hsiang-ping huang math 1220-90, spring 2008 5. webwork … · 2008-08-20 · hsiang-ping huang math...

Hsiang-Ping HuangMath 1220-90, Spring 2008WeBWorK Assignment 1 due 01/29/2008 at 10:59pmMSTLogs and Exponentials.This assignment will cover the material from Sections 6.1 – 6.5.

1. (1 pt) set1/p1-1.pg

Evaluate the following expressions.(a) log2

(18

)=

(b) log51 =(c) log5

√125=

(d) 3log3 15 =Correct Answers:

• -3• 0• 1.5• 15

2. (1 pt) set1/p1-2.pg

ln(r5s4 2√

r9s7)is equal to

Aln r +Blns

whereA = and whereB =Correct Answers:

• 9.5• 7.5

3. (1 pt) set1/p1-3.pgR 41

ln(12x)x dx=

Correct Answers:

• 4.40571810434682652

4. (1 pt) set1/p1-4.pg

If f (x) = 5√

xln(x), find f ′(x).

Find f ′(5).

Correct Answers:

• .5*5*xˆ(-.5)*ln(x)+ 5*xˆ(-.5)• 4.03547426638379092

5. (1 pt) set1/p1-5.pg

Evaluate the integrals

f (x) =Z

e6x+2

e10x dx

f (x) = +C

g(x) =Z

ex+2

ex dx

g(x) = +CCorrect Answers:

• exp(6*x+2-10*x)/(6-10)• exp(2)*x

6. (1 pt) set1/p1-7.pg

Find the integral Z 1

0

t +12t2 +4t +3

dt.

Answer:Correct Answers:

• 0.274653072167027423

7. (1 pt) set1/p1-8.pg

Supposey = e1/x2+1/ex2

. FindDxy.Dxy =Correct Answers:

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

8. (1 pt) set1/p1-9.pg

A curve is given by the equation:

y3 +68= (ex +1)2.

Find the slope of the tangent line at the point(0,−4).

Correct Answers:

• 0.0833333333333333333

9. (1 pt) set1/p1-10.pg

The region bounded byy = e−x2, y = 0, x = 0, andx = 1 is re-

volved about they-axis. Find the volume of the resulting solid.Answer:Correct Answers:

• 1.98586530405817165

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

1

Hsiang-Ping HuangMath 1220-90, Spring 2008WeBWorK Assignment 2 due 02/05/2008 at 10:59pmMSTExponential Growth and Decay, Inverse Functions,Circular (Trigonometric) and Hyperbolic functionsand their inverses.This assignment will cover the material from Sections 6.6 – 6.9.

1. (1 pt) set2/p2-1.pg

In 460 days unknown radioactive substance decay to 74 percentof its size.

(a) What is the half life of this substance?t = (days)(b) How long will it take for a sample of 100mg to decay to

86 mg?T =Correct Answers:

• 1058.92497569406985• 230.413005095519066

2. (1 pt) set2/p2-3.pg

A bacteria culture starts with 420 bacteria and grows at a ratepropotional to its size. After 6 hours there are 2520 bacteria.

(a) Find the population after t hoursy(t) = (function of t)(b) Find the population after 9 hours.y(9) =(c) When will the population reach 2650 ?T =Correct Answers:

• 420*(2.71828182845904524ˆ(0.298626578204675833*t))• 6172.71415181360881• 6.16844025999696267

3. (1 pt) set2/p2-4.pg

The loudness of sound is measured in decibels in honor ofAlexander Graham Bell (1847-1922), inventor of the telephone.If the variation in pressure isP pounds per square inch, then theloudnessL in decibels is

L = 20log10(121.3P).Find the variation in pressure caused by a rock band at 115

decibels.Answer: pounds per square inch.Correct Answers:

• 4635.95486554286134

4. (1 pt) set2/p2-5.pg

The count in a bacteria culture was 800 after 10 minutes and1600 after 30 minutes. What was the initial size of the cul-ture? Find the doubling period in minutes. Find thepopulation after 75 minutes. When (in minutes) will thepopulation reach 11000?

Correct Answers:

• 565.68542494923802• 20

• 7610.92553601741483• 85.6271942704931921

5. (1 pt) set2/p2-5a.pg

The Hustler Bank Mutual Fund pays interest at a rate of 7.4%,compounded continuously. How much should be invested so asto have 12 thousand dollars in 6 years?

Correct Answers:

• 7697.58504992783836

6. (1 pt) set2/p2-6.pg

The rat population in a major metropolitan city is given bythe formulan(t) = 42e0.035t wheret is measured in years since1991 andn(t) is measured in millions.

What was the rat population in 1991 ?What is the rat population going to be in the year 2008 ?

Correct Answers:

• 42000000• 76147299.6883258219

7. (1 pt) set2/p2-7.pg

The half-life of Radium-226 is 1590 years. If a sample con-tains 200 mg, how many mg will remain after 2000 years?

Correct Answers:

• 83.6326605774975656

8. (1 pt) set2/p2-9.pg

Solve the inital value problem fory(x);xy

′+4y = 4x4

with the initial condition:y(1) = 10.y(x) =Correct Answers:

• (10 - (4/(4+4)))/xˆ{4} + (4/(4+4))*xˆ{4}

9. (1 pt) set2/p2-11.pg

Use logarithmic differentiation to finddy/dx, where

y =(x−4)−3(sin(x))4

(x2−2x)2ex3

y′ = y∗Correct Answers:

• ( -3/(x-4) + 4*cos(x)/sin(x) - 2*(2*x-2)/(x*x-2*x) - 3*x*x )

10. (1 pt) set2/srw210 17a.pg

(a) If f is one-to-one andf (9) = 9, then f−1(9) = and( f (9))−1 = .

