g)(x) =

1

8.(1 pt) Let f (x) = 3x + 4 and g(x) = 5x2 + 4x.Match the statements defined below with the letterslabeling their equivalent expressions.You must get all of the answers correct to receivecredit.

1. f ◦g2. g◦g3. g◦ f4. f ◦ f

A. 45x2 +132x+96B. 15x2 +12x+4C. 9x+16D. 125x4 +200x3 +100x2 +16x

9.(1 pt) Let

f (x) =x+3x+8

f−1(−1) =

10.(1 pt) Below is the graph of a function f : (Clickon image for a larger view )

Graph A

Graph B

Graph CThe inverse of the function f is (A, B or C):

2

11.(1 pt) This problem tests calculating new functions from old ones:From the table below calculate the quantities asked for:

x −12 0 −2 2 −1 −7f (x) −277 −1 −7 −11 −2 −92g(x) −1882 2 −12 8 −1 −397

f (g(−2))f (−2)/(g(−2)+5)( f g)(−2)

Tip: Sometimes webwork will do arithmetic for you. For example you can type in 4*11 instead of 44 andwebwork will do the calculation for you. This works with many numerical problems, although not all of them.

12.(1 pt) This is a graph of the function F(x):(Click on image for a larger view )

Enter the letter of the graph below which corre-sponds to the transformation of the function.

1. F(x+3)2. 5F(x)3. F(x−3)4. −F(−x)

A B C D

13.(1 pt) Below is the graph of a function f : (Clickon image for a larger view )

Graph A

3

Graph B

Graph C

Graph DThe inverse of the function f is (A, B, C or D):


Peter Alfeld MATH 1050-1 Fall 2006Homework Set 4 due 10/4/06 at 11:59 PM

Thought: ”Do or do not, there is no try”. -Yoda

1.(1.5 pts) Find the standard form of the parabolaequation,

y = a(x−h)2 + kwith vertex (-7,1) and passing through the point (-

4,9).Then h is

and k is

and a is:2.(2.5 pts) Find the x and y intercepts and the coor-

dinates of the vertex of the parabola

f (x) =18(x2 −5x−5).

(1) Then the smaller one of the x intercept is

and the larger one of the x intercept is

(2) y intercept is

(3) x coordinate of the vertex (h,k):h =

and y coordinate of the vertex (h,k):k =

3.(4 pts) (a) Find two quadratic functions, one thatopens upward

f (x) = x2 +b1x+ c1.

and one that opens downward,

g(x) = −x2 +b2x+ c2.

whose graphs have x-intercepts (20,0), (11,0).b1 =

c1 =b2 =c2 =(b) Find the vertex of both parabolas in part (a).

The vertex (h1,k1) of the open upward quadraticfunction ish1 =

k1 =and the vertex (h2,k2) of the open downward qua-

dratic function ish2 =

k2 =4.(1 pt) Find the turning point of the graph of

f (x) = x2 −26xThe turning point (h,k) has the valuesh =

and k =5.(1.5 pts) Use the long division to find A,B,C so

that

x3 −14x2 +48x−15 = (Ax2 +Bx+C)(x−5)

A =B =C =6.(3.5 pts)Let

p(x) = 2x3 +3x2 +4x+5.

As discussed in class, p(2) can be computed via syn-thetic division by filling in for the letters a through gin the table

2 3 4 5b d f

a c e gwhere

• = a,• = b,• = c,• = d,• = e,• = f ,• = g.

Moreover, p(2) = and p(x) can be written asp(x) = (x−2)q(x)+ r

whereq(x) = is a quadratic polynomial andr = is a real number.

1

7.(2.5 pts) Find the x and y intercepts and the coor-dinates of the vertex of the parabola

f (x) =1012

(x2 +10x−5).

(1) Then the smaller one of the x intercept is

and the larger one of the x intercept is

(2) y intercept is

(3) x coordinate of the vertex (h,k):h =

and y coordinate of the vertex (h,k):k =

8.(2.5 pts)

The Figure above shows the graphs of five functions,distinguished by color. For every function, enter be-low the letter p if the graph may be that of a poly-nomial, and the letter n if it cannot be the graph of apolynomial. (Of course we assume that figure showsall the relevant aspects of the graph.)

