Calc ii complete_assignments

CALCULUS II Assignment Problems

Paul Dawkins

Calculus II

Table of Contents Preface ........................................................................................................................................... iii Outline ........................................................................................................................................... iii Integration Techniques ................................................................................................................. 5

Introduction ................................................................................................................................................ 5 Integration by Parts .................................................................................................................................... 5 Integrals Involving Trig Functions ............................................................................................................. 6 Trig Substitutions ....................................................................................................................................... 8 Partial Fractions ........................................................................................................................................11 Integrals Involving Roots ..........................................................................................................................12 Integrals Involving Quadratics ..................................................................................................................13 Integration Strategy ...................................................................................................................................13 Improper Integrals .....................................................................................................................................14 Comparison Test for Improper Integrals ...................................................................................................15 Approximating Definite Integrals .............................................................................................................16

Applications of Integrals ............................................................................................................. 17 Introduction ...............................................................................................................................................17 Arc Length ................................................................................................................................................17 Surface Area ..............................................................................................................................................18 Center of Mass ..........................................................................................................................................20 Hydrostatic Pressure and Force .................................................................................................................20 Probability .................................................................................................................................................23

Parametric Equations and Polar Coordinates .......................................................................... 25 Introduction ...............................................................................................................................................25 Parametric Equations and Curves .............................................................................................................25 Tangents with Parametric Equations .........................................................................................................27 Area with Parametric Equations ................................................................................................................28 Arc Length with Parametric Equations .....................................................................................................28 Surface Area with Parametric Equations...................................................................................................29 Polar Coordinates ......................................................................................................................................30 Tangents with Polar Coordinates ..............................................................................................................32 Area with Polar Coordinates .....................................................................................................................32 Arc Length with Polar Coordinates ...........................................................................................................32 Surface Area with Polar Coordinates ........................................................................................................33 Arc Length and Surface Area Revisited ....................................................................................................33

Sequences and Series ................................................................................................................... 34 Introduction ...............................................................................................................................................34 Sequences ..................................................................................................................................................34 More on Sequences ...................................................................................................................................35 Series – The Basics ...................................................................................................................................36 Series – Convergence/Divergence ............................................................................................................37 Series – Special Series ..............................................................................................................................38 Integral Test ..............................................................................................................................................39 Comparison Test / Limit Comparison Test ...............................................................................................40 Alternating Series Test ..............................................................................................................................41 Absolute Convergence ..............................................................................................................................42 Ratio Test ..................................................................................................................................................42 Root Test ...................................................................................................................................................43 Strategy for Series .....................................................................................................................................44 Estimating the Value of a Series ...............................................................................................................44 Power Series ..............................................................................................................................................45 Power Series and Functions ......................................................................................................................45 Taylor Series .............................................................................................................................................46 Applications of Series ...............................................................................................................................47

© 2007 Paul Dawkins i http://tutorial.math.lamar.edu/terms.aspx

Calculus II

Binomial Series .........................................................................................................................................48 Vectors .......................................................................................................................................... 48

Introduction ...............................................................................................................................................48 Vectors – The Basics .................................................................................................................................49 Vector Arithmetic .....................................................................................................................................49 Dot Product ...............................................................................................................................................49 Cross Product ............................................................................................................................................49

Three Dimensional Space............................................................................................................ 49 Introduction ...............................................................................................................................................49 The 3-D Coordinate System ......................................................................................................................50 Equations of Lines ....................................................................................................................................50 Equations of Planes ...................................................................................................................................50 Quadric Surfaces .......................................................................................................................................50 Functions of Several Variables .................................................................................................................50 Vector Functions .......................................................................................................................................51 Calculus with Vector Functions ................................................................................................................51 Tangent, Normal and Binormal Vectors ...................................................................................................51 Arc Length with Vector Functions ............................................................................................................51 Curvature ...................................................................................................................................................51 Velocity and Acceleration .........................................................................................................................51 Cylindrical Coordinates ............................................................................................................................51 Spherical Coordinates ...............................................................................................................................51

© 2007 Paul Dawkins ii http://tutorial.math.lamar.edu/terms.aspx

Calculus II

Preface Here are a set of problems for my Calculus I notes. These problems do not have any solutions available on this site. These are intended mostly for instructors who might want a set of problems to assign for turning in. I try to put up both practice problems (with solutions available) and these problems at the same time so that both will be available to anyone who wishes to use them.

Outline Here is a list of sections for which problems have been written. Integration Techniques

Integration by Parts Integrals Involving Trig Functions Trig Substitutions Partial Fractions Integrals Involving Roots Integrals Involving Quadratics Using Integral Tables Integration Strategy Improper Integrals Comparison Test for Improper Integrals Approximating Definite Integrals

Applications of Integrals

Arc Length – No problems written yet. Surface Area – No problems written yet. Center of Mass – No problems written yet. Hydrostatic Pressure and Force – No problems written yet. Probability – No problems written yet.

Parametric Equations and Polar Coordinates

Parametric Equations and Curves – No problems written yet. Tangents with Parametric Equations – No problems written yet. Area with Parametric Equations – No problems written yet. Arc Length with Parametric Equations – No problems written yet. Surface Area with Parametric Equations – No problems written yet. Polar Coordinates – No problems written yet. Tangents with Polar Coordinates – No problems written yet. Area with Polar Coordinates – No problems written yet. Arc Length with Polar Coordinates – No problems written yet. Surface Area with Polar Coordinates – No problems written yet. Arc Length and Surface Area Revisited – No problems written yet.

© 2007 Paul Dawkins iii http://tutorial.math.lamar.edu/terms.aspx

Calculus II

Sequences and Series

Sequences – No problems written yet. More on Sequences – No problems written yet. Series – The Basics – No problems written yet. Series – Convergence/Divergence – No problems written yet. Series – Special Series – No problems written yet. Integral Test – No problems written yet. Comparison Test/Limit Comparison Test – No problems written yet. Alternating Series Test – No problems written yet. Absolute Convergence – No problems written yet. Ratio Test – No problems written yet. Root Test – No problems written yet. Strategy for Series – No problems written yet. Estimating the Value of a Series – No problems written yet. Power Series – No problems written yet. Power Series and Functions – No problems written yet. Taylor Series – No problems written yet. Applications of Series – No problems written yet. Binomial Series – No problems written yet.

Vectors

Vectors – The Basics – No problems written yet. Vector Arithmetic – No problems written yet. Dot Product – No problems written yet. Cross Product – No problems written yet.

Three Dimensional Space

The 3-D Coordinate System – No problems written yet. Equations of Lines – No problems written yet. Equations of Planes – No problems written yet. Quadric Surfaces – No problems written yet. Functions of Several Variables – No problems written yet. Vector Functions – No problems written yet. Calculus with Vector Functions – No problems written yet. Tangent, Normal and Binormal Vectors – No problems written yet. Arc Length with Vector Functions – No problems written yet. Curvature – No problems written yet. Velocity and Acceleration – No problems written yet. Cylindrical Coordinates – No problems written yet. Spherical Coordinates – No problems written yet.