(b) If g is one-to-one andg(−2) = 2, theng−1(2) = and(g(−2))−1 = .

Correct Answers:

• 9• 0.111111111111111111• -2• 0.5

1

11. (1 pt) set2/mec6.pg

Letf (x) = 5+2x+5ex

f−1(10) =Correct Answers:

• 0

12. (1 pt) set2/osutr 4 2.pg

Simplify the expression

tan(2cos−1(x/5)

)answer =Correct Answers:

• 2*x*sqrt(5ˆ2-xˆ2)/(2*xˆ2-5ˆ2)

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -The next three problems deal with two new functions.The first is called the HYPERBOLIC SINE FUNCTION and

is denoted assinh(x).The second is called the HYPERBOLIC COSINE FUNC-

TION and is denoted ascosh(x).These two functions are both defined using either the differ-

ence or sum of exponential functions and then dividing by 2:

sinh(x) =ex−e−x

2

cosh(x) =ex +e−x

213. (1 pt) set2/srw41 33.pg

sinh(0) =cosh(0) =Correct Answers:

• 0• 1

14. (1 pt) set2/invtrigs2.pg

Evaluate the definite integral.Z 7

0

1√36+x2

dx

[NOTE: Remeber to enter all necessary *, (, and ) !!Enter arctan(x) for tan−1x , sin(x) for sinx ... ]

Correct Answers:

• 0.994457493817766288

15. (1 pt) set2/p3-1.pg

Letf (x) = 8cos(x)sin−1(x)

f ′(x) =NOTE: The webwork system will accept arcsin(x) and notsin−1(x) as the inverse of sin(x).

Correct Answers:

• 8*(-sin(x)*arcsin(x)+cos(x)/sqrt(1-x*x))

16. (1 pt) set2/ur in 10 1.pg

For each of the indefinite integrals below, choose which of thefollowing substitutions would be most helpful in evaluating theintegral. Enter the appropriate letter (A,B, or C) in each blank.DO NOT EVALUATE THE INTEGRALS.A. x = 6tanθB. x = 6sinθC. x = 6secθ

1.Z

dx

(36−x2)3/2

2.Z

dx(36+x2)3

3.Z

(x2−36)5/2dx

4.Z

x2√

36+x2dx

5.Z √

x2−36dx

Correct Answers:

• B• A• C• A• C

17. (1 pt) set2/ur in 10 2.pg

Match each of the trigonometric expressions below with theequivalent non-trigonometric function from the following list.Enter the appropriate letter (A,B,C,D, or E) in each blank.A. tan(arcsin(x/5))B. cos(arcsin(x/5))C. (1/2)sin(2arcsin(x/5))D. sin(arctan(x/5))E. cos(arctan(x/5))

1.x√

25−x2

2.

√25−x2

53.

x25

√25−x2

4.x√

25+x2

5.5√

25+x2

Correct Answers:

• A• B• C• D• E


2

Hsiang-Ping HuangMath 1220-90, Spring 2008WeBWorK Assignment 3 due 02/12/2008 at 10:59pmMSTIntegration by Substitution, Trigonometric Integrals,Integration by Parts.This assignment will cover the material from Sections 7.1 – 7.3.

1. (1 pt) set3/p3-2.pg

Let

f (x) = tan−1(sin(3x))

f ′(x) =Correct Answers:

• 3*cos(3*x)/(1+(sin(3*x))**2)

2. (1 pt) set3/p3-3.pg


0

31+1x2 dx

[NOTE: Remeber to enter all necessary *, (, and ) !!Enter arctan(x) for tan−1x , sin(x) for sinx . ]

Correct Answers:

• 2.35619449019234493

3. (1 pt) set3/p3-4.pg

Evaluate the indefinite integral.Z7x

x4 +1dx

+ C[NOTE: Remeber to enter all necessary *, (, and ) !!Enter arctan(x) for tan−1x , sin(x) for sinx . ]

Correct Answers:

• 3.5 * arctan(xˆ2)

4. (1 pt) set3/p3-5.pg

Let

f (x) = x3 tan−1(6x)

f ′(x) =NOTE: The WeBWorK system will accept arctan(x) but nottan−1(x) as the inverse of tan(x).

Correct Answers:

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

5. (1 pt) set3/p3-6.pg

Evaluate the integral

f (x) =Z

dx√52−12 +2x−x2

f (x) = + CCorrect Answers:

• arcsin((x-1)/5)

6. (1 pt) set3/p3-7.pg

Perform the following integration:

Z 1

0

e2x−e−2x

e2x +e−2x dx

Answer: .Correct Answers:

• 0.662501373678932216

7. (1 pt) set3/p3-8.pg

Perform the following integration:Z1

x2−4x+9dx

Answer: + C.Correct Answers:

• arctan((x-2)/sqrt(5))/sqrt(5)

8. (1 pt) set3/p3-9.pg

Perform the following integration:Zcos3x dx


• sin(x)-((sin(x))**3)/3

9. (1 pt) set3/sc55 100.pg

Evaluate the indefinite integral.Z69cos2(15x)dx

+ CCorrect Answers:

• 69*x/2 + 69*sin(2*15*x)/(4*15)

10. (1 pt) set3/c4s5p3.pg

Find the value ofZ π/6

0sin(2x)sin(x) dx.

Correct Answers:

• 0.083333333074262322

11. (1 pt) set3/c4s5p4.pg

Find the value ofZ π/5

0cos(x)sin(sin(x))dx.