• : blue• : green• : purple (magenta)• : red• : yellow

9.(1 pt)The vertex of the parabola defined by

f (x) = −5x2 +4x+1is the point( , ).


Peter Alfeld MATH 1050-1 Fall 2006Homework Set 5 due 10/11/06 at 11:59 PM

Thought: Time flies like an arrow, fruit flies like a banana.

1.(1.5 pts) Find all the zeros of the function

f (x) = x5 +17x3 −38x,where the three zeros,A,B,C, are real values withA < B < C.A =

B =C =2.(1.5 pts) Find a,b,c ∈ R so that the function

f (x) = x3 +ax2 +bx+ chas zeros at 0,169,−52.a =

b =c =3.(3 pts) (1) Find real numbers a and b such that

18+(−158+28i)−21i = a+biThen a is

and b is

(2) Find real numbers a and b such that

5i(2+3i)2 = a+bi

Then a is

and b is

(3) Find real numbers a and b such that

1+ ii − 3

4− i = a+bi

Then a is

and b is

4.(2 pts) Find the complex roots of the equation

22x2 +3x+5 = 0.

(1) The first root of the equation can be written asa+bi for positive real number b , where

a=b=(2) And the second root of the equation can be

written as a+bi for negative real number b , wherea=b=5.(1 pt) Simplify the following

(√−88)3

Answer = i6.(2 pts) Find all the real zeros of the function

f (x) = x4 +1x3 −13x2 −1x+12.

There are four real zeros x1 < x2 < x3 < x4x1 =x2 =x3 =x4 =7.(3.5 pts) Let

f (x) = 4x3 −12x2 − x+15. Find 0 < a < b < c < d and 0 < l1 < l2 < l3 so that

± al1

,± al2

,± al3

,

± bl1

,± bl2

,± bl3

,

± cl1

,± cl2

,± cl3

,

± dl1

,± dl2

,± dl3

,

are all the possible rational zeros of f (x)a =

b =c =d =l1 =l2 =l3 =8.(1.5 pts) (1) Find the horizontal asymptote of

1

h(x) =8

x2(x−6)y =

(2) Find the vertical asymptotes of

h(x) =8

x2(x−6)x = x1,x = x2 with x1 < x2.x1 =

x2 =9.(1 pt) Find the equation of the slant asymptote

for

g(x) =x3 +12x2 +22x+220

x2 +19.The slant asymptote can be written in the form

y = Ax+Bfor constants A,B.

A =B =10.(2 pts) A rectangular page is designed to con-

tain 64 square inches of print. The margins at the topand bottom of the page are each 1 inch deep. Themargins on each side are 1.5 inch wide. What shouldthe dimensions of the page be so the least amount ofpaper is used ? Use a graphing calculator to estimate.The smaller length side of the pagex=

The larger length side of the pagey=

11.(1.5 pts) Find all the zeros of the functionf (x) = x5 +9x3 −162x

,where the three zeros,A,B,C, are real values withA < B < C.A =

B =C =12.(8 pts) Let

u = 2−3i and v = −3+ i.Write the following complex numbers in the standardform of a complex number.

• u+ v = + i.• u− v = + i.

• uv = + i.• u

v = + i.• v

u = + i.• v2 = + i.• v3 = + i.• v4 = + i.

13.(1 pt)Two zeros of the polynomial

p(x) = x4 −4x3 −24x2 +20x+7

are 1 and 7. The other two zeros are real, but irra-tional.The smaller is , and the larger is

.

14.(1 pt)The equation

x4 −9x3 +31x2 −49x+30 = 0

has the conjugate complex pair of solutions x = 2± i.The equation also has two real solutions. The smalleris , and the larger is .

15.(1.5 pts)Match the graphs shown above with the functionslisted below. Enter ”r” for red, ”g” for green, and”y” for yellow.

• : f (x) = xx−1 .

• : f (x) = x2

5(x+1) .

• : f (x) = xx2+1 .

2

16.(1.5 pts) Find the followings for

f (x) = − 1(x−9)2

(a) y intercept, y =(b) Vertical Asymptote , x =(c) Horizontal Asymptote y =

17.(2 pts) Find the followings for

g(x) =x3 +1

7x2 −343(a) x intercept, x =

(b) y intercept, y =(c) Vertical Asymptote x = ±(d) Slant Asymptote y =



Assignment 6 due October 17.Thought: The early bird may get the worm, but thesecond mouse gets the cheese.

1.(3.5 pts) Consider the inequality2x3 +7x2 −2x−1 ≤ 6

The solution of this inequality consists one or moreof the following intervals: (−∞,A), (A,B), (B,C),and(C,∞) where A < B < C.

Find AFind BFind CFor each interval, answer YES or NO to whether

the interval is included in the solution.(−∞,A)(A,B)(B,C)(C,∞)

2.(3.5 pts) Consider the inequality5

(x−7)≤ 3

(x+2)

The solution of this inequality consists one or moreof the following intervals: (−∞,A), (A,B), (B,C),and(C,∞) where A < B < C.

Find AFind BFind CFor each interval, answer YES or NO to whether

the interval is included in the solution.(−∞,A)(A,B)(B,C)(C,∞)

3.(3 pts) Simplify each expression and enter ”dne”if the result is not real. Practice these problems with-out using a calculator.(a) 43/2 =

(b) 64−4/3 =(c) −8−2/3 =(d) −644/3 =(e) 93/2 =(f) (−4)3/2 =

4.(3 pts) Solve for x without a calculator:(a) 3x = 1

27 x =(b) 8x = 1

2 x =(c) 23x = 64 x =(d) 64x+2 = 4 x =(e) 2x = 1

256 x =(f) 4x = 27 x =

5.(6 pts) Practice these law of exponent problemswithout a calculator. Each answer should be of theform aeb. For each one determine the value of a andb.(a) exex−1

a =b =

(b) ex+2/ex

a =b =

(c) (2ex)3

a =b =

(d) 3√−8e6x

a =b =

(e) (e2xe3x)/e5

a =b =

(f) (16e6x)−1/2

a =b =

6.(2 pts)1

The Figure above shows the graphs of four exponen-tial functions, listed below. Match the functions withthe colors, using b for blue, r for red, g for green, andy for yellow.

• : f (x) = 2x.• : f (x) = 3x.• : f (x) =

(12)x

.

• : f (x) =(1

3)x

.

7.(2 pts)The lessons we learned about shifting graphs apply toexponential functions just as they apply to any otherfunctions.

The Figure above shows the graphs of five exponen-tial functions, listed below. Match the functions withthe colors, using b for blue, r for red, g for green, pfor purple, and y for yellow.

• : f (x) = 2x.• : f (x) = 2x +1.• : f (x) = 2x+1.• : f (x) = 2x −1.• : f (x) = 2x−1.

8.(2.5 pts) You invest 10,000 dollars at 4.5% inter-est. What is in your account after 20 years...(a) if compounded annually?

(b) if compounded monthly?

(c) if compounded daily?

(d) if compounded continuously?

(e) Use your calculator to estimate during whichyear it will double if the compounding is continuous?

9.(2 pts) The population of Mathtown grows ac-cording to this equation:

P(t) = 2500e0.025t ,

where t is measured in years, with t = 0 correspond-ing to the year 2000.(a) How many were in the town in 1990?

(b) How many are in the town in 2005?

(c) How many are in the town in 2020?

(d) In what year will the population double?


Peter Alfeld

Math 1050-1, Fall 2006

Assignment 7 due October 26.Thought: You may be on the right track, but if youdon’t keep moving you will be run over by the train.More practice on logs. Still no calculator.

1.(3 pts) Logs. Determine the value of x without acalculator.(a) log3 x = 4x =

(b) log5 125 = xx =

(c) logx 9 = 2x =

(d) logx 27 = −3x =

(e) log21

16 = xx =

(f) logx 1000 = 3x =

2.(4 pts) Determine the value of each without a cal-culator. If it is not defined, write ND.(a) log2 1 =

(b) log4 4 =(c) log2 0 =(d) log2 64 =(e) log3 1/

√3 =

(f) lne−6 =(g) log1000 =(h) lne9x =(i) lne13x =(j) log0.001 =(k) log3 1 =(l) log5−5 =

3.(2.5 pts)The lessons we learned about shifting graphs apply tologarithmic functions just as they apply to any otherfunctions.