© 2007 Paul Dawkins iv http://tutorial.math.lamar.edu/terms.aspx

Calculus II

Integration Techniques

Introduction Here are a set of problems for which no solutions are available. The main intent of these problems is to have a set of problems available for any instructors who are looking for some extra problems. Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the problems although this will vary from section to section. Here is a list of topics in this chapter that have problems written for them. Integration by Parts Integrals Involving Trig Functions Trig Substitutions Partial Fractions Integrals Involving Roots Integrals Involving Quadratics Using Integral Tables Integration Strategy Improper Integrals Comparison Test for Improper Integrals Approximating Definite Integrals

Integration by Parts Evaluate each of the following integrals. 1. 78 tt dt∫ e

2. ( ) ( )2

121 3 sinx x dx

π

π−∫

3. 2 2 4

1

ww dw−∫ e

4. ( ) ( )3 2

12 ln 4x x dx−∫

5. ( ) ( )6 3 cos 1 4z z dz+ +∫

© 2007 Paul Dawkins 5 http://tutorial.math.lamar.edu/terms.aspx

Calculus II

6. ( )22 cos 9y y dy∫ 7. ( ) ( )23 sinz z z dz+∫

8. ( )3 3lnx x dx∫

9. ( )2 7 12 ww w dw−−∫ e 10. ( )29 sec 2t t dt∫

11. ( )8

0sin 4x x dx

π−∫ e

12. ( )18 tan 2y dy−∫ 13. ( )6 cos 2t t dt∫e 14. ( )13sin 10x dx−−∫ 15. ( )3 sin 2z z dz− +∫e

16. 90 17 1

12 xx dx+

−∫ e

17. ( )11 69 cos 1t t dt−∫

18. 7

4 1x dx

x +

⌠⌡

19. ( ) ( )4 1

25 sinx x dx+∫ 20. 5 12 zz dz−∫ e 21. ( ) ( )3 55 2 cos 3w w w dw+ −∫

Integrals Involving Trig Functions Evaluate each of the following integrals.


Calculus II

1. ( ) ( )5 2cos 2 sin 2t t dt∫ 2. ( )3cos 12x dx∫ 3. ( ) ( )2 4cos sinz z dz∫

4. ( ) ( )3

5 63 34 4sin cosw w dw

π

π

∫

5. ( ) ( )11 3

0cos 5 sin 5z z dz

π

∫

6. ( )2sin 7x dx∫

7. ( ) ( )6 3 3

0tan 8 sec 8x x dx

π

∫

8. ( ) ( )8 51 1

2 2sec tant t dt∫ 9. ( ) ( )2 3sec 9 tan 9z z dz∫

10. ( ) ( )34

6 4sec 10 tan 10t t dtπ

π

∫

11. ( ) ( )12 6tan 2 sec 2w w dw∫ 12. ( ) ( )2 6cot 3 csc 3x x dx∫

13. ( ) ( )23

3

3 31 14 4csc cotw w dw

π

π∫

14. ( )4csc 6w dw∫ 15. ( ) ( )12 5csc cotx x dx∫ 16. ( )cot x dx∫ 17. ( )3cot x dx∫


Calculus II

18. ( )csc x dx∫ 19. ( )3csc x dx∫

20. ( ) ( )4

2sin 8 cos 15x x dx

−∫

21. ( ) ( )cos 2 cos 24x x dx∫ 22. ( ) ( )sin 13 sin 9z z dz∫

23. ( )( )

5

3

cos 2sin 2

tdt

t⌠⌡

24. ( )( )

3

2

sin 2cos 2

xdx

x−−

⌠⌡

25. ( )( )

6 12

8 12

sectan

zdz

z⌠⌡

26. ( )( )

5

2

tansec

xdx

x⌠⌡

27. ( )

( )

5

2

1 9cos 8sin 8

wdw

w+⌠

⌡

28. ( )( ) ( )3 23 7cos sinx x dx+∫ 29. ( ) ( )3 2sin 9 sec 9y y dy∫ 30. ( ) ( )5 5tan cosz z dz∫ 31. ( ) ( ) ( )3 3tan 2 sin 2 sec 2t t t dt − ∫

Trig Substitutions For problems 1 – 15 use a trig substitution to eliminate the root.


Calculus II

1. 264 1t +

2. 24 49z −

3. 27 w−

4. ( )72216 81x−

5. 26 9y+

6. ( )3221 8z−

7. ( )29 16 3 1x− −

8. ( )( )522211 1t+ +

9. ( )2144 8 3z + −

10. 24 24 43x x− +

11. ( )11222 24 36z z− +

12. 24 10 5t t− − −

13. ( )29sin 4 1t −

14. 336 9 z− e 15. 16x + For problems 16 – 42 use a trig substitution to evaluate the given integral.

16. 5 23 16x x dx−∫

17. ( )523 225 81t t dt+∫


Calculus II

18.

14 3

20 1 9

w dww−

⌠⌡

19. ( )

32

5

29 25

z dzz −

⌠⌡

20. 1 3 2

349 4y y dy

−

−−∫

21. 5

2 21

54

dxx x +

⌠⌡

22. 2

2

3 4t dtt−⌠

⌡

23. 5

28 1w dww +

⌠⌡

34. 2

3

15x dxx−⌠

⌡

35. ( )6 2

23 6 5

dxx x x− − + −

⌠⌡

36. ( ) ( )

322 2

1

1 2 4 34dz

z z z+ + −

⌠⌡

37. ( )

2

6

4 16 192

y ydy

y− +

−

⌠⌡

38. ( )12 3

29

4

8 7

tdt

t t

−

− +

⌠⌡

39. 6 2

05 10 6x x dx+ +∫

40. 7 49x x dx−∫


Calculus II

41. 12

64 1

t

tdt

−

⌠⌡

ee

42. ( ) ( ) ( )3 2sin cos 16 cosz z z dz+∫

Partial Fractions Evaluate each of the following integrals.