Correct Answers:

• 0.167829160210847734

12. (1 pt) set3/sc55 97.pg


3sin2(3x)cos2(3x)dx

Correct Answers:

• 0.989116016243862771

1

13. (1 pt) set3/osuin 5 7.pgZ π/6

0sin4(6x)dx =

Correct Answers:

• 0.196349540875

14. (1 pt) set3/sc55 29.pg

Evaluate the indefinite integral.Zx+4

x2 +8xdx

+ CCorrect Answers:

• 1/2 * ln(abs(xˆ2 + 8 * x))

15. (1 pt) set3/sc55 57.pg

Verify that1

x2−1=

12

(1

x−1− 1

x+1

)and use this equation to evaluateZ 5

2

1x2−1

dx

Correct Answers:

• 0.346573590279972655

16. (1 pt) set3/sc56 26.pg

First make a substitution and then use integration by parts toevaluate the integral. Z

x5cos(x3)dx

+CCorrect Answers:

• 1/3 * xˆ3 * sin(xˆ3) + 1/3 * cos(xˆ3)

17. (1 pt) set3/sc56 41.pg

A particle that moves along a straight line has velocity

v(t) = t2e−2t

meters per second after t seconds. How many meters will ittravel during the first t seconds?

Correct Answers:

• - e**(- 2 * t)*(t**2/2 + 2*t/4 + 2/8) + 2/8

18. (1 pt) set3/osuin 15 3.pg

Note: You can get full credit for this problem by just enteringthe final answer (to the last question) correctly. The initial ques-tions are meant as hints towards the final answer and also allowyou the opportunity to get partial credit.

Consider the definite integralZ 1/3

0xsin−1(3x)dx

The first step in evaluating this integral is to apply integrationby parts: Z

udv= uv−Z

vdu

whereu =anddv= h(x)dx whereh(x) =Note: Use arcsin(x) for sin−1(x).

After integrating by parts, we obtain the integralZ 1/3

0vdu=Z 1/3

0f (x)dx on the right hand side where

f (x) =The most appropriate substitution to simplify this integral isx = g(t) whereg(t) =Note: We are usingt as variable for angles instead ofθ, sincethere is no standard way to typeθ on a computer keyboard.

After making this substitution and simplifying (using trig

identities), we obtain the integralZ b

ak(t)dt where

k(t) =a =b =

After evaluating this integral and plugging back into the in-tegration by parts formula we obtain:Z 1/3

0xsin−1(3x)dx =

Correct Answers:

• asin(3*x)• x• 3*xˆ2/(2*sqrt(1-3ˆ2*xˆ2))• (sin(t))/3• (sin(t))ˆ2/(2*3ˆ2)• 0• 1.570796327• 0.0436332313055555556


2

Hsiang-Ping HuangMath 1220-90, Spring 2008WeBWorK Assignment 4 due 02/19/2008 at 10:59pmMSTIntegration Rational Functions by Substitutions andPartial Fractions, and Strategies for Integration.This assignment will cover the material from Sections 7.4 – 7.6.

1. (1 pt) set4/p4-1.pg

Evaluate the definite integral.Z 15sin( π8 )

0

x3√

225−x2dx


Correct Answers:

• 19.0596486834868969

2. (1 pt) set4/p4-2.pg


1

dxx2 +9


Correct Answers:

• 0.358124524463338244

3. (1 pt) set4/p4-3.pg

Evaluate the indefinite integral.Z √18x−x2dx

+C[NOTE: Remeber to enter all necessary *, (, and ) !!Enter arctan(x) for tan−1x , sin(x) for sinx.... ]

Correct Answers:

• (x-9)*((18*x-x**2)**.5)/2 + 9*9*arcsin((x-9)/9)/2

4. (1 pt) set4/p4-4.pg

Use integration by parts to evaluate the integral.Zxe4xdx

+CCorrect Answers:

• 0.25 * (x * eˆ(4 * x) - 0.25 * eˆ(4 * x))

5. (1 pt) set4/p4-5.pg

Find the integral Zxln(9x)dx

Answer:+C

Correct Answers:

• x**2.0*ln(9*x)/2-x**2.0/4.0

6. (1 pt) set4/p4-6.pg

Use integration by parts to evaluate the integral.Z4xcos3xdx

+CCorrect Answers:

• 4 * 0.333333333333333333 * (x * sin(3 * x) + 0.333333333333333333 * cos(3 * x))

7. (1 pt) set4/p4-7.pg

Evaluate the following integralZcos(lnx) dx


• (x/2)*( cos(ln(x))+sin(ln(x)) )

8. (1 pt) set4/p4-8.pg

Evaluate the integral.Z1

(x−3)(x+3)dx

+CCorrect Answers:

• (ln(x + 3))/(-6) - (ln(x - 3))/(-6)

9. (1 pt) set4/p4-9.pg

Write out the form of the partial fraction decomposition of thefunction appearing in the integral:Z −5x−90

x2 +3x−54dx

Determine the numerical values of the coefficients, A and B,where A≤ B.

Adenominator

+B

denominatorA = B =Correct Answers:

• -8• 3

10. (1 pt) set4/p4-10.pg

Evaluate the integral.Z 6

5

7x−9x2−2x−3

dx

Correct Answers:

• 1.83299804363352636

11. (1 pt) set4/p10-3.pg

Evaluate the following integral.Z 0.5

0sin√

x dx

Note, enter your answer symbolically.Answer:Correct Answers:

• 0.224125658254899694

1

12. (1 pt) set4/p3-10.pg

Evaluate the following integral:

Z 1

0

√t

t +1dt

Answer: .Correct Answers:

• 0.429203673

13. (1 pt) set4/ur in 25 12.pg

The form of the partial fraction decomposition of a rationalfunction is given below.

−1x2 +5x+16(x−1)(x2 +4)

=A

x−1+

Bx+Cx2 +4

A = B = C =Now evaluate the indefinite integral.Z −1x2 +5x+16

(x−1)(x2 +4)dx

+ CCorrect Answers:

• 4• -5• 0• 4*ln(abs(x+-1))+-5*ln(x**2+4)/2

14. (1 pt) set4/osuin 25 9a.pg

Consider the integralZx21−4x14+7x7−12

(x3−6x2 +5x)3 (x4−625)2 dx

Enter a T or an F in each answer space below to indicate whetheror not a term of the given type occurs in the general form ofthe complete partial fractions decomposition of the integrand.A1,A2,A3 . . . andB1,B2,B3, . . . denote constants.You must get all of the answers correct to receive credit.

1. A3x+B3

(x2−25)2

2. B4

(x+5)3

3. A8x+B8

(x−5)2

4. A2x+B2

(x2+25)2

5. B1x+1

6. B6

(x−5)3

7. B5

(x−5)2

8. B7

(x+5)2

Correct Answers:

• F• F• F• T• F• T• T• T


2

Hsiang-Ping HuangMath 1220-90, Spring 2008WeBWorK Assignment 5 due 02/26/2008 at 10:59pmMSTIndeterminate Limits and Improper Integrals.This assignment will cover the material from Sections 8.1 – 8.4.

1. (1 pt) set5/p1-6.pg

Evaluate the definite integral.Z e5

1

dx

x√

lnx

Correct Answers:

• 4.47213595499957939

2. (1 pt) set5/p5-1.pg

Find the indicated limit. Make sure that you have an indetermi-nate form before you apply l’Hopital’s Rule.

limx→π/2cosxπ2 −x

= .

Instruction: If your answer is∞, enter ”INF”; if it is −∞,enter ”-INF”.

Correct Answers:

• 1

3. (1 pt) set5/p5-2.pg


limx→0x3−3x2 +x

x3−2x= .


Correct Answers:

• -0.5

4. (1 pt) set5/p5-3.pg


limx→0sinx−tanx

x2 sinx= .


Correct Answers:

• -0.5

5. (1 pt) set5/p5-4.pg


limx→0+x2

sinx−x= .


Correct Answers:

• -INF

6. (1 pt) set5/p5-5.pg

Findlimx→0

xcos(ax)tan(bx) = .

Instruction: If your answer is∞, enter ”INF”; if it is −∞, en-ter ”-INF”. Note that b must be nonzero for the expression to bedefined.

Correct Answers:

• 1/b

7. (1 pt) set5/p5-6.pg

Evaluate the limit

limx→∞

5+10x5−2x

Correct Answers:

• -5

8. (1 pt) set5/p5-7.pg

Evaluate the limit

limx→∞

9x3−9x2−3x9−2x−8x3

Correct Answers:

• -1.125

9. (1 pt) set5/p5-8.pg

Evaluate

limx→∞

√x4−8x3 +52x2−3

Correct Answers:

• 0.5

10. (1 pt) set5/p5-9.pg

Determine the infinite limit of the following functions. EnterINF for ∞ and -INF for−∞.

limx→π/2−

(x−π/2) tan(x) =

limx→π/2−

π/2−xcos(x)

=

limx→4+

x−4ln(x/4)

=

limx→4−

ln(x/4)

(x−4)2 =

Correct Answers:

• -1• 1• 4• -INF

1

11. (1 pt) set5/p5-9a.pg

Determine the limit of the following functions. Enter INF for∞and -INF for−∞.limx→0

√xlnx =

limx→∞

1√x

lnx =

Correct Answers:

• 0• 0

12. (1 pt) set5/p5-9b.pg

Determine the limit of the following function. Enter INF for∞and -INF for−∞.limx→∞

x−2ex =Correct Answers:

• INF

13. (1 pt) set5/p5-10.pg

Evaluate the following limit. If needed, enter ’INF’ for∞ and’-INF’ for −∞.

limx→∞

(√x2 +1x+1−x

)=

Correct Answers:

• 0.5

14. (1 pt) set5/p6-2.pg

Determine whether the integral is divergent or convergent. If itis convergent, evaluate it. If it diverges to infinity, state youranswer as ”INF” (without the quotation marks). If it divergesto negative infinity, state your answer as ”MINF”. If it divergeswithout being infinity or negative infinity, state your answer as”DIV”. Z ∞

4

1

x8/7dx

Correct Answers:

• 5.74234749205346552

15. (1 pt) set5/p6-4.pg

Determine whether the integral is divergent or convergent. If itis convergent, evaluate it. If it diverges to infinity, state youranswer as ”INF” (without the quotation marks). If it divergesto negative infinity, state your answer as ”MINF”. If it divergeswithout being infinity or negative infinity, state your answer as”DIV”. Z 5

0

1x1.6 dx

Correct Answers:

• INF

16. (1 pt) set5/p6-1.pg

Determine whether the integral is divergent or convergent. If itis convergent, evaluate it. If not, state your answer as ”diver-gent.” Z ∞

2

2

(x+3)3/2dx

Correct Answers:

• 1.78885438199983176

17. (1 pt) set5/p6-3.pg

Determine whether the integral is divergent or convergent. If itis convergent, evaluate it. If it is divergent, enter your answer as”-1”. Z 3

−∞

1x2 +1

dx

Correct Answers:

• 2.81984209919315105

18. (1 pt) set5/p6-5.pg

Determine whether the integral is divergent or convergent. If itis convergent, evaluate it. If it diverges to infinity, state youranswer as ”INF” (without the quotation marks). If it divergesto negative infinity, state your answer as ”MINF”. If it divergeswithout being infinity or negative infinity, state your answer as”DIV”. Z 8

1

83√

x−1dx

Correct Answers:

• 43.9116685202756582

19. (1 pt) set5/p6-6.pg

Evaluate the following improper integral:Z ∞

10

x1+x2 dx

Answer: .If the integral diverges, enter ”diverge” as answer.Correct Answers:

• DIVERGE

20. (1 pt) set5/p6-7.pg


1xe−x dx


• 0.735758882342884643

2

21. (1 pt) set5/p6-8.pg


4

1

(π−x)2/3dx


• DIVERGE

22. (1 pt) set5/p6-9.pg

Find the area of the region under the curve

y =1

x2 +xto the right ofx = 1.Answer: .Correct Answers:

• 0.693147180559945309

23. (1 pt) set5/p6-10.pg


e

lnxx

dx

Answer: .

If the integral diverges, enter ”diverge” as answer.Correct Answers:

• DIVERGE

24. (1 pt) set5/osuin 12 4.pg

Find the indicated integrals (if they exist)Zx2√

5x+4dx =

+CZ ∞

−∞

e4x

e8x +1dx =Z

6x+55x2 +21x+4

dx =

+CZln(x)

x7 dx =

+CCorrect Answers:

• (1/5ˆ3)*((2/7)*(5*x+4)ˆ(7/2)-(4/5)*4*(5*x+4)ˆ(5/2)+(2/3)*4ˆ2*(5*x+4)ˆ(3/2))• 0.39269908175• ln(5*x+1)/5 + ln(x+4)• (xˆ-6/-6)*(ln(x)-1/-6)


3

Hsiang-Ping HuangMath 1220-90, Spring 2008WeBWorK Assignment 6 due 03/04/2008 at 10:59pmMSTSequences and Series.This assignment will cover the material from Sections 9.1 – 9.2.

1. (1 pt) set6/p7-1.pg

Determine the sum of the following series.∞

∑n=1

(−3)n−1

5n

Correct Answers:

• 0.125

2. (1 pt) set6/p7-2.pg

Consider the sequence

an =ln(1/n)√

2n.

Write the first five terms ofan, and find limn→∞ an. If thesequence diverges, enter ”divergent” in the answer box for itslimit.

a) First five terms: , , , , .b) limn→∞ an = .Correct Answers:

• 0• -0.346573590279972655• -0.448506588731347769• -0.490129071734273596• -0.508948955591838792• 0

3. (1 pt) set6/p7-3.pg

Suppose

a1 =1

2− 12

,a2 =2

3− 13

,a3 =3

4− 14

,a4 =4

5− 15

,a5 =5

6− 16

.

a) Find an explicit formula foran: .b) Determine whether the sequence is convergent or diver-

gent: .(Enter ”convergent” or ”divergent” as appropriate.)

c) If it converges, find limn→∞ an = .Correct Answers:

• (n**2+n)/(n**2+2*n)• CONVERGENT• 1

4. (1 pt) set6/p7-4.pg

Determine whether the sequences are increasing, decreasing, ornot monotonic. If increasing, enter 1 as your answer. If decreas-ing, enter -1 as your answer. If not monotonic, enter 0 as youranswer.

1. an = cosn3n

2. an = n−3n+3

3. an = 13n+6

4. an =√

n+36n+3

Correct Answers:

• 0• 1• -1• -1

5. (1 pt) set6/p7-5.pg


∑n=1

(3n +8n

12n )

Correct Answers:

• 2.33333333333333333

6. (1 pt) set6/p7-6.pg

If the following series converges, compute its sum. Otherwise,enter INF if it diverges to infinity, MINF if it diverges to minusinfinity, and DIV otherwise.

∞

∑n=1

2n(n+2)

(Hint: try breaking the summands up partial fractions-style.)Correct Answers:

• 1.5

7. (1 pt) set6/p7-8.pg

Match each of the following with the correct statement.C stands for Convergent, D stands for Divergent.

1. ∑∞n=1

4n3+n8

2. ∑∞n=1ne−n2

3. ∑∞n=1

n2

n3+7

4. ∑∞n=2

48nln(n)

5. ∑∞n=1

2+9n

10n

Correct Answers:

• C• C• D• D• C

8. (1 pt) set6/p7-9.pg

A ball is dropped from a height of 91 feet. Each time it hits

the floor, it rebounds to23

its previous height. Find the total

distance it travels before coming to rest.Answer: feet.Correct Answers:

• 455

1

9. (1 pt) set6/p7-10.pg

Decide the convergence or divergence of the following series:

∞

∑k=1

(3π

)k

.

Answer: (Enter ”convergent” or ”divergent”.)Correct Answers:

• CONVERGENT


2

Hsiang-Ping HuangMath 1220-90, Spring 2008WeBWorK Assignment 7 due 03/11/2008 at 10:59pmMDTConvergence Tests.This assignment will cover the material from Sections 9.3 – 9.5.

1. (1 pt) set7/p8-1.pg

Use the Integral Test to decide the convergence or divergenceof the following series:

∞

∑k=1

k2

ek .

Answer: (Enter ”converge” or ”diverge”.)Correct Answers:

• CONVERGE

2. (1 pt) set7/p8-2.pg

Use the Integral Test to decide the convergence or divergenceof the following series:

∞

∑k=1

1000k2

1+k3 .

Answer: (Enter ”convergent” or ”divergent”.)Correct Answers:

• DIVERGENT

3. (1 pt) set7/p8-3.pg


∑n=1

(3)n−1

7n

Correct Answers:

• 0.25

4. (1 pt) set7/p8-4.pg

Determine the convergence or divergence of the following se-ries.

∞

∑n=1

√2n+1n2

• A. convergent• B. divergent

Correct Answers:

• A

5. (1 pt) set7/p8-5.pg

Determine whether the following series is

∞

∑n=1

(−1)n+1 15n1.1

• A. conditionally convergent• B. absolutely convergent• C. divergent

Correct Answers:

• B

6. (1 pt) set7/p8-6.pg

Match each of the following with the correct statement.C stands for Convergent, D stands for Divergent.

1. ∑∞n=1

1

7+ 5√n3

2. ∑∞n=1

2n(n+5)

3. ∑∞n=1

1+10n

10+8n

4. ∑∞n=1

2n10−9

5. ∑∞n=1

ln(n)10n

Correct Answers:

• D• C• D• C• D

7. (1 pt) set7/p8-7.pg

Select the FIRST correct reason on the list why the given seriesconverges.

A. Geometric series.B. Comparison with a convergent p series.C. Integral test.D. Ratio test.E. Alternating series test.

1. ∑∞n=1

(cos(nπ)ln(4n)

2. ∑∞n=1

sin2(7n)n2

3. ∑∞n=1

(n+1)(32−1)n

32n

4. ∑∞n=0

en

n!

5. ∑∞n=1

n2+√

nn4−8

6. ∑∞n=1(

−eπ )n

Correct Answers:

• E• B• D• D• B• A

1

8. (1 pt) set7/p8-8.pg

Select the FIRST correct reason on the list why the given seriesconverges.

A. Geometric series.B. Comparison with a convergent p series.C. Integral test.D. Ratio test.E. Alternating series test.

1.∞

∑n=1

1

n(ln(n))2

2.∞

∑n=1

sin2(4n)n2

3.∞

∑n=1

(n+1)(35)n

62n

4.∞

∑n=1

3(4)n

92n

5.∞

∑n=1

n2 +√

nn4−2

6.∞

∑n=1

(−1)n ln(en)n4cos(nπ)

Correct Answers:

• C• B• D• A• B• B

9. (1 pt) set7/p8-9.pg

Select the FIRST correct reason why the given series diverges.

A. Diverges because the terms don’t have limit zeroB. Divergent geometric seriesC. Divergent p seriesD. Integral testE. Comparison with a divergent p seriesF. Diverges by limit comparison testG. Diverges by alternating series test

1. ∑∞n=1

1√n

2. ∑∞n=1

ln(n)n

3. ∑∞n=1

(n+1)(102+1)n

102n

4. ∑∞n=1(n)−

13

5. ∑∞n=1

2n+3(−1)n

6. ∑∞n=1

1nln(n)

Correct Answers:

• C• D• A• C• A• D

10. (1 pt) set7/p8-10.pg

Here are some series and sequences. Enter the letter C if thereis convergence, and the letter D if not

1. limn→∞

n!(2n)!

2. limn→∞

2n

n2

3. limn→∞

3n2−2n+14n3 +1

4.∞

∑n=1

n!(2n)!

5.∞

∑n=1

nn2 +1

6.∞

∑n=1

n(n−1)(n−3)2n

7.∞

∑n=1

n(ln(n))2

n3 +1

8.∞

∑n=1

1(2n+1)(2n+2)

Correct Answers:

• C• D• C• C• D• D• C• C


2

Hsiang-Ping HuangMath 1220-90, Spring 2008WeBWorK Assignment 8 due 03/25/2008 at 10:59pmMDTPower and Taylor Series.This assignment will cover the material from Sections 9.6 – 9.9.

1. (1 pt) set8/p9-1.pg

Find the interval of convergence for the given power series.∞

∑n=1

(x−10)n

n(−4)n

The series is convergentfrom x = , left end included (Y,N):to x = , right end included(Y,N):

Correct Answers:

• 6• N• 14• Y

2. (1 pt) set8/p9-2.pg

Match each of the power series with its interval of convergence.

1.∞

∑n=1

(3x)n

n3

2.∞

∑n=1

n!(3x−3)n

3n

3.∞

∑n=1

(x−3)n

(3)n

4.∞

∑n=1

(x−3)n

(n!)3n

A. [−13 , 1

3]B. {3/3}C. (0,6)D. (−∞,∞)

Correct Answers:

• A• B• C• D

3. (1 pt) set8/p9-3.pg

Suppose that 5x(9+x) = ∑∞

n=0cnxn.

Find the first few coefficients.c0 =

c1 =c2 =c3 =c4 =Find the radius of convergenceRof the power series.

R= .Correct Answers:

• 0• 0.555555555555555556• -0.0617283950617283951• 0.00685871056241426612

• -0.000762078951379362902• 9

4. (1 pt) set8/p9-3a.pg

The functionf (x) = 4(1−9x)2 is represented as a power series

f (x) = ∑∞n=0cnxn.

Find the first few coefficients in the power series.c0 =

c1 =c2 =c3 =c4 =Find the radius of convergenceRof the series.

R= .Correct Answers:

• 4• 72• 972• 11664• 131220• 0.111111111111111111

5. (1 pt) set8/p9-4.pg

Find a formula for the sum of the series∞

∑0

(n+1)xn

2n+2

for −2 < x < 2.

Correct Answers:

• 1/((2-x)ˆ2)

6. (1 pt) set8/p9-5.pg

Find the power series representation for

f (x) =1

(1+x)2

and specify the radius of convergence. (Note: To determineen uniquely, we requirean is positive anden is eithern−1 or n. )

f (x) =∞

∑n=1

(−1)en anxpn,

whereen = ,wherean = ,andpn = .Radius of convergence: .Correct Answers:

• n-1• n• n-1• 1

1

7. (1 pt) set8/p9-6.pg


f (x) = xex2.

f (x) =∞

∑n=0

1an!

xpn,

wherean = andpn = .Correct Answers:

• n• 2*n+1

8. (1 pt) set8/p9-7.pg


f (x) =Z x

0

tan−1 tt

dt.

f (x) =∞

∑n=1

(−1)en anxpn,

whereen = ,andan = ,andpn = .Correct Answers:

• n-1• 1/((2*n-1)**2)• 2*n-1

9. (1 pt) set8/p9-8.pg

Find the sum of∑∞

n=1n(n+1)xn =for < x < .Correct Answers:

• 2*x/((1-x)**3)• -1• 1

10. (1 pt) set8/p9-9.pg

Find the terms throughx5 in the Maclaurin series for

f (x) = (x3−x+1)e−x.

f (x) = +O(x6).Correct Answers:

• 1-2x + (3/2)*xˆ2 + (1/3)*xˆ3 - (19/24)*xˆ4 + (9/20)*xˆ5

11. (1 pt) set8/p10-1.pg

Let F(x) =Z x

0sin(4t2) dt.

Find the MacLaurin polynomial of degree 7 forF(x).

Use this polynomial to estimate the value ofZ 0.61

0sin(4x2) dx.

Correct Answers:

• 4 * xˆ3 / 3 - 4ˆ3 * xˆ7 / 42• 0.25475191868920381


2

Hsiang-Ping HuangMath 1220-90, Spring 2008WeBWorK Assignment 9 due 04/01/2008 at 10:59pmMDTConics.This assignment will cover the material from Sections 10.1 –10.4.

1. (1 pt) set9/p10-4.pg

The parabolay = x2 +12x has its focus at the point(b,c) whereb =c =

Correct Answers:

• -6• -35.75

2. (1 pt) set9/p10-5.pg

The ellipse 3x2 + 2x+ y2 = 1 has its center at the point(b,c)whereb =c =The length of the major diameter of this ellipse is

Correct Answers:

• -0.333333333333333333• 0• 2.30940107675850306

3. (1 pt) set9/p10-6.pg

Determine the distance D between the vertices of−9x2 +18x+4y2 +24y−9 = 0.

D =Correct Answers:

• 6

4. (1 pt) set9/p10-7.pg

Match each equation below to the curve it represents. Each an-swer should be A, B, C, D, E, F, or G.CURVES

A. Circle,B. Ellipse,C. Point,D. Parabola,E. Empty Set,F. Intersecting lines,G. Hyperbola,

EQUATIONS

1. 4x2−4y2 +8x+12y−5 = 0

2. 9x2 +4y2 +72x−16y+124= 03. 9x2 +4y2 +72x−16y+160= 04. 3x2 +3y2−6x+12y+60= 05. x2 +y2−2x+2y+1 = 06. y2−5x−4y−6 = 07. x2 +xy+y2−6 = 08. 4x2−3xy−18= 0

Correct Answers:

• F• B• C• E• A• D• B• G

5. (1 pt) set9/p10-8.pg

A bridge underpass in the shape of an elliptical arch, that is, halfof an ellipse, is 20 feet wide and 13 feet high. An eight foot widerectangular truck is to drive (safely) underneath. How high canit be?h =

Correct Answers:

• 11.914696806885184

6. (1 pt) set9/p10-10.pg

Match each polar equation below to the best description. Eachanswer should be C,F,I,L,M,O,or T.DESCRIPTIONS

C. Cardioid, F. Rose with four petals, I. Inwardly spiralingspiral, L. Lemacon, M. Lemniscate, O. Outwardly spiraling spi-ral, T. Rose with three petalsPOLAR EQUATIONS

1. r = 4−4sinθ2. r = 16+16cosθ3. r = 4cos3θ4. r = 16sin2θ5. r2 = 8cos2θ6. r = 4θ, r > 0

Correct Answers:

• C• C• T• F• M• O


1

Hsiang-Ping HuangMath 1220-90, Spring 2008WeBWorK Assignment 10 due 04/08/2008 at 10:59pmMDTPolar Coordinates and Calculus.This assignment will cover the material from Sections 10.5 –10.7.

1. (1 pt) set10/p11-1.pg

Find the area of the region inside:r = 5sinθ but outside:r = 3

Correct Answers:

• 9.24553326300564281

2. (1 pt) set10/p11-2.pg

Find the area of the region bounded by the given curve:r = 5eθ

on the interval25π≤ θ≤ 2π.

Correct Answers:

• 1792118.5490794632

3. (1 pt) set10/p11-3.pg

Find the area of the region bounded by:r = 10−4sinθ

Correct Answers:

• 339.29200658769767

4. (1 pt) set10/p11-4.pg

A circle C has center at the origin and radius 8. Another circleK has a diameter with one end at the origin and the other endat the point(0,12). The circles C and K intersect in two points.Let P be the point of intersection of C and K which lies in thefirst quadrant. Let(r,θ) be the polar coordinates of P, chosen sothatr is positive and 0≤ θ≤ π/2. Find r andθ.

r =θ =

Correct Answers:

• 8• 0.729727656226966363

5. (1 pt) set10/p11-5.pg

Find the area inside the inner loop of the following limacon:r = 7−14sinθ

Correct Answers:

• 26.6323056695873876

6. (1 pt) set10/p11-6.pg

Find the exact length of the polar curve described by:r = 7e−θ

on the interval710π≤ θ≤ 4π.

Note, you can use the standard arclength formula found inproblem 26 on page 551.

Correct Answers:

• 1.09783212070211391

7. (1 pt) set10/p11-8.pg

A curve with polar equation

r =34

8sinθ+25cosθrepresents a line. This line has a Cartesian equation of the form

y = mx + b ,where m and b are constants. Give the formulafor y in terms of x. For example, if the line had equationy = 2x+3 then the answer would be 2*x + 3.

Correct Answers:

• (34/8) - ( 25/8)*x

8. (1 pt) set10/p11-9.pg

Find the length of the curver = θ2 from θ = 0 to θ = 4.

Note, you can use the standard arclength formula found inproblem 26 on page 555.

Correct Answers:

• 27.1475730333305293

9. (1 pt) set10/p11-10.pg

Find the slope of the tangent to the curver =−3+1cosθ at thevalueθ = π/2

Correct Answers:

• -0.333333333333333333

10. (1 pt) set10/p10-9.pg

Match each polar equation below to the best description. Possi-ble answers are C,E,H,L,P,R,S,V,and Z.DESCRIPTIONSC. Circle centered at origin, E. Ellipse, H. Hyperbola, L. Lineneither vertical nor horizontal, P. Parabola, R. Circle not cen-tered at origin, S. Spiral, V. Vertical Line, Z. Horizontal LinePOLAR EQUATIONS

1. r2 = 11sin2θ

2. r =−73. r = 1

14cosθ4. r = 14sinθ5. r = 11

7sinθ+14cosθCorrect Answers:

• H• C• V• R• L


1

Hsiang-Ping HuangMath 1220-90, Spring 2008WeBWorK Assignment 11 due 04/16/2008 at 10:59pmMDTLinear DEs and Applications.This assignment will cover the material from Sections 15.1 –15.3.

1. (1 pt) set11/p12-1.pg

Solve the following differential equation:

y′′−3y′−10y = 0; y = 1,y′ = 10 atx = 0

Answer:y(x)= .Correct Answers:

• (12/7)*exp(5*x) - (5/7)*exp(-2*x)

2. (1 pt) set11/p12-2.pg


y′′+10y′+25y = 0

Answer:y(x) = C1 +C2 .NOTE: The order of your answers is important in this prob-

lem. For example, webwork may expect the answer ”A+B” butthe answer you give is ”B+A”. Both answers are correct butwebwork will only accept the former.

Correct Answers:

• exp(-5*x)• x*exp(-5*x)

3. (1 pt) set11/p12-3.pg


y′′+9y = 0; y = 3, y′ = 3 atx = π/3

Answer:y(x)= .Correct Answers:

• -sin(3*x)-3*cos(3*x)

4. (1 pt) set11/p12-4.pg


y′′+y′+y = 0

Answer:y(x) = C1 +C2 .NOTE: The order of your answers is important in this prob-


Correct Answers:

• exp(-x/2)*cos(sqrt(3)*x/2)• exp(-x/2)*sin(sqrt(3)*x/2)

5. (1 pt) set11/p12-5.pg


y′′−2y′+2y = 0

and express your answer in the form

ceαx sin(βx+ γ)Answer:α = , β = .Correct Answers:

• 1• 1

6. (1 pt) set11/p12-6.pg

Use the method of undetermined coefficients to solve the fol-lowing differential equation:

y′′+y′ = 4x

Answer: y(x) = +C1

+C2 .NOTE: The order of your answers is important in this prob-


Correct Answers:

• 2*x**2 - 4*x• 1• exp(-x)

7. (1 pt) set11/p12-7.pg

Use the method of undetermined coefficients to solve the fol-lowing differential equation:

y′′+6y′+9y = 2e−x

Answer: y(x) = +C1

+C2 .NOTE: The order of your answers is important in this prob-


Correct Answers:

• exp(-x)/2• exp(-3*x)• x*exp(-3*x)

8. (1 pt) set11/p12-8.pg

Let y be the solution of the initial value problem

y′′+2y′+2y = 0,y(0) = 0,y′(0) = 4

The maximum value ofy, for t > 0, is .Correct Answers:

• 1.2895877677793378


1

WeBWorK demonstration assignment

The main purpose of this WeBWorK set is to familiarizeyourself with WeBWorK.

Here are some hints on how to use WeBWorK effectively:

• After first logging into WeBWorK change your pass-word.

• Find out how to print a hard copy on the computer sys-tem that you are going to use. Print a hard copy of thisassignment.

• Get to work on this set right away and answer thesequestions well before the deadline. Not only will thisgive you the chance to figure out what’s wrong if an an-swer is not accepted, you also will avoid the likely rushand congestion prior to the deadline.

• The primary purpose of the WeBWorK assignments inthis class is to give you the opportunity to learn by hav-ing instant feedback on your active solution of relevantproblems. Make the best of it!

1. (1 pt) setDemo/demopr1.pg

Evaluate the expression1(7−5) = .

Correct Answers:

• 2


Evaluate the expression5/(5+6) = .Enter you answer as a decimal number listing at least 4 decimaldigits. (WeBWorK will reject your answer if it differs by morethan one tenth of 1 percent from what it thinks the answer is.)

Correct Answers:

• 0.454545454545454545


Let r = 3.Evaluate 4/π∗ r = .

Next, enter the expression 4/(π∗ r) = and let WeB-WorK compute the result.

Correct Answers:

• 3.81971863420548806• 0.424413181578387562


Enter here the expression1a + 1b.

Enter here the expression1a+b.

Correct Answers:

• 1/a+1/b• 1/(a+b)


Enter here the expression

a+12+b


a+bc+d

If WeBWorK rejects your answer use the preview button tosee what it thinks you are trying to tell it.

Correct Answers:

• (a+1)/(2+b)• (a+b)/(c+d)


Enter here the expression√

a+b


a√a+b


a+b√a+b

Correct Answers:

• sqrt(a+b)• a/sqrt(a+b)• (a+b)/sqrt(a+b)


Enter here the expression√x2 +y2


x√

x2 +y2


x+y√x2 +y2

Correct Answers:

• sqrt(x**2+y**2)• x*sqrt(x**2+y**2)• (x+y)/sqrt(x**2+y**2)

1



−b+√

b2−4ac2a

Note: this is an expression that gives the solution of aquadraticequationby thequadratic formula.

Correct Answers:

• (-b+sqrt(b**2-4*a*c))/(2a)


Simplify the expression

1−sin2(x)

Correct Answers:

• (cos(x))ˆ2


Evaluate the expression√(12 +12)

Correct Answers:

• sqrt(2)


2

hsiang-ping huang math 1220-90, spring 2008 5. webwork … · 2008-08-20 · hsiang-ping huang math...

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