The Figure above shows the graphs of five logarith-mic functions, listed below. Match the functions withthe colors, using b for blue, r for red, g for green, pfor purple, and y for yellow.

• : f (x) = ln(x).• : f (x) = ln(x−1).• : f (x) = ln(x+1).• : f (x) = ln(x)−1.• : f (x) = ln(x)+1.

4.(1 pt) The following few problems ask you tosolve exponential equations of increasing complex-ity. Your answer needs to be a decimal number withat least four digits. The first problem is easy:

The solution of the equation5x = 100

is x = .5.(1 pt) The solution of the equation

3×43t+1 = 5is t = .

6.(1 pt) The equationln(x+1)− ln(x) = 2

has the solution x = .7.(1 pt) The equation

ln(x+1)− ln(x−2) = lnxhas the solution x = .

8.(1 pt) The next few questions ask you to com-pute the inverse of an exponential or a logarithmic

1

function. Remember how we do this: we solve theequation y = f (x) for x and then switch x and y.

The purpose of this first problem is just to makesure you understand the WeBWorK notation. You canenter the exponential with the natural base as exp(x)or eˆx and the natural logarithm as ln(x) or log(x).WeBWorK does not have a base 10 logarithm builtinto it, if you need a logarithm with a base other thane, let’s say a, then enter something like log(x)/log(a).

The inverse of the function

f (x) = ex

isf−1(x) =


f (x) = lnx

isf−1(x) =

9.(1 pt) The inverse of the function

f (x) = 10x

isf−1(x) =


f (x) = log10 x

isf−1(x) =

10.(1 pt) The inverse of the function

f (x) = ln(x−1)

isf−1(x) =The inverse of the function

f (x) = (lnx)−1

isf−1(x) =

11.(3.5 pts) The remaining problems are true/falsequestions concerning logarithmic and exponentialidentities. You don’t need to memorize these, theyall flow from two facts:

• Logarithms and Exponentials are inversesof each other. (Of course they need to havethe same base.)

• Exponential functions are just powers andlogarithms are just exponents.

For the following proposed identities enter T ifthey are true, and F if they are false. We assume thatthe expressions involved make sense. For exampleany base is positive and not equal to 1, and logarithmsare taken only of positive numbers.

• loga(uv) = loga u+ loga v.• loga(u+ v) = (loga u)(loga v).• loga

(uv)

= loga u− loga v.• loga(u− v) =

loga uloga v .

• loga (uv) = (loga u)(loga v).• loga (uv) = u(loga v).• loga (uv) = v(loga u).

12.(2 pts) For the following proposed identities en-ter T if they are true, and F if they are false.

• eln(x−1) = x−1.• e(lnx)−1 = x

e .• ln((x−1)(x−2)) = ln(x−1)+ ln(x−2).• lnx2 = 2lnx.

13.(2.5 pts) For the following proposed identitiesenter T if they are true, and F if they are false.

• ln10x = x ln10• log5(x) = lnx

ln5 .• ln10x = x

log10 e .

• lnx2 = (lnx)2.

• lnxln5 =

log10 xlog10 5 = log5(x).

14.(3.5 pts) For the following proposed identitiesenter T if they are true, and F if they are false.

• lnex = x• ln

(

(ex)2)

= x2.

• ln(

(ex)2)

= 2x.

• ln(

1ex

)

= −x.

• ln(eu + ev) = u+ v.• ln(eu + ev) = uv.• ln(eu + ev) = eu+v.



Assignment 8 due November 07.Quote: Few people think more than two or threetimes a year. I have made an international reputationfor myself by thinking once or twice a week.— George Bernard Shaw

1.(1 pt)A group of eight people goes to the movies. Tick-

ets are $6.- each for adults and $3 each for kids. To-gether they pay $33.- for the tickets. There areadults in that group

and kids.2.(1 pt)You buy a total of 4 pounds of two kinds of candy.

One is for your upcoming party and costs $2.- perpound. However, you also treat yourself to a fancykind that costs $10 per pound, to be enjoyed in soli-tude. You spend a total of $12.- and buy poundsof the gourmet candy.

3.(1 pt)The solution of the linear system

x +y = 73x −y = 1

isx = and y = .4.(1 pt)The solution of the linear system

x +y = 13x −y = 2

isx = and y = .5.(1.5 pts)The solution of the linear system

x +y +z = 2x +2y −z = −1x −3y +2z = 7

isx = , y = , and z = .

Note: Keep a careful record of your calculationssince the next problem differs from this one only in

the right hand side, and you don’t want to redo yourcalculations.

6.(1.5 pts) This problem differs from the preced-ing one only in the right hand side. You have alreadydone many of the necessary calculations and don’tneed to redo them if you have good notes for yoursolution of the previous problem.

The solution of the linear system

x +y +z = 1x +2y −z = 2x −3y +2z = 3

isx = , y = , and z = .7.(1.5 pts) For the following linear systems enter

the letter U if the system has a unique solution, theletter N if it has no solution, and the letter I if it hasinfinitely many solutions.

• .3x + 4y = 13x − 4y = 1

• .3x + 4y = 16x + 8y = 1

• .3x + 4y = 16x + 8y = 2

8.(1.5 pts) For the following linear systems enterthe letter U if the system has a unique solution, theletter N if it has no solution, and the letter I if it hasinfinitely many solutions.

• .3x + 4y + 5z = 13x − 4y + 5z = 1

x + y + z = 1• .

3x + 4y + 5z = 1x + 2y + 3z = 1

4x + 6y + 8z = 2• .

3x + 4y + 5z = 1x + 2y + 3z = 1

4x + 6y + 8z = 41

9.(1.5 pts) Find the equation of the parabola

y = ax2 +bx+ cthat passes through the following given points:

(−5,−149), (−2,−29), (1,1)

Answer:a =

b =c =

10.(3 pts) Find the minimum and maximim valuesof the following objective function and where theyoccur, subject to the indicated constraints.

Objective function:z = 2x+8y

Constraints:x ≥ 0

y ≥ 0

2x+ y ≤ 4Minimum: ( , )

Minimum value:Maximum: ( , )

Maximum value:11.(3 pts) Find the minimum and maximim values

of the following objective function and they occur,subject to the indicated constraints.

Objective function:z = 4x+5y

Constraints:x ≥ 0

2x+3y ≥ 6

3x− y ≤ 9

x+4y ≤ 16Minimum: ( , )

Minimum value:Maximum: ( , )

Maximum value:12.(2 pts) (a) Find the numbers A,B such that

116x2 −36 =

A4x+6 +

B4x−6

A =B =(b) Find the numbers A,B such that

x+1x2 +6x+5 =

Ax+5 +

Bx+1

A =B =13.(2 pts) (a) Find the numbers A,B such that

19x2 −64

=A

3x+8+

B3x−8

A =B =(b) Find the numbers A,B such that

x+3x2 +6x+9 =

Ax+3 +

Bx+3

A =B =14.(1.5 pts) Find the numbers A,B,C such that

x(x−17)(x2 + x+1)

=A

x−17+

Bx+Cx2 + x+1

A =B =C =15.(2 pts) Solve the system using substitution

{

x−6y=07x−y2=0

There are two solutions (x1,y1),(x2,y2) for x1 <x2.

x1 =y1 =

x2 =y2 =

16.(2 pts) Solve the system using substitution{

x2 +y2=252x+10y=10

There are two solutions (x1,y1),(x2,y2) for y1 <y2.

x1 =2

y1 =x2 =

y2 =

17.(2 pts) Solve the system using substitution{

y= x+3y=

√x+3

There are two solutions (x1,y1),(x2,y2) for y1 < y2.x1 =y1 =

x2 =y2 =



Assignment 9 due November 20.Thought: People who never make a mistake end upnever doing anything worthwhile.

1.(1 pt)A group of eight people goes to the movies. Tick-

ets are $6.- each for adults and $3 each for kids. To-gether they pay $33.- for the tickets. There areadults in that group

and kids.2.(1 pt)You buy a total of 4 pounds of two kinds of candy.

One is for your upcoming party and costs $2.- perpound. However, you also treat yourself to a fancykind that costs $10 per pound, to be enjoyed in soli-tude. You spend a total of $12.- and buy poundsof the gourmet candy.

3.(3 pts) You’ll need to use formatted text modein order to do these problems: (Click the ”formattedtext” radio button at the bottom of the page and thenclick ”submit answer”.)

If

A =

4 0 3−1 2 42 3 −2

and B =

1 −2 42 −1 4−3 −4 0

Then 3A−B =

4.(2 pts) You’ll need to use formatted text mode inorder to do this problem: click the ”formatted text”button at the bottom of the page and then click ”sub-mit answer”.

If

A =

[

−2 −6−1 −3

]

and B =

[

6 910 −1

]

then AB =( )

5.(1.5 pts) You’ll need to use formatted text modein order to do this problem: click the ”formatted text”button at the bottom of the page and then click ”sub-mit answer”.

If

A =

−10 −9−4 50 −8

and B =

[

8−10

]

then AB =

6.(3 pts) You’ll need to use formatted text mode inorder to do this problem: click the ”formatted text”button at the bottom of the page and then click ”sub-mit answer”.

If

A =

5 −5 1−1 2 −1−1 1 0

then A−1 =

7.(1 pt)Compute the determinant of the following matrix:

A =

[

−5 −72 −10

]

detA = .8.(1 pt)Compute the determinant of the following matrix:

A =

−3 −2 4−1 −2 −4−5 −1 0

detA = .9.(1 pt)Solve the following equation for x:

∣

∣

∣

∣

x−2 −1−3 x

∣

∣

∣

∣

= 0

x = or10.(6 pts) If

A =

0 4 −1−3 2 3−2 2 −4

B =

0 1 40 −3 0−2 1 −4

Then AB =1

and BA =

11.(2 pts) You’ll need to use typeset mode in orderto do this problem: click the ”typeset” button at thebottom of the page and then click ”submit answer”.

IfA =

[

−3 4−8 −9

]

then A−1 =

( )



Assignment 10 due November 30.Thought: Don’t be content to be average; average isas close to the bottom as it is to the top.

1.(1.5 pts)Consider the arithmetic sequence:

a1 = 6, ak+1 = ak +5(a) Write the first five terms of the above arithmetic

sequence:, , , , .

(b) Find the common difference:(c) Write the nth term of the sequence as function

of n:2.(2 pts)Write the first five terms of the following geomet-

ric sequence:

a1 = 6, r = −14

Answer: , , , , .3.(1 pt)Find the nth term of the following geometric se-

quence:

a1 = 5, r =32, n = 8

Answer:4.(1 pt)Find the sum of the following finite geometric se-

ries:

9∑n=1

(−2)n−1

Answer:

5.(1 pt)Find the sum of the following infinite geometric

series:∞

∑n=0

2(

−23

)n

Answer:6.(3 pts)Expand the following binomial:

(x+2y)5

Answer: x5 + x4y + x3y2 + x2y3 +xy4 + y5.

7.(1 pt)Consider the arithmetic sequence:

a1 = 15, d = 4Write the nth term of the sequence as function of

n:8.(1 pt)Find the indicated nth partial sum of the following

arithmetic sequence:

40,37,34,31, . . . n = 10Answer:9.(1 pt)Find the following partial sum

400∑n=1

(2n−1)

Answer:10.(1 pt)Find the coefficient a of the indicated term in the

expansion of the following binomial:

(2x−3y)8, ax6y2

Answer: a =



Assignment 11 due December 12.Thought: We have not succeeded in solving all ofour problems. The answers we have found only serveto raise a whole set of new questions. In some wayswe feel we are as confused as ever, but we believe weare confused on a higher level and about more impor-tant things.

These problems are for review.1.(1.5 pts) Equations of straight lines and graphs

of linear equations. Know how to obtain an equationof a straight line given two pieces of information, forexample, two points, or a point and slope. Interceptslead to special cases of points.

Let L be the line through the points (1,−2) and(−2,−3). The slope of L is . Using x and y asthe variables as usually, its equation in slope interceptform is y = .

The x-intercept of L is x = .2.(1 pt) Quadratic Equations. Know how to solve

quadratic equations, by completing the square, or ap-plying the quadratic formula.

The equation2x2 +3x−3 = 0

has two real solutions. The smaller is , and thelarger is .

3.(3 pts) Quadratic Equations. Quadratic Equa-tions don’t always look like such. To obtain one youmay have to carry out some sort of substitution ormanipulate expressions suitably.

The equation2x−5

√x +2 = 0

has two real solutions. The smaller is , and thelarger is .The equation

2x4 −5x2 +2 = 0has two positive real solutions. The smaller is ,and the larger is .The equation

1x−1

− xx+1

−2 = 0

has two real solutions. The smaller is , and thelarger is .

4.(2.5 pts) Functions. Understand the concepts ofdomain, range, and inverse of a function, and knowhow to evaluate and compose functions.

Letf (x) =

x+5x+2 .

Then the domain of f is the set of all real num-bers except , and its range is the set of all realnumbers except .Moreover,f (2) = ,f (t +1) = , andf ( f (t2)) = .This is hard to do in WeBWorK, but you should alsobe able to draw the graph of f , showing clearly allintercepts and asymptotes.

5.(1.5 pts) Linear Systems. Know how to solve alinear system.

The solution of the linear system

2x+3y+3z = 54x−3y+ z = 8

−2x+6y− z = −6

isx = ,y = , andz = , and

6.(1 pt) Rules of Exponents. Understand how tocombine powers.

(6y2)2(2y3)−1 = ayb

wherea = and

b = .7.(1 pt) Simplifying Rational Expressions. Un-

derstand how to manipulate rational expressions.They work just like fractions!

x−5x2 −25

− 3x+5

=AB

where A and B are polynomials of degree as low aspossible and the leading coefficient of B is 1.A = andB = .

1

8.(1 pt) Graphing. Understand the interplay be-tween algebra and geometry when graphs are shiftedhorizontally and vertically.For example, the graph of

f (x) = ex−2 −3is the graph of the exponential y = ex shiftedunits horizontally and units vertically. (Distin-guish left and right and up and down with the appro-priate signs of the shift. You should also draw thegraphs of the original and the shifted exponential inone coordinate system.)

9.(2 pts) More Graphing. Know how to graph ra-tional functions and to compute asymptotes and in-tercepts. For example the graph of

f (x) =2x−1x−1

has a horizontal asymptote y = and a verticalasymptote x = .Its y intercept is y = and its x intercept is x =

. You should also draw the graph.10.(1.5 pts) Logarithm Rules. Understand the

rules for logarithms. You should be able to writelogarithms as sums, differences or multiples of log-arithms when appropriate, or expressions int terms ofseveral logarithms in terms of single logarithms. Forexample,log2 32(a + 1)−4 = A + B log2(C) where A =andB = are numbers andC = is an expression in terms of a.

11.(1 pt) More Logarithm Rules. Similarly

ln(x+1)− ln(x2) = ln(C)

whereC = is an expression in terms of x.

12.(1 pt) Logarithmic Equations. Understandhow to solve equations involving logarithms. For ex-ample, the solution of the equation

ln12− ln(x−1) = ln(x−2)

isx =

13.(1 pt) Exponential Equations. Understandhow to solve equations involving exponentials. Forexample, the largest solution of the equation

2x2−5x+9 = 8isx =

14.(2 pts) Matrices. Know how to add, subtractand multiply matrices. For example, if

A =

[

1 23 4

]

and B =

[

2 −11 2

]

,

thenAB−B2 =

[

a bc d

]

wherea = ,b = ,c = ,d = ,

15.(3 pts) Matrices Inverses. Know how to com-pute matrix inverses and use them to solve linear sys-tems. For example, if

A =

[

1 2−1 2

]

thenA−1 =

[

a bc d

]

wherea = ,b = ,c = ,d = ,Use it to solve the linear system

x+ y = 1−x+2y = 2

You obtainx = ,y = ,

16.(1 pt) Exponentials. Go over past word prob-lems, particularly those involving exponentials andlogarithms.Suppose a population doubles every 20 years. Theneach year it increases by percent.

17.(3 pts) The Binomial Theorem. Letp(x) = (2x−1)5 = ax5 +bx4 + cx3 +dx2 + ex+ f .

Thena = ,b = ,

2

c = ,d = ,

e = , andf = .


peter alfeld. webwork assignment number 0. due 8/29/06 at ... · u of u math 1050-1 fall 2006 peter...

Documents