1. 2

912

dzz z−

⌠⌡

2. 2

714 40

x dxx x+ +

⌠⌡

3. 4

20

8 12 15 8

y dyy y

−− −

⌠⌡

4. ( )( )( )

291 3 5 4

w dww w w

−+ − +

⌠⌡

5. 8

3 21

122 63

dzz z z− −

⌠⌡

6. ( )( )( )

27 24 2 3 2 1

x x dxx x x

+− + +

⌠⌡

7. ( )( )2

4 102 1x dx

x x+

− −

⌠⌡

8. 2

4 31

246

dtt t−

⌠⌡

9. ( ) ( )2 2

10 21 3

z dzz z

+

+ −

⌠⌡

10. ( )( )

2

2

87 16w w dw

w w+

− +

⌠⌡


Calculus II

11. ( )( )2

6 72 1 4 1

y dyy y

−+ +

⌠⌡

12. ( )( )3 2

2 2

8 5 72 102 9

t t t dtt t− + −+ +

⌠⌡

13. ( )( )

3 2

2 2

16 6 12 219 4 3

w w w dww w+ + ++ +

⌠⌡

14. ( )

4 3

22

5 20 16

4

x x x dxx x

+ + +

+

⌠⌡

15. 2

2

62 21

z dzz z

−+ −

⌠⌡

16. 3

2

430

x x dxx x

−− −

⌠⌡

17. ( )( )

3

283 1

t dtt t

−

− +

⌠⌡

18. 6 5 4 3 2

4 2

6 3 10 9 12 273

x x x x x x dxx x

− + − − + −+

⌠⌡

Integrals Involving Roots Evaluate each of the following integrals.

1. 5

4 3dz

z− −⌠⌡

2. 1

4 3dx

x x+ −⌠⌡

3. 4

7 1dt

t t− + +⌠⌡


Calculus II

4. 2 3 2 1 9

z dzz z+ + −

⌠⌡

5. 2 6 3 5

w dww w− − −

⌠⌡

6. ( )cos x dx∫

Integrals Involving Quadratics Evaluate each of the following integrals.

1. 2

110 18x dx

x x+

+ +⌠⌡

2. 2

153 4 4

dyy y− +

⌠⌡

3. 2

12 91 4 4

z dzz z−

− −⌠⌡

4. ( )22

3 7

12 40

t dtt t

−

+ +

⌠⌡

5. ( )22

11 4

3 6

w dww w

+

+ −

⌠⌡

6. ( )

722

3

2 10 4dx

x x+ +

⌠⌡

Integration Strategy Problems have not yet been written for this section. I was finding it very difficult to come up with a good mix of “new” problems and decided my time was better spent writing problems for later sections rather than trying to come up with a sufficient number of problems for what is essentially a review section. I intend to come back at a later date when I have more time to devote to this section and add problems then.


Calculus II

Improper Integrals Determine if each of the following integrals converge or diverge. If the integral converges determine its value.

1. 2

42 4 6x x dx

∞− +∫

2. 5

0

14 20

dww−

⌠⌡

3. 2

61

34 2

dzz− −

⌠⌡

4. 0 2 3xx dx+

−∞∫ e

5. 2 3

0

xx dx∞ +∫ e

6. 22

11

dxx

∞

+⌠⌡

7. 3

20

14

dzz z−

⌠⌡

8. 1

2 1x dx

x−∞ +⌠⌡

9. 2

21

12 3

dyy y− − −

⌠⌡

10. ( )0

cos w dw−∞∫

11. ( )2

10

15 2

dzz

∞

−

⌠⌡

12. 3

4 1z dz

z

∞

−∞ +⌠⌡


Calculus II

13. 4

1

16 2

dyy −

⌠⌡

14. 5

31

12

dww−

⌠⌡

15.

11

2

2

xdx

x−

⌠⌡

e

16. 32 xx dx

∞

−∞∫ e

17. ( )32 1

y dyy

∞

−∞ +

⌠⌡

18. 3 3

20 9

w dww−

⌠⌡

19. 1

23

12

dww w− +

⌠⌡

20.

1

2

0

xdx

x

∞⌠⌡

e

21. ( ) 2

0

1ln

dzz z

∞

⌠⌡

22. 0

11

dww

∞

−⌠⌡

Comparison Test for Improper Integrals Use the Comparison Test to determine if the following integrals converge or diverge.

1. 5

4

12

dzz

∞

−⌠⌡


Calculus II

2. 6

0 2w dw

w

∞

+

⌠⌡

3. ( )4

2

12 3

dww

∞

+

⌠⌡

4. 2

12

4 27

y y dyy

∞− +−

⌠⌡

5. ( )2

1ln

dxx

∞⌠⌡

Hint : Sketch the graph of y x= and ( )lny x= on the same axis system.

6. ( )2

32

4sinz zdz

z

∞ −⌠⌡

7. ( )( )

23

220

2 sincos

x xdx

x x

∞+

−

⌠⌡

8. 30 1

zz dzz

∞ −

+⌠⌡

e

Approximating Definite Integrals For each of the following integrals use the given value of n to approximate the value of the definite integral using

(a) the Midpoint Rule, (b) the Trapezoid Rule, and (c) Simpson’s Rule.

Use at least 6 decimal places of accuracy for your work.

1. ( )4 2

2sin 2x dx

−+∫ using 6n =

2. 4 3 4

06x dx+∫ using 6n =

3. ( )5 cos

1

x dx∫ e using 8n =


Calculus II

4. ( )

5

3

11 ln

dxx−

⌠⌡

using 6n =

5. ( ) ( )1 2

3sin cosx x dx

−∫ using 8n =

Applications of Integrals

Introduction Here are a set of problems for which no solutions are available. The main intent of these problems is to have a set of problems available for any instructors who are looking for some extra problems. Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the problems although this will vary from section to section. Here is a list of topics in this chapter that have problems written for them. Arc Length – No problems written yet. Surface Area – No problems written yet. Center of Mass – No problems written yet. Hydrostatic Pressure and Force – No problems written yet. Probability – No problems written yet.

Arc Length 1. Set up, but do not evaluate, an integral for the length of 14 9y x= − , 22 31y− ≤ ≤ using,

(a) 2

1 dyds dxdx = +

(b) 2

1 dxds dydy

= +

2. Set up, but do not evaluate, an integral for the length of 2 yx = e , 1 0y− ≤ ≤ using,

(a) 2

1 dyds dxdx = +


Calculus II

(b) 2

1 dxds dydy

= +

3. Set up, but do not evaluate, an integral for the length of ( )tan 2y x= , 30 x π≤ ≤ using,

(a) 2

1 dyds dxdx = +

(b) 2

1 dxds dydy

= +

4. Set up, but do not evaluate, an integral for the length of 2

29 116x y+ = .

5. For 6 1x y= + , 2 8y− ≤ ≤ (a) Use an integral to find the length of the curve. (b) Verify your answer from part (a) geometrically. 6. Determine the length of 4

3 2y x= + , 0 9x≤ ≤ .

7. Determine the length of ( )328 3y x= + ,

3 32 211 27y≤ ≤ .

8. Determine the length of ( )3210 2x y= − , 1 2y− ≤ ≤ .

9. Determine the length of ( )22 1x y= + − , 2 5y≤ ≤ . 10. Determine the length of ( )23 2y x= + , 1 4x≤ ≤ .

Surface Area 1. Set up, but do not evaluate, an integral for the surface area of the object obtained by rotating

7 2y x= + , 5 0y− ≤ ≤ about the x-axis using,

(a) 2

1 dyds dxdx = +

(b) 2

1 dxds dydy

= +


Calculus II

2. Set up, but do not evaluate, an integral for the surface area of the object obtained by rotating 51 2y x= + , 0 1x≤ ≤ about the x-axis using,

(a) 2

1 dyds dxdx = +

(b) 2

1 dxds dydy

= +

3. Set up, but do not evaluate, an integral for the surface area of the object obtained by rotating

2 yx = e , 1 0y− ≤ ≤ about the y-axis using,

(a) 2

1 dyds dxdx = +

(b) 2

1 dxds dydy

= +

4. Set up, but do not evaluate, an integral for the surface area of the object obtained by rotating

( )12cosy x= , 0 x π≤ ≤ about

(a) the x-axis (b) the y-axis. 5. Set up, but do not evaluate, an integral for the surface area of the object obtained by rotating

3 7x y= + , 0 1y≤ ≤ about (a) the x-axis (b) the y-axis. 6. Find the surface area of the object obtained by rotating 1

4 6 2y x= + , 522 2y≤ ≤ about the

x-axis. 7. Find the surface area of the object obtained by rotating 4y x= − , 1 6x≤ ≤ about the y-axis. 8. Find the surface area of the object obtained by rotating 22 5x y= + , 1 2x− ≤ ≤ about the y-axis. 9. Find the surface area of the object obtained by rotating 21x y= − , 0 3y≤ ≤ about the x-axis. 10. Find the surface area of the object obtained by rotating 2 yx = e , 1 0y− ≤ ≤ about the y-axis. 11. Find for the surface area of the object obtained by rotating ( )1

2cosy x= , 0 x π≤ ≤ about the x-axis.


Calculus II

Center of Mass Find the center of mass for each of the following regions. 1. The region bounded by 3y x= , 2x = − and the x-axis. 2. The triangle with vertices (-2, -2), (4, -2) and (4,4). 3. The region bounded by ( )22y x= − and 4y = . 4. The region bounded by ( )cosy x= and the x-axis between 2 2xπ π− ≤ ≤ . 5. The region bounded by 2y x= and 6y x= − . 6. The region bounded by 2xy = e and the x-axis between 1 1x− ≤ ≤ . 7. The region bounded by 2xy = e and ( )cosy xπ= − between 1 1

2 2x− ≤ ≤ .

Hydrostatic Pressure and Force Find the hydrostatic force on the following plates submerged in water as shown in each image. In each case consider the top of the blue “box” to be the surface of the water in which the plate is submerged. Note as well that the dimensions in many of the images will not be perfectly to scale in order to better fit the plate in the image. The lengths given in each image are in meters. 1.

2.


Calculus II

3.

4.

5.


Calculus II

6.

7.


Calculus II

Probability 1. Let,

( ) ( )23 2 if 0 240 otherwise

x x xf x

− ≤ ≤=

(a) Show that ( )f x is a probability density function.

(b) Find ( )0.25P X ≤ .

(c) Find ( )1.4P X ≥ .

(d) Find ( )0.1 1.2P X≤ ≤ . (e) Determine the mean value of X.

2. Let,

( ) ( )( )2

4 if 1 6ln 3 4

0 otherwise

xx xf x

≤ ≤ +=


(b) Find ( )1P X ≤ .

(c) Find ( )5P X ≥ .

(d) Find ( )1 5P X≤ ≤ . (e) Determine the mean value of X.

3. Let,

( )1 1 sin if 0 10

10 20 otherwise

x xf x

ππ + − ≤ ≤ =


(b) Find ( )3P X ≤ .

(c) Find ( )5P X ≥ .

(d) Find ( )2.5 7P X≤ ≤ . (e) Determine the mean value of X.


Calculus II

4. The probability density function of the life span of a battery is given by the function below, where t is in years.

( )0.81.25 if 0

0 if 0

t tf t

t

− ≥=

<

e

(a) Verify that ( )f t is a probability density function. (b) What is the probability that a battery will have a life span less than 10 months? (c) What is the probability that a battery will have a life span more than 2 years? (d) What is the probability that a battery will have a life span between 1.5 and 4 years? (e) Determine the mean value of the life span of the batteries.

5. The probability density function of the successful outcome from some experiment is given by the function below, where t is in minutes.

( )636 if 0

0 if 0

tt tf t

t

− ≥=

<

e

(a) Verify that ( )f t is a probability density function. (b) What is the probability of a successful outcome happening in less than 12 minutes? (c) What is the probability of a successful outcome happening after 25 minutes? (d) What is the probability of a successful outcome happening between 10 and 75 minutes? (e) What is the mean time of a successful outcome from the experiment?

6. Determine the value of c for which the function below will be a probability density function.

( ) ( )4 512 if 0 12

0 otherwise

c x x xf x

− ≤ ≤=

7. Use the function below for this problem.

( )1

00 0

ax

c xf xx

− ≥= <

e

(a) Determine the value of c for which this function will be a probability density function. (b) Using the value of c found in the first part determine the mean value of the probability

density function.


Calculus II

Parametric Equations and Polar Coordinates

Introduction Here are a set of problems for which no solutions are available. The main intent of these problems is to have a set of problems available for any instructors who are looking for some extra problems. Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the problems although this will vary from section to section. Here is a list of topics in this chapter that have problems written for them. Parametric Equations and Curves Tangents with Parametric Equations Area with Parametric Equations Arc Length with Parametric Equations Surface Area with Parametric Equations Polar Coordinates Tangents with Polar Coordinates Area with Polar Coordinates Arc Length with Polar Coordinates Surface Area with Polar Coordinates Arc Length and Surface Area Revisited

Parametric Equations and Curves For problems 1 – 9 eliminate the parameter for the given set of parametric equations, sketch the graph of the parametric curve and give any limits that might exist on x and y. 1. 22 4 7x t y t t= + = − + 2. 22 4 7 3 1x t y t t t= + = − + − ≤ ≤ 3. 21 3 2x t y t= − = + 4. 21 3 2 2 3x t y t t= − = + − ≤ ≤ 5. 1

2 6 0x t y t t t= = − − ≥ 6. 1

2 6 8 20x t y t t t= = − − ≤ ≤


Calculus II

7. ( ) ( ) 4 86cos 4 2sin 4x t y t tπ π= − = − ≤ ≤ 8. ( ) ( )1 1

2 21 3sin 4cosx t y t= − = 9. 2 26 7 4 3t tx y− −= − = +e e 10. Answer each of the questions about the following set of parametric equations ( ) ( )3cos 3sin 0 2x at y at t π= = ≤ ≤

(a) Sketch the graph of the parametric curve for 1a = . (b) Sketch the graph of the parametric curve for 6a = . (c) Sketch the graph of the parametric curve for 1

5a = . (d) In general, for 0a > , how does the value of a affect the graph of the parametric curve?

For problems 11 – 21 the path of a particle is given by the set of parametric equations. Completely describe the path of the particle. To completely describe the path of the particle you will need to provide the following information. (i) A sketch of the parametric curve (including direction of motion) based on the equation you get by eliminating the parameter. (ii) Limits on x and y. (iii) A range of t’s for a single trace of the parametric curve. (iv) The number of traces of the curve the particle makes if an overall range of t’s is provided in the problem. 11. ( ) ( )1 1

3 36cos 2 sin 0 75x t y t t π= = + ≤ ≤ 12. ( ) ( )7 3sin 2 4 2cos 2x t y t= − = + 13. ( ) ( )2 2 5

66cos 3 2 3sin 3 3x t y t tπ π= = − − ≤ ≤

14. ( ) ( )2 7 714 3 42 cos sinx t y t= + =

15. ( ) ( )36 sin 4 2sin 4 127 201x t y t tπ π= − = − ≤ ≤ 16. ( ) ( )21 1

6 63 cos 4 cos 90 216x t y t tπ π= + = + − ≤ ≤ 17. 4 122t tx y−= =e e 18. 3 61 1 6t tx y t= + = − ≤ ≤e e

19. ( ) ( ) 21 ln ln 0x t y t t= − = >


Calculus II

20. ( ) ( )1 12 2cos secx t y t tπ π= = − < <

21. ( ) ( ) ( )2 91 17

4 4sin 2 sin 2 4sin 2x t y t t tπ π= = − − ≤ ≤ For problems 22 – 27 write down a set of parametric equations for the given equation that meets the given extra conditions (if any). 22. ( ) ( )2 2sin 3 cosx x x= − +

23. ( )

2

6cos 89x

yx x

−=

+

24. 2 2 100x y+ = and the parametric curve resulting from the parametric equations should be at

( )0,10 when 0t = and the curve should have a clockwise rotation. 25. 2 2 100x y+ = and the parametric curve resulting from the parametric equations should be at

( )0,10 when 0t = and the curve should have a counter clockwise rotation.

26. 2

2 125x y+ = and the parametric curve resulting from the parametric equations should be at

( )5,0− when 0t = and the curve should have a counter clockwise rotation.

27. 2

2 125x y+ = and the parametric curve resulting from the parametric equations should be at

( )5,0− when 0t = and the curve should have a clockwise rotation. 28. Eliminate the parameter for the following set of parametric equations and identify the resulting equation. ( ) ( )cos sinx h a t x k b tω ω= + = +

Tangents with Parametric Equations

For problems 1 – 3 compute dydx

and 2

2

d ydx

for the given set of parametric equations.

1. 2 6 27 9 2x t t y t t= − = + 2. ( ) ( ) ( )tan 2 12 3sin 2 sec 2 4x t y t t t= − = + +


Calculus II

3. ( ) ( ) ( )2 4 2ln 3 8 ln 6lnx t t y t t= + = − For problems 4 – 7 find the equation of the tangent line(s) to the given set of parametric equations at the given point. 4. ( ) ( )3 cos 4 sin 2 6x t t y t tπ= + = + + at 3t = − 5. 2 3 22 1 7 8x t t y t t t= + − = + + at 1t = 6.

4 29 3 26 3 18 2t tx y t t t−= − = + − +e at ( )5,2 7. ( ) 2

26sin 2 8x t y t tπ= = + − at ( )6,7− For problems 8 and 9 find the values of t that will have horizontal or vertical tangent lines for the given set of parametric equations. 8. 3 2 4 3 25 1 8 3x t t t y t t t= − + + = + + 9. ( )22 2 1

27 10sin 1tx t y t−= + = −e

Area with Parametric Equations For problems 1 – 3 determine the area of the region below the parametric curve given by the set of parametric equations. For each problem you may assume that each curve traces out exactly once from right to left for the given range of t. For these problems you should only use the given parametric equations to determine the answer. 1. 2 25 1 40 2 5x t t y t t= + − = − − ≤ ≤ 2. ( ) ( ) ( )2 2

23cos sin 6 cos 0x t t y t tπ= − = + − ≤ ≤

3. 1 1 14 4 22 4 6 1t t tx y t= − = + − − ≤ ≤e e e

Arc Length with Parametric Equations For all the problems in this section you should only use the given parametric equations to determine the answer. For problems 1 – 5 determine the length of the parametric curve given by the set of parametric equations. For these problems you may assume that the curve traces out exactly once for the given range of t’s.


Calculus II

1. 3 9 10 15 5 8x t y t t= + = − − ≤ ≤

2. ( )3 32 26 3 3 2 1x t y t t= + = − − ≤ ≤

3. 24 3 3 0 5x t y t t= − = ≤ ≤ 4. ( )23 6 1 1 3x t y t t= + = + − ≤ ≤ 5. 2 41 5 0 1x t y t t= − = + ≤ ≤ For problems 6 and 7 a particle travels along a path defined by the following set of parametric equations. Determine the total distance the particle travels and compare this to the length of the parametric curve itself. 6. ( ) ( )2 2 31 42

4 56cos 3 2 3sin 3x t y t tπ π= = − − ≤ ≤ 7. ( ) ( )2 171 1 21

6 6 2 23 cos 4 cosx t y t tπ π= + = + − ≤ ≤ For problems 8 – 10 set up, but do not evaluate, an integral that gives the length of the parametric curve given by the set of parametric equations. For these problems you may assume that the curve traces out exactly once for the given range of t’s. 8. ( ) ( )cos 2 sin 3 2 3x t t y t t= = ≤ ≤ 9. ( ) ( )1 sin 1 sin 1 4tx t y t−= − + = ≤ ≤e

10. ( ) 1ln 2 1 27

x t y tt

= + = − ≤ ≤+

Surface Area with Parametric Equations For all the problems in this section you should only use the given parametric equations to determine the answer. For problems1 – 4 determine the surface area of the object obtained by rotating the parametric curve about the given axis. For these problems you may assume that the curve traces out exactly once for the given range of t’s. 1. Rotate 2 23 2 0 5x t y t t= − = + ≤ ≤ about the x-axis. 2. Rotate 28 6 3 0x t y t t= − = + − ≤ ≤ about the y-axis.


Calculus II

3. Rotate 2 4 2 0 2x t y t t= = − ≤ ≤ about the y-axis.

4. Rotate 122 4 1 2tx t y t−= + = − ≤ ≤e about the x-axis.

For problems 5 – 7 set up, but do not evaluate, an integral that gives the surface area of the object obtained by rotating the parametric curve about the given axis. For these problems you may assume that the curve traces out exactly once for the given range of t’s. 5. Rotate ( )cos 22 1 2 0tx y t t= + = + − ≤ ≤e about the x-axis. 6. Rotate ( ) ( ) ( )2cos 2cos 2 sin 0 1x t y t t t= = − ≤ ≤ about the y-axis. 7. Rotate ( )2 ln 3 0 2tx t y t−= = + ≤ ≤e about the x-axis.

Polar Coordinates

1. For the point with polar coordinates ( )3

79, π− determine three different sets of coordinates for

the same point all of which have angles different from 23π− and are in the range 2 2π θ π− ≤ ≤ .

2. For the point with polar coordinates ( )2

37, π− determine three different sets of coordinates for

the same point all of which have angles different from 37π and are in the range 2 2π θ π− ≤ ≤ .

3. The polar coordinates of a point are ( )14, 2.48 . Determine the Cartesian coordinates for the point. 4. The polar coordinates of a point are ( )3

10 , 5.29− − . Determine the Cartesian coordinates for the point. 5. The Cartesian coordinate of a point are ( )3,5− . Determine a set of polar coordinates for the point. 6. The Cartesian coordinate of a point are ( )4, 7− . Determine a set of polar coordinates for the point. 7. The Cartesian coordinate of a point are ( )3, 12− − . Determine a set of polar coordinates for the point. For problems 8 and 9 convert the given equation into an equation in terms of polar coordinates. 8. 2 2 27 8 3 6 6x y y x y+ = − −


Calculus II

9. 22 2

7 98

y yx y x

= ++ −

For problems 10 – 12 convert the given equation into an equation in terms of Cartesian coordinates.

10. 8sin 2cosr

rθ θ− =

11. 3 csc 5cos 6r θ θ= − 12. ( )28 sin 2r r θ− = 13. 2 cos 2 sinr a bθ θ= + For problems 14 – 27 sketch the graph of the given polar equation. 14. 7 sinr θ− =

15. 57πθ =

16. 95πθ = −

17. cos 4r θ = 18. 6sinr θ= 19. 100r = 20. 24cosr θ= 21. 15sinr θ= − 22. 4 12cosr θ= + 23. 7 7sinr θ= − 24. 1 3sinr θ= + 25. 5 4cosr θ= − 26. 8 3sinr θ= +


Calculus II

27. 1 cosr θ= −

Tangents with Polar Coordinates

1. Find the tangent line to ( )sin 3r θ θ= at 2πθ = .

2. Find the tangent line to ( ) ( )cos 2 sinr θ θ= − at 4πθ = − .

3. Find the tangent line to ( )2cosr θ θ= − at θ π= .

Area with Polar Coordinates 1. Find the area inside the inner loop of 3 10sinr θ= + . 2. Find the area inside the inner loop of 5 12cosr θ= + . 3. Find the area inside the graph of 8 cosr θ= + and to the right of the y-axis. 4. Find the area inside the graph of 5 4sinr θ= − and the below the x-axis. 5. Find the area that is inside 4r = and outside 4 2sinr θ= − . 6. Find the area that is inside 7 3cosr θ= − and outside 4r = . 7. Find the area that is inside 6 6cosr θ= + and outside 4 3cosr θ= − . 8. Find the area that is inside 4 2sinr θ= + and outside 5 sinr θ= − . 9. Find the area that is inside 5 sinr θ= − and outside 4 2sinr θ= + . 10. Find the area that is inside both 6 4sinr θ= − and 5r = . 11. Find the area that is inside both 3 2cosr θ= + and 3 cosr θ= − .

Arc Length with Polar Coordinates For problems 1 – 3 determine the length of the given polar curve. For these problems you may assume that the curve traces out exactly once for the given range of θ .


Calculus II

1. 1

cosr

θ= , 30 πθ≤ ≤

2. 2r θ= , 0 3θ π≤ ≤ 3. 6cos 3sinr θ θ= − , 0 θ π≤ ≤ For problems 4 – 6 set up, but do not evaluate, an integral that gives the length of the given polar curve. For these problems you may assume that the curve traces out exactly once for the given range of θ . 4. ( )2sinr θ= , 0 θ π≤ ≤ 5. ( )cos 1 sinr θ= + , 0 2θ π≤ ≤

6. 14 cosr θ θ−= e , 0 3θ π≤ ≤

Surface Area with Polar Coordinates For problems 1 – 4 set up, but do not evaluate, an integral that gives the surface area of the curve rotated about the given axis. For these problems you may assume that the curve traces out exactly once for the given range of θ .

1. ( )14cosr θ−= e , 20 πθ≤ ≤ rotated about the x-axis.

2. sinr θ θ= , 20 πθ≤ ≤ rotated about the y-axis. 3. ( ) ( )cos sin 2r θ θ= , 60 πθ≤ ≤ rotated about the x-axis. 4. sinr θ θ= + , 2

π θ π≤ ≤ rotated about the y-axis.

Arc Length and Surface Area Revisited Problems have not yet been written for this section and probably won’t be to be honest since this is just a summary section.


Calculus II

Sequences and Series

Introduction Here are a set of problems for which no solutions are available. The main intent of these problems is to have a set of problems available for any instructors who are looking for some extra problems. Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the problems although this will vary from section to section. Here is a list of topics in this chapter that have problems written for them. Sequences More on Sequences Series – The Basics Series – Convergence/Divergence Series – Special Series Integral Test Comparison Test/Limit Comparison Test Alternating Series Test Absolute Convergence Ratio Test Root Test Strategy for Series Estimating the Value of a Series Power Series Power Series and Functions Taylor Series Applications of Series Binomial Series

Sequences For problems 1 – 3 list the first 5 terms of the sequence.

1. ( ){ }2

11 n

n

n ∞

=− e

2. 20

6 89 n

nn n

∞

=

− +


Calculus II

3. ( ) 1

1

42 3n n

n

∞

+

=

− +

For problems 4 – 10 determine if the given sequence converges or diverges. If it converges what is its limit?

4. 3

20

52 8 1 n

nn n

∞

=

+ − +

5. 4 2

4 211

6 99 8 7 n

n nn n

∞

=

+ − +

6. ( ) ( )7

2

2

1 2 89

n

n

nn

∞+

=

− − +

7. ( ){ } 0cos

nnπ

∞

=

8. ( )

2

1ln 6

n

nn

∞

=

+

9. 1

3cos1 nn

∞

=

+

10. ( ) ( ){ } 0ln 4 1 ln 2 7

nn n

∞

=+ − +

More on Sequences For each of the following problems determine if the sequence is increasing, decreasing, not monotonic, bounded below, bounded above and/or bounded.

1. 31

11 nn

∞

=

+

2. { }3

0n

n ∞

=e


Calculus II

3. ( ){ }0

3 n

n

∞

=−

4. ( ){ } 4sin

nn

∞

=

5. 2

1lnnn

∞

=

4. 1

31 3 n

nn

∞

=

− −

5. 0

2 14 3 n

nn

∞

=

+ +

6. ( ){ }3

1n

nn∞

=− e

7. 2

21

403 1 n

nn n

∞

=

+ + +

8. 20

5100,000 n

nn

∞

=

+ +

Series – The Basics For problems 1 – 4 perform an index shift so that the series starts at 4n = .

1. 8

25n

nn

∞

=

+−∑

2. 2

3 12

31 4

n

nn

+∞

+= −∑

3. ( )2 3

02 n

n

n∞

−

=

−∑ e

4. ( )3 2

25

12 1

n

n

nn n

+∞

=

−− +∑


Calculus II

5. Strip out the first 4 terms from the series 2

03 6n n

n

∞−

=∑ .

6. Strip out the first 2 terms from the series 23

41n n n

∞

= + +∑ .

7. Given that 4

4 0.02257n

nn

∞−

=

=∑ determine the value of 1

4 n

nn

∞−

=∑ .

8. Given that 33

1 0.47199n

nn

∞

=

+=∑ determine the value of 3

5

1n

nn

∞

=

+∑ .

Series – Convergence/Divergence For problems 1 – 4 compute the first 3 terms in the sequence of partial sums for the given series.

1. 0

11 3n

n

∞

= +∑

2. ( )1

2 3n n

n

∞

=

−∑

3. 1

12n

nn

∞

=

+∑

4. 010

n

∞

=∑

For problems 5 – 7 assume that the nth term in the sequence of partial sums for the series 0

nn

a∞

=∑ is

given below. Determine if the series 0

nn

a∞

=∑ is convergent or divergent. If the series is convergent

determine the value of the series. 5. ( )2 24n

ns n n −= + e

6. 2

2

1 2 34 5 6n

n nsn n+ +

=+ +


Calculus II

7. ( )ln 2n

nsn

=+

8. Let 7 84 3n

ndn

−=

+

(a) Does the sequence { } 0n nd ∞

= converge or diverge?

(b) Does the series 0

nn

d∞

=∑ converge or diverge?

9. Let 1n

nd n −= + e

(c) Does the sequence { } 0n nd ∞

= converge or diverge?

(d) Does the series 0

nn

d∞

=∑ converge or diverge?

For problems 10 – 12 show that the series is divergent.

10. 2

21

9 21 4n

nn n

∞

=

−+ +∑

11. 0

5 13

n

nn

∞

=

+∑

12. ( )1cos

nn

∞

=∑

Series – Special Series For each of the following series determine if the series converges or diverges. If the series converges give its value.

1. 22

2n n n

∞

=

−+∑

2. 1

12n n

∞

=∑


Calculus II

3. 3

15 4n n

n

∞+

=∑

4. 1 10

34 5n n

n

∞

+ −=∑

5. 1

114n n

∞

=∑

6. 20

75 6n n n

∞

= + +∑

7. 1 2 2 3

14 3n n

n

∞+ −

=∑

8. 1 2 2 3

44 3n n

n

∞+ −

=∑

9. 23

51n n

∞

= −∑

10. 24

14 3n n n

∞

= − +∑

11. 3

2 30

52

n

nn

+∞

+=∑

Integral Test For each of the following series determine if the series converges or diverges.

1. ( )3

2

4n n

∞

=∑

2. 7 2 61

1n n n

∞

=∑


Calculus II

3. 2

12 1n n

∞

= +∑

4. ( )2

0

810n n

∞

= +∑

5. 20

11n n

∞

= +∑

6. ( )

2

ln

n

nn

∞

=∑

7. 3

40 1n

nn

∞

= +∑

8. ( )

3

240 1n

n

n

∞

= +∑

9. 24

46n n n

∞

= − −∑

10. 21

95 4n n n

∞

= + +∑

11. 0n

nn∞

−

=∑ e

Comparison Test / Limit Comparison Test For each of the following series determine if the series converges or diverges.

1. 10

32

n

nn

n∞

+=

−∑

2. 51

4 32n

nn

∞

=

−∑


Calculus II

3. ( )( )4

12 1 3n n n

∞

= − −∑

4. ( )2

8

ln

n

nn

∞

=∑

5. ( )3

1

41n

nn

∞

= +∑

6. ( )2

0

41 n

n

nn

∞

=

−+

∑ e

7. ( )2

22

2 cos 5

1n

n

n n

∞

=

+

− −∑

8. 3

3

13n

nn n

∞

=

−

+ +∑

9. 2

40

3 7 13n

n nn n

∞

=

+ −− +∑

10. ( )( ) ( )( )

21

1 sin 1 sin8 1n

n nn n

∞

=

− +

+ +∑

Alternating Series Test For each of the following series determine if the series converges or diverges.

1. ( ) 7

20

13

n

n n

+∞

=

−+∑

2. ( ) 2

0

13 3

n

nn n

−∞

=

−+∑

3. ( ) 1

3 22

14 8

n

n n n

−∞

=

−+ +∑


Calculus II

4. ( ) ( )1

12 6 1n

n n

∞

= − +∑

5. ( )

23

4 cos2 1n

n nn

π∞

= +∑

6. ( ) 10 2

3 20

14

n

n

nn n

−∞

=

−+ +∑

7. ( ) ( )5

21

1 2 18

n

n

nn

+∞

=

− ++∑

Absolute Convergence

For each of the following series determine if they are absolutely convergent or conditionally convergent.

1. ( ) 2

32

11

n

n n

−∞

=

−

−∑

2. ( )

43

cos

n

nnπ∞

=∑

3. ( ) 3

20

14 3

n

n

nn

−∞

=

−+∑

4. ( ) ( )6 2

41

1 1n

n

nn

+∞

=

− +∑

5. ( )3

32

cos

n

nn n

∞

= −∑

Ratio Test For each of the following series determine if the series converges or diverges.


Calculus II

1. ( )

3 2

0 1 !n

n nn

∞

=

++∑

2. ( )1

1

25 1n

n

nn

∞

−=

++∑

3. ( )( )0

2 1 !3 !n

nn

∞

=

−∑

4. ( )4

20

23 1

n

n n

+∞

=

−+∑

5. ( )

121 2

22

43 3

n

nn

nn

+∞

+= +∑

6. ( ) ( )2 2

1

41 1n

n n n

∞

+= − + +∑

7. ( )( )

2

3 2 23

6 44 2

n

nn

nn

−∞

−=

−

−∑

8. ( ) ( )

22

1 11

n

n

nn

∞

=

− ++∑

Root Test For each of the following series determine if the series converges or diverges.

1. ( )

1 3

1 20

23

n

nn

−∞

−= −∑

2. 2

22

5 2 13 3

n

n

n nn n

−∞

=

− + + −

∑

3. 2

3 5

11

10 n

nnn

+∞

+=∑


Calculus II

4. 12

1

6 92 4

n

n

nn

∞

=

− +

∑

Strategy for Series Problems have not yet been written for this section. I was finding it very difficult to come up with a good mix of “new” problems and decided my time was better spent writing problems for later sections rather than trying to come up with a sufficient number of problems for what is essentially a review section. I intend to come back at a later date when I have more time to devote to this section and add problems then.

Estimating the Value of a Series

1. Use the Integral Test and 8n = to estimate the value of ( )2

2

1lnn n n

∞

=∑ .

2. Use the Integral Test and 14n = to estimate the value of ( )

50 2

1

2 1n n

∞

= +∑ .

3. Use the Comparison Test and 10n = to estimate the value of 20

3 22 1

n

nn

∞

=

−+∑ .

4. Use the Comparison Test and 8n = to estimate the value of 2

41 1n

nn

∞

= +∑ .

5. Use the Alternating Series Test and 12n = to estimate the value of ( )

0

11

n

n n

∞

=

−+∑ .

6. Use the Alternating Series Test and 18n = to estimate the value of ( )

1

13 4

n

n

nn

∞

=

−+∑ .

7. Use the Ratio Test and 10n = to estimate the value of ( )1 2

21

27n

n

nn

+∞

=

−∑ .


Calculus II

8. Use the Ratio Test and 5n = to estimate the value of ( )1 1 !n

nn

∞

= −∑ .

Power Series For each of the following power series determine the interval and radius of convergence.

1. ( )

( )1

3 20

6 42

nn

nn

x−∞

−=

+−

∑

2. ( )

30

10 1 n

nn

xn

∞

+=

−∑

3. ( )

( ) ( )0

3 !6 9

2 2 !n

n

nx

n

∞

=

−−∑

4. ( ) ( )

2

0

15 20

4 1

nn

n

nx

n

∞

=

−+

+∑

5. ( ) ( )

1

0

1 87

4

n nn

nx

n

+∞

=

−−

+∑

6. ( )

( )1 2

1 2 20

2 4 23

nn

nn

xn

+∞

+=

+−

∑

7. ( )( )

2

50

1216

n

nn

x +∞

+=

+

−∑

Power Series and Functions For problems 1 – 4 write the given function as a power series and give the interval of convergence.

1. ( )1 8

xf xx

=−


Calculus II

2. ( )2

7

121 6

xf xx

−=

+

3. ( )7

38xf x

x=

+

4. ( )5 2

24 3xf x

x=

−

For problems 5 & 6 give a power series representation for the derivative of the following function.

5. ( )10

22xg x

x=

−

6. ( )5

6

91 3

xg xx

=+

For problems 7 & 8 give a power series representation for the integral of the following function.

7. ( ) 73 6

xh xx

=−

8. ( )4

92xh x

x=

+

Taylor Series For problems 1 – 3 use one of the Taylor Series derived in the notes to determine the Taylor Series for the given function. 1. ( ) ( )4sinf x x= about 0x = 2. ( ) 4 129 xf x x −= e about 0x = 3. ( ) ( )2 56 cos 7f x x x= about 0x = For problem 4 – 13 find the Taylor Series for each of the following functions. 4. ( ) ( )sinf x x= about 3

2x π=


Calculus II

5. ( ) 1 8xf x −= e about 3x = 6. ( ) ( )ln 1f x x= − about 2x = − 7. ( ) ( )ln 2 9f x x= + about 1x =

8. ( )( )7

16

f xx

=−

about 4x =

9. ( )( )2

14 9

f xx

=+

about 2x = −

10. ( ) 2f x x= + about 1x = 11. ( ) 1 4f x x= − about 3x = − 12. ( ) 23 10f x x x= − − + about 8x = − 13. ( ) 3 29 10 2f x x x x= + − + about 3x =

Applications of Series 1. Determine a Taylor Series about 0x = for the following integral.

( )cos 1x

dxx

−⌠⌡

2. Write down ( )2T x , ( )4T x and ( )6T x for the Taylor Series of ( ) ( )sinf x x= about 3

2x π=

. Graph all three of the Taylor polynomials and ( )f x on the same graph for the interval

[ ], 2π π− . 3. Write down ( )2T x , ( )3T x and ( )4T x for the Taylor Series of ( ) ( )ln 1f x x= − about

2x = − . Graph all three of the Taylor polynomials and ( )f x on the same graph for the interval

[ ]4,0− .


Calculus II

4. Write down ( )1T x , ( )3T x and ( )5T x for the Taylor Series of ( )( )7

16

f xx

=−

about 4x =

. Graph all three of the Taylor polynomials and ( )f x on the same graph for the interval [ ]1,5 . 5. Write down ( )2T x , ( )4T x and ( )6T x for the Taylor Series of ( ) 2f x x= + about 1x = .

Graph all three of the Taylor polynomials and ( )f x on the same graph for the interval [ ]2,4− .

Binomial Series For problems 1 – 3 use the Binomial Theorem to expand the given function. 1. ( )41 2

2 3 x+ 2. ( )51 8x− 3. ( )63 7x+ For problems 4 – 6 write down the first four terms in the binomial series for the given function. 4. ( ) 48 4x −+ 5. 16 25x+ 6. 4 16 4x−

Vectors

Introduction Here are a set of problems for which no solutions are available. The main intent of these problems is to have a set of problems available for any instructors who are looking for some extra problems. Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the problems although this will vary from section to section.


Calculus II

Here is a list of topics in this chapter that have problems written for them. Vectors – The Basics – No problems written yet. Vector Arithmetic – No problems written yet. Dot Product – No problems written yet. Cross Product – No problems written yet.

Vectors – The Basics Problems have not yet been written for this section.

Vector Arithmetic Problems have not yet been written for this section.

Dot Product Problems have not yet been written for this section.

Cross Product Problems have not yet been written for this section.

Three Dimensional Space

Introduction Here are a set of problems for which no solutions are available. The main intent of these problems is to have a set of problems available for any instructors who are looking for some extra problems. Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the problems although this will vary from section to section. Here is a list of topics in this chapter that have problems written for them.


Calculus II

The 3-D Coordinate System – No problems written yet. Equations of Lines – No problems written yet. Equations of Planes – No problems written yet. Quadric Surfaces – No problems written yet. Functions of Several Variables – No problems written yet. Vector Functions – No problems written yet. Calculus with Vector Functions – No problems written yet. Tangent, Normal and Binormal Vectors – No problems written yet. Arc Length with Vector Functions – No problems written yet. Curvature – No problems written yet. Velocity and Acceleration – No problems written yet. Cylindrical Coordinates – No problems written yet. Spherical Coordinates – No problems written yet.

The 3-D Coordinate System Problems have not yet been written for this section.

Equations of Lines Problems have not yet been written for this section.

Equations of Planes Problems have not yet been written for this section.

Quadric Surfaces Problems have not yet been written for this section.

Functions of Several Variables Problems have not yet been written for this section.


Calculus II

Vector Functions Problems have not yet been written for this section.

Calculus with Vector Functions Problems have not yet been written for this section.

Tangent, Normal and Binormal Vectors Problems have not yet been written for this section.

Arc Length with Vector Functions Problems have not yet been written for this section.

Curvature Problems have not yet been written for this section.

Velocity and Acceleration Problems have not yet been written for this section.

Cylindrical Coordinates Problems have not yet been written for this section.

Spherical Coordinates Problems have not yet been written for this section.


Calculus II


Technology

Calc ii complete_assignments