Chapter 11 Practice Exercises

840

Chapter 11: Infinite Sequences and Series

Chapter 11 Practice Exercises

Convergent or Divergent Sequences

Which of the sequences whose nth terms appear in Exercises 1–18 converge, and which diverge? Find the limit of each convergent se- quence.

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

13. 14.

15. 16.

17. 18.

Convergent Series

Find the sums of the series in Exercises 19–24.

19. 20.

21. 22.

23. 24. a

q n=1s-1dⁿ3

4ⁿ a

q n=0e^-n

a

q n=3

-8 s4n- 3ds4n + 1d a

q n=1

9 s3n - 1ds3n + 2d

a

q n=2

-2 nsn+ 1d a

q n=3

1 s2n - 3ds2n - 1d

an = s-4dⁿ an= sn + 1d! n!

n!

an = 2ⁿ2n + 1 an= ns2^1>n - 1d

a_n = a3 nb^1>n a_n= A

n3ⁿ n

a_n = a1 + 1 nb^-n a_n= an- 5

n bⁿ

a_n = lns2n³ + 1d a_n= n + lnn n

n

a_n = lns2n + 1d a_n= lnsn²d n

n

an = sinnp an= sinnp

2

a_n = 1 + s0.9dⁿ a_n= 1 - 2ⁿ

2ⁿ

a_n = 1 - s-1dⁿ 2n a_n= 1+ s-1dⁿ

n

Convergent or Divergent Series

Which of the series in Exercises 25–40 converge absolutely, which converge conditionally, and which diverge? Give reasons for your answers.

25. 26. 27.

28. 29. 30.

31. 32.

33. 34.

35. 36.

37. 38.

39. 40.

Power Series

In Exercises 41–50, (a)find the series’ radius and interