Hsiang-Ping Huang 1.61418970655352 Math 1220-90, Summer 2008 5… · · 2009-08-2131= (c) log 5...

Hsiang-Ping HuangMath 1220-90, Summer 2008WeBWorK Assignment 1 due 05/21/2008 at 10:59pmMDTLogs and Exponentials.This assignment will cover the material from Sections 6.1 – 6.4.

1. (1 pt) set1/p1-1.pgEvaluate the following expressions.

(a) log2( 1

8

)=

(b) log3 1 =(c) log5

√3125 =

(d) 7log7 10 =Correct Answers:

• -3• 0• 2.5• 10

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

ln(r8s2 10√r7s10)

is equal toA lnr +B lns

where A = and where B =Correct Answers:

• 8.7• 3

3. (1 pt) set1/p1-3.pgR 121

ln(15x)x dx =

Correct Answers:• 9.81663248175892

4. (1 pt) set1/p1-4.pgIf f (x) = 2

√x ln(x), find f ′(x).

Find f ′(5).

Correct Answers:• .5*2*xˆ(-.5)*ln(x)+ 2*xˆ(-.5)

• 1.61418970655352

5. (1 pt) set1/p1-5.pgEvaluate the integrals

f (x) =Z e3x+8

e2x dx

f (x) = +C

g(x) =Z ex+8

ex dx

g(x) = +CCorrect Answers:

• exp(3*x+8-2*x)/(3-2)• exp(8)*x

6. (1 pt) set1/p1-7.pgFind the integral Z 1

0

t +12t2 +4t +3

dt.

Answer:Correct Answers:

• 0.274653072167027

7. (1 pt) set1/p1-8.pgSuppose y = e1/x2

+1/ex2. Find Dxy.

Dxy =Correct Answers:

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

8. (1 pt) set1/p1-9.pgA curve is given by the equation:

y3 +129 = (ex +1)2.

Find the slope of the tangent line at the point (0,−5).

Correct Answers:• 0.0533333333333333

9. (1 pt) set1/p1-10.pgThe region bounded by y = e−x2

, y = 0, x = 0, and x = 1 is re-volved about the y-axis. Find the volume of the resulting solid.

Answer:Correct Answers:

• 1.98586530405817

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

1

Hsiang-Ping HuangMath 1220-90, Summer 2008WeBWorK Assignment 10 due 07/16/2008 at 10:59pmMDTPolar Coordinates and Calculus.This assignment will cover the material from Sections 10.5 –10.7.

1. (1 pt) set10/p11-1.pgFind the area of the region inside: r = 5sinθ but outside: r = 2

Correct Answers:

• 14.4364512811388

2. (1 pt) set10/p11-2.pgFind the area of the region bounded by the given curve: r = 5eθ

on the interval 65 π≤ θ≤ 2π.

Correct Answers:

• 1780436.35846173

3. (1 pt) set10/p11-3.pgFind the area of the region bounded by: r = 6−1sinθ

Correct Answers:

• 114.668131856027

4. (1 pt) set10/p11-4.pgA circle C has center at the origin and radius 6. Another circleK has a diameter with one end at the origin and the other endat the point (0,14). 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 sothat r is positive and 0≤ θ≤ π/2. Find r and θ.

r =

θ =Correct Answers:

• 6• 0.442911044073639

5. (1 pt) set10/p11-5.pgFind the area inside the inner loop of the following limacon:r = 7−14sinθ

Correct Answers:

• 26.6323056695874

6. (1 pt) set10/p11-6.pgFind the exact length of the polar curve described by:

r = 5e−θ

on the interval 510 π≤ θ≤ 9π.

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

Correct Answers:

• 1.46993058107439

7. (1 pt) set10/p11-8.pgA curve with polar equation

r =37

3sinθ+16cosθ

represents 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:

• (37/3) - ( 16/3)*x

8. (1 pt) set10/p11-9.pgFind the length of the curve r = θ2 from θ = 0 to θ = 10.

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

Correct Answers:

• 350.865352942433

9. (1 pt) set10/p11-10.pgFind the slope of the tangent to the curve r = −10− 5cosθ atthe value θ = π/2

Correct Answers:

• 0.5

1

10. (1 pt) set10/p10-9.pgMatch each polar equation below to the best description. Possi-ble answers are C,E,H,L,P,R,S,V,and Z.DESCRIPTIONS

C. 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 Line

POLAR EQUATIONS

1. r =−42. r = 1

16cosθ

3. r2 = 40sin2θ

4. r = 16sinθ

5. r = 404sinθ+16cosθ

Correct Answers:

• C• V• H• R• L


2

Hsiang-Ping HuangMath 1220-90, Summer 2008WeBWorK Assignment 11 due 07/23/2008 at 10:59pmMDTLinear DEs and Applications.This assignment will cover the material from Sections 15.1 –15.3.

1. (1 pt) set11/p12-1.pgSolve the following differential equation:

y′′−3y′−10y = 0; y = 1,y′ = 10 at x = 0

Answer: y(x)= .Correct Answers:

• (12/7)*exp(5*x) - (5/7)*exp(-2*x)


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)


y′′+9y = 0; y = 3, y′ = 3 at x = π/3

Answer: y(x)= .Correct Answers:

• -sin(3*x)-3*cos(3*x)


y′′+ y′+ y = 0

Answer: y(x) = C1 +C2 .


Correct Answers:

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


y′′−2y′+2y = 0and express your answer in the form

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

• 1• 1

6. (1 pt) set11/p12-6.pgUse the method of undetermined coefficients to solve the fol-lowing differential equation:

y′′+ y′ = 4xAnswer: y(x) = +C1

+C2 .


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

7. (1 pt) set11/p12-7.pgUse the method of undetermined coefficients to solve the fol-lowing differential equation:

y′′+6y′+9y = 2e−x

Answer: y(x) = +C1+C2 .


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

8. (1 pt) set11/p12-8.pgLet y be the solution of the initial value problem

y′′+2y′+2y = 0,y(0) = 0,y′(0) = 10

The maximum value of y, for t > 0, is .


1


2

Hsiang-Ping HuangMath 1220-90, Summer 2008WeBWorK Assignment 2 due 05/28/2008 at 10:59pmMDTExponential Growth and Decay, Inverse Functions,Circular (Trigonometric) and Hyperbolic functionsand their inverses.This assignment will cover the material from Sections 6.5 – 6.9.

1. (1 pt) set2/p2-1.pgIn 620 days unknown radioactive substance decay to 44 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 to81 mg?

T =

Correct Answers:

• 523.460940534694• 159.135364517858

2. (1 pt) set2/p2-3.pgA bacteria culture starts with 180 bacteria and grows at a ratepropotional to its size. After 2 hours there are 360 bacteria.

(a) Find the population after t hours

y(t) = (function of t)

(b) Find the population after 5 hours.

y(5) =

(c) When will the population reach 2730 ?

T =

Correct Answers:

• 180*(2.71828182845905ˆ(0.346573590279973*t))• 1018.23376490863• 7.84566427895508

3. (1 pt) set2/p2-4.pgThe loudness of sound is measured in decibels in honor ofAlexander Graham Bell (1847-1922), inventor of the telephone.If the variation in pressure is P pounds per square inch, then theloudness L 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:

• 4635.95486554286

4. (1 pt) set2/p2-5.pgThe count in a bacteria culture was 500 after 10 minutes and1400 after 30 minutes. What was the initial size of the culture?

Find the doubling period in minutes. Find the pop-ulation after 100 minutes. When (in minutes) will thepopulation reach 14000?

Correct Answers:

• 298.807152333598• 13.4641435270832• 51425.8105269329• 74.7269166562136

5. (1 pt) set2/p2-5a.pgThe Hustler Bank Mutual Fund pays interest at a rate of 5.3%,compounded continuously. How much should be invested so asto have 20 thousand dollars in 6 years?

Correct Answers:

• 14552.0557682919

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

The rat population in a major metropolitan city is given bythe formula n(t) = 28e0.035t where t is measured in years since1990 and n(t) is measured in millions.

What was the rat population in 1990 ?What is the rat population going to be in the year 2001 ?

Correct Answers:

• 28000000• 41149201.0003518

7. (1 pt) set2/p2-7.pgThe half-life of Radium-226 is 1590 years. If a sample con-tains 300 mg, how many mg will remain after 3000 years?

Correct Answers:

• 81.1222767322651

8. (1 pt) set2/p2-9.pgSolve the inital value problem for y(x);

xy′+7y = 2x4

with the initial condition: y(1) = 18.

y(x) =

Correct Answers:

• (18 - (2/(7+4)))/xˆ{7} + (2/(7+4))*xˆ{4}

1

9. (1 pt) set2/p2-11.pgUse logarithmic differentiation to find dy/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/srw2 10 17a.pg(a) If f is one-to-one and f (−13) = 11, then f−1(11) = and( f (−13))−1 = .

(b) If g is one-to-one and g(−3) = 9, then g−1(9) = and(g(−3))−1 = .

Correct Answers:

• -13• 0.0909090909090909• -3• 0.111111111111111

11. (1 pt) set2/mec6.pgLet

f (x) = 2+1x+3ex

f−1(5) =Correct Answers:

• 0

12. (1 pt) set2/osu tr 4 2.pgSimplify the expression

tan(2cos−1(x/4)

)answer =Correct Answers:

• 2*x*sqrt(4ˆ2-xˆ2)/(2*xˆ2-4ˆ2)

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

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

TION and is denoted as cosh(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/srw4 1 33.pg

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

• 0• 1

14. (1 pt) set2/p3-1.pgLet

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

15. (1 pt) set2/ur in 10 2.pgMatch 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/3))B. cos(arcsin(x/3))C. (1/2)sin(2arcsin(x/3))D. sin(arctan(x/3))E. cos(arctan(x/3))

1.x√

9− x2

2.3√

9+ x2

3.x√

9+ x2

4.x9

√9− x2

5.

√9− x2

3Correct Answers:

• A• E• D• C• B


2

Hsiang-Ping HuangMath 1220-90, Summer 2008WeBWorK Assignment 3 due 06/04/2008 at 10:59pmMDTIntegration by Substitution, Trigonometric Integrals,Integration by Parts.This assignment will cover the material from Sections 7.1 – 7.3.


f (x) = tan−1(sin(8x))

f ′(x) =Correct Answers:

• 8*cos(8*x)/(1+(sin(8*x))**2)

2. (1 pt) set3/p3-3.pgEvaluate the definite integral.

Z 13

0

21+9x2 dx

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

Correct Answers:

• 0.523598775598299

3. (1 pt) set3/p3-4.pgEvaluate the indefinite integral.

Z 6xx4 +1

dx

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

Correct Answers:

• 3 * arctan(xˆ2)


f (x) = x4 tan−1(8x)

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

Correct Answers:

• 4*x**(4-1)*arctan(8*x) +x**4*8/(1+8*8*x**2)

5. (1 pt) set3/p3-6.pgEvaluate the integral

f (x) =Z dx√

82−12 +2x− x2

f (x) = + CCorrect Answers:

• arcsin((x-1)/8)

6. (1 pt) set3/p3-7.pgPerform the following integration:

Z 1

0

e2x− e−2x

e2x + e−2x dx

Answer: .Correct Answers:

• 0.662501373678932

7. (1 pt) set3/p3-8.pgPerform the following integration:Z 1

x2−4x+9dx

Answer: + C.Correct Answers:

• arctan((x-2)/sqrt(5))/sqrt(5)

8. (1 pt) set3/p3-9.pgPerform the following integration:Z

cos3 x dx


• sin(x)-((sin(x))**3)/3

9. (1 pt) set3/sc5 5 100.pgEvaluate the indefinite integral.Z

12cos2(86x)dx

+ C

Correct Answers:

• 12*x/2 + 12*sin(2*86*x)/(4*86)

10. (1 pt) set3/c4s5p3.pg

Find the value ofZ

π/3

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

Correct Answers:

• 0.433012700994771

1

11. (1 pt) set3/c4s5p4.pg

Find the value ofZ

π/2

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

Correct Answers:

• 0.45969769413186

12. (1 pt) set3/sc5 5 97.pgEvaluate the definite integral.

Z 13

8sin2(8x)cos2(8x)dx

Correct Answers:

• 0.617322915687565

13. (1 pt) set3/osu in 5 7.pgZπ/4

0sin4(4x)dx =

Correct Answers:

• 0.2945243113125

14. (1 pt) set3/sc5 5 29.pgEvaluate the indefinite integral.Z x+3

x2 +6xdx

+ C

Correct Answers:

• 1/2 * ln(abs(xˆ2 + 6 * x))

15. (1 pt) set3/sc5 5 57.pgVerify that

1x2−1

=12

(1

x−1− 1

x+1

)and use this equation to evaluateZ 3

2

3x2−1

dx

Correct Answers:

• 0.608197662162247

16. (1 pt) set3/sc5 6 26.pgFirst make a substitution and then use integration by parts toevaluate the integral. Z

x13 cos(x7)dx

+CCorrect Answers:

• 1/7 * xˆ7 * sin(xˆ7) + 1/7 * cos(xˆ7)

17. (1 pt) set3/sc5 6 41.pgA particle that moves along a straight line has velocity

v(t) = t2e−3t

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

Correct Answers:

• - e**(- 3 * t)*(t**2/3 + 2*t/9 + 2/27) + 2/27

18. (1 pt) set3/osu in 15 3.pgNote: 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 1/8

0xsin−1(8x)dx

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

udv = uv−Z

vdu

whereu =and dv = h(x)dx where h(x) =Note: Use arcsin(x) for sin−1(x).

After integrating by parts, we obtain the integralZ 1/8

0vdu =Z 1/8

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 using t as variable for angles instead of θ, sincethere is no standard way to type θ 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 =

2

After evaluating this integral and plugging back into the in-tegration by parts formula we obtain:Z 1/8

0xsin−1(8x)dx =

Correct Answers:

• asin(8*x)

• x• 8*xˆ2/(2*sqrt(1-8ˆ2*xˆ2))• (sin(t))/8• (sin(t))ˆ2/(2*8ˆ2)• 0• 1.570796327• 0.00613592315234375


3

Hsiang-Ping HuangMath 1220-90, Summer 2008WeBWorK Assignment 4 due 06/11/2008 at 10:59pmMDTIntegration 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.pgEvaluate the definite integral.Z 5sin( π

7 )

0

x3√

25− x2dx

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


2. (1 pt) set4/p4-2.pgEvaluate the definite integral.Z 2

1

dxx2 +9


Correct Answers:• 0.0887506830503084

3. (1 pt) set4/p4-3.pgEvaluate the indefinite integral.Z √

12x− x2dx

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

Correct Answers:• (x-6)*((12*x-x**2)**.5)/2 + 6*6*arcsin((x-6)/6)/2

4. (1 pt) set4/p4-4.pgUse integration by parts to evaluate the integral.Z

xe4xdx

+C

Correct Answers:• 0.25 * (x * eˆ(4 * x) - 0.25 * eˆ(4 * x))

5. (1 pt) set4/p4-5.pgFind the integral

Zx ln(10x)dx

Answer:

+C

Correct Answers:

• x**2.0*ln(10*x)/2-x**2.0/4.0

6. (1 pt) set4/p4-6.pgUse integration by parts to evaluate the integral.

Z4xcos4xdx

+C

Correct Answers:

• 4 * 0.25 * (x * sin(4 * x) + 0.25 * cos(4 * x))

7. (1 pt) set4/p4-7.pgEvaluate the following integral

Zcos(lnx) dx


• (x/2)*( cos(ln(x))+sin(ln(x)) )

8. (1 pt) set4/p4-8.pgEvaluate the integral.

Z 1(x−3)(x+2)

dx

+CCorrect Answers:

• (ln(x + 2))/(-5) - (ln(x - 3))/(-5)

1

9. (1 pt) set4/p4-9.pgWrite out the form of the partial fraction decomposition of thefunction appearing in the integral:

Z 11x−134x2 +5x−84

dx

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

Adenominator

+B

denominator

A = B =Correct Answers:

• -3• 14

10. (1 pt) set4/p4-10.pgEvaluate the integral.

Z 1

0

2x+5x2 +5x+6

dx

Correct Answers:

• 0.693147180559945

11. (1 pt) set4/invtrigs2.pgEvaluate the definite integral.Z 7

0

1√289+ x2

dx


Correct Answers:

• 0.400936292441021

12. (1 pt) set4/ur in 10 1.pgFor 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 = 8tanθ

B. x = 8sinθ

C. x = 8secθ

1.Z dx

(64+ x2)3

2.Z dx

(64− x2)3/2

3.Z

x2√

64+ x2 dx

4.Z x2 dx√

64− x2

5.Z

(x2−64)5/2 dx

Correct Answers:

• A• B• A• B• C

13. (1 pt) set4/p10-3.pgEvaluate the following integral.Z 0.5

0sin√

x dx

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

• 0.2241256582549

14. (1 pt) set4/p3-10.pgEvaluate the following integral:Z 1

0

√t

t +1dt

Answer: .Correct Answers:

• 0.429203673

15. (1 pt) set4/ur in 25 12.pgThe form of the partial fraction decomposition of a rationalfunction is given below.

−5x2 +3x−60(x−5)(x2 +9)

=A

x−5+

Bx+Cx2 +9

A = B = C =Now evaluate the indefinite integral.

Z −5x2 +3x−60(x−5)(x2 +9)

dx

+ CCorrect Answers:

• -5• 0• 3• -5*ln(abs(x+-5))+3*arctan(x/3)/3

2

16. (1 pt) set4/osu in 25 9a.pgConsider the integralZ x21−8x14 +12x7−20

(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 . . . and B1,B2,B3, . . . denote constants.You must get all of the answers correct to receive credit.

1. B4(x+5)3

2. B6(x−5)3

3. A2x+B2

(x2+25)2

4. B1x+1

5. A8x+B8(x−5)2

6. B5(x−5)2

7. A3x+B3

(x2−25)2

8. B7(x+5)2

Correct Answers:

• F• T• T• F• F• T• F• T


3

Hsiang-Ping HuangMath 1220-90, Summer 2008WeBWorK Assignment 5 due 06/18/2008 at 10:59pmMDTIndeterminate Limits and Improper Integrals.This assignment will cover the material from Sections 8.1 – 8.4.

1. (1 pt) set5/p1-6.pgEvaluate the definite integral.

Z e4

1

dxx√

lnx

Correct Answers:

• 4

2. (1 pt) set5/p5-1.pgFind the indicated limit. Make sure that you have an indetermi-nate form before you apply l’Hopital’s Rule.

limx→π/2cosxπ

2 − x= .

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

Correct Answers:

• 1


limx→0x3−3x2 + x

x3−2x= .


Correct Answers:

• -0.5


limx→0sinx−tanx

x2 sinx = .


Correct Answers:

• -0.5


limx→0+x2

sinx− x= .


Correct Answers:

• -INF

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

limx→0xcos(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.pgEvaluate the limit

limx→∞

3+8x10−8x

Correct Answers:

• -1

8. (1 pt) set5/p5-7.pgEvaluate the limit

limx→∞

4x3−6x2−6x9−2x−6x3

Correct Answers:

• -0.666666666666667

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

limx→∞

√x4−5x3−97x2−10

Correct Answers:

• 0.142857142857143

1

10. (1 pt) set5/p5-9.pgDetermine 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→2+

x−2ln(x/2)

=

limx→2−

ln(x/2)

(x−2)2 =

Correct Answers:

• -1• 1• 2• -INF

11. (1 pt) set5/p5-9a.pgDetermine the limit of the following functions. Enter INF for ∞

and -INF for −∞.limx→0

√x lnx =

limx→∞

1√x

lnx =

Correct Answers:

• 0• 0

12. (1 pt) set5/p5-9b.pgDetermine the limit of the following function. Enter INF for ∞

and -INF for −∞.lim

x→+∞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 +10x+1− x

)=

Correct Answers:

• 5

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

1x3/2 dx

Correct Answers:

• 0.707106781186548


Z 3

0

1x1.3 dx

Correct Answers:

• INF

16. (1 pt) set5/p6-1.pgDetermine whether the integral is divergent or convergent. If itis convergent, evaluate it. If not, state your answer as ”diver-gent.”

Z∞

2

9(x+3)3/2 dx

Correct Answers:

• 8.04984471899924

17. (1 pt) set5/p6-3.pgDetermine whether the integral is divergent or convergent. If itis convergent, evaluate it. If it is divergent, enter your answer as”-1”.

Z 1

−∞

1x2 +1

dx

Correct Answers:

• 2.35619449019234

2


Z 12

5

123√

x−5dx


19. (1 pt) set5/p6-6.pgEvaluate the following improper integral:Z

∞

10

x1+ x2 dx

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

• diverge


∞

1xe−x dx


• 0.735758882342885


∞

4

1(π− x)2/3 dx


• diverge

22. (1 pt) set5/p6-9.pgFind the area of the region under the curve

y =1

x2 + x

to the right of x = 1.Answer: .Correct Answers:

• 0.693147180559945

23. (1 pt) set5/p6-10.pgEvaluate the following improper integral:

Z∞

e

lnxx

dx


• diverge

24. (1 pt) set5/osu in 12 4.pgFind the indicated integrals (if they exist)Z

x2√4x+5dx =

+CZ∞

−∞

e5x

e10x +1dx =Z 5x+6

4x2 +21x+5dx =

+CZ ln(x)x5 dx =

+CCorrect Answers:

• (1/4ˆ3)*((2/7)*(4*x+5)ˆ(7/2)-(4/5)*5*(4*x+5)ˆ(5/2)+(2/3)*5ˆ2*(4*x+5)ˆ(3/2))• 0.3141592654• ln(4*x+1)/4 + ln(x+5)• (xˆ-4/-4)*(ln(x)-1/-4)


3

Hsiang-Ping HuangMath 1220-90, Summer 2008WeBWorK Assignment 6 due 06/25/2008 at 10:59pmMDTSequences and Series.This assignment will cover the material from Sections 9.1 – 9.2.

1. (1 pt) set6/p7-1.pgDetermine the sum of the following series.

∞

∑n=1

(2)n−1

8n

Correct Answers:

• 0.166666666666667

2. (1 pt) set6/p7-2.pgConsider the sequence

an =ln(1/n)√

2n.

Write the first five terms of an, 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• -0.346573590279973• -0.448506588731348• -0.490129071734274• -0.508948955591839• 0

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

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 for an: .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.pgDetermine 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 = n−5n+5

2. an = 15n+8

3. an = cosn5n

4. an =√

n+58n+5

Correct Answers:

• 1• -1• 0• -1


∞

∑n=1

(4n +6n

11n )

Correct Answers:

• 1.77142857142857

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

9n(n+2)

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

• 6.75

7. (1 pt) set6/p7-8.pgMatch each of the following with the correct statement.C stands for Convergent, D stands for Divergent.

1. ∑∞n=1 ne−n2

2. ∑∞n=2

96n ln(n)

3. ∑∞n=1

4+3n

7n

4. ∑∞n=1

9n7+n6

5. ∑∞n=1

n4

n7+2Correct Answers:

• C• D• C• C• C

1

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

A ball is dropped from a height of 108 feet. Each time it

hits the floor, it rebounds to23

its previous height. Find the totaldistance it travels before coming to rest.

Answer: feet.Correct Answers:

• 540

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, Summer 2008WeBWorK Assignment 7 due 06/30/2008 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


∞

∑n=1

(2)n−1

8n

Correct Answers:

• 0.166666666666667

4. (1 pt) set7/p8-4.pgDetermine 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.pgDetermine whether the following series is

∞

∑n=1

(−1)n+1 15n1.1

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

Correct Answers:

• A

6. (1 pt) set7/p8-6.pgMatch each of the following with the correct statement.C stands for Convergent, D stands for Divergent.

1. ∑∞n=1

14+ 2√n3

2. ∑∞n=1

9+6n

1+2n

3. ∑∞n=1

5n(n+7)

4. ∑∞n=1

ln(n)3n

5. ∑∞n=1

5n3−81

Correct Answers:

• C• D• C• D• C

7. (1 pt) set7/p8-7.pgSelect 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(4n)n2

3. ∑∞n=1

(n+1)(42−1)n

42n

4. ∑∞n=0

en

n!

5. ∑∞n=1

n2+√

nn4−6

6. ∑∞n=1(

−eπ

)n

Correct Answers:

• E• B• D• D• B• A

1

8. (1 pt) set7/p8-8.pgSelect 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=2

1

n(ln(n))2

2.∞

∑n=1

sin2(7n)n2

3.∞

∑n=1

(n+1)(15)n

42n

4.∞

∑n=1

4(4)n

62n

5.∞

∑n=1

n2 +√

nn4−5

6.∞

∑n=1

(−1)n ln(en)n4 cos(nπ)

Correct Answers:• C• B• D• A• B• B

9. (1 pt) set7/p8-9.pgSelect 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 test1. ∑

∞n=1

1√n

2. ∑∞n=1(−1)n (2n)!

(n!)2

3. ∑∞n=1

1n ln(n)

4. ∑∞n=1

ln(n)n

5. ∑∞n=1

4n+3(−1)n

6. ∑∞n=1

cos(nπ)ln(5)

Correct Answers:• C• A• D• D• A• A

10. (1 pt) set7/p8-10.pgHere 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, Summer 2008WeBWorK Assignment 8 due 07/02/2008 at 10:59pmMDTPower and Taylor Series.This assignment will cover the material from Sections 9.6 – 9.9.

1. (1 pt) set8/p9-1.pgFind the interval of convergence for the given power series.

∞

∑n=1

(x−1)n

n(−6)n

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

Correct Answers:

• -5• N• 7• Y

2. (1 pt) set8/p9-2.pgMatch each of the power series with its interval of convergence.

1.∞

∑n=1

(x−8)n

(n!)8n

2.∞

∑n=1

n!(10x−8)n

8n

3.∞

∑n=1

(x−8)n

(8)n

4.∞

∑n=1

(10x)n

n8

A. (−∞,∞)B. {8/10}C. [−1

10 , 110 ]

D. (0,16)Correct Answers:

• A• B• D• C

3. (1 pt) set8/p9-3.pgSuppose that 7x

(10+x) = ∑∞n=0 cnxn.

Find the first few coefficients.c0 =

c1 =

c2 =

c3 =

c4 =

Find the radius of convergence R of the power series.R = .

Correct Answers:

• 0• 0.7• -0.07• 0.007• -0.0007• 10

4. (1 pt) set8/p9-3a.pgThe function f (x) = 1

(1−8x)2 is represented as a power seriesf (x) = ∑

∞n=0 cnxn.

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

c1 =

c2 =

c3 =

c4 =

Find the radius of convergence R of the series.R = .

Correct Answers:

• 1• 16• 192• 2048• 20480• 0.125

5. (1 pt) set8/p9-4.pgFind a formula for the sum of the series

∞

∑0

(n+1)xn

15n+2

for −15 < x < 15.

Correct Answers:

• 1/((15-x)ˆ2)

1

6. (1 pt) set8/p9-5.pgFind the power series representation for

f (x) =1

(1+ x)2

and specify the radius of convergence. (Note: To determineen uniquely, we require an is positive and en is either n−1 or n. )

f (x) =∞

∑n=1

(−1)en anxpn ,

where en = ,where an = ,and pn = .Radius of convergence: .Correct Answers:

• n-1• n• n-1• 1


f (x) = xex2.

f (x) =∞

∑n=0

1an!

xpn ,

where an = and pn = .Correct Answers:

• n• 2*n+1


f (x) =Z x

0

tan−1 tt

dt.

f (x) =∞

∑n=1

(−1)en anxpn ,

where en = ,and an = ,and pn = .Correct Answers:

• n-1• 1/((2*n-1)**2)• 2*n-1

9. (1 pt) set8/p9-8.pgFind the sum of

∑∞n=1 n(n+1)xn =

for < x < .Correct Answers:

• 2*x/((1-x)**3)• -1• 1

10. (1 pt) set8/p9-9.pgFind the terms through x5 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(2t2) dt.

Find the MacLaurin polynomial of degree 7 for F(x).

Use this polynomial to estimate the value ofZ 0.78

0sin(2x2) dx.

Correct Answers:

• 2 * xˆ3 / 3 - 2ˆ3 * xˆ7 / 42• 0.282909773609691


2

Hsiang-Ping HuangMath 1220-90, Summer 2008WeBWorK Assignment 9 due 07/09/2008 at 10:59pmMDTConics.This assignment will cover the material from Sections 10.1 –10.4.

1. (1 pt) set9/p10-4.pgThe parabola y = x2 +10x has its focus at the point (b,c) where

b =

c =

Correct Answers:

• -5• -24.75

2. (1 pt) set9/p10-5.pgThe ellipse 6x2 + 2x + y2 = 1 has its center at the point (b,c)where

b =c =The length of the major diameter of this ellipse is

Correct Answers:

• -0.166666666666667• 0• 2.16024689946929

3. (1 pt) set9/p10-6.pgDetermine the distance D between the vertices of −9x2 +18x+4y2 +24y−9 = 0.

D =

Correct Answers:

• 6

4. (1 pt) set9/p10-7.pgMatch each equation below to the curve it represents. Eachanswer 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,

EQUATIONS1. 4x2−4y2 +8x+12y−5 = 02. 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.pgA bridge underpass in the shape of an elliptical arch, that is,half of an ellipse, is 30 feet wide and 14 feet high. An eightfoot wide rectangular truck is to drive (safely) underneath. Howhigh can it be?

h =


6. (1 pt) set9/p10-10.pgMatch 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 = 14+14cosθ

2. r = 6−6sinθ

3. r2 = 12cos2θ

4. r = 14sin2θ

5. r = 6cos3θ

6. r = 6θ,r > 0Correct Answers:

• C• C• M• F• T• O

1


2

WeBWorK demonstration assignmentThe main purpose of this WeBWorK set is to familiarize

yourself 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/demo pr1.pg

Evaluate the expression9(9−5) = .

Correct Answers:

• 36


Evaluate the expression2/(8+5) = .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:

• 0.153846153846154

3. (1 pt) setDemo/demo pr3.pgLet r = 7.

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

Correct Answers:

• 8.91267681314614• 0.181891363533595

4. (1 pt) setDemo/demo pr4.pgEnter here the expression 1

a + 1b .

Enter here the expression 1a+b .

Correct Answers:

• 1/a+1/b• 1/(a+b)

5. (1 pt) setDemo/demo pr5.pgEnter here the expression

a+12+b

Enter here the expression

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)

6. (1 pt) setDemo/demo pr6.pgEnter 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 a quadraticequation by the quadratic 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 1.61418970655352 Math 1220-90, Summer 2008 5… · · 2009-08-2131= (c) log 5...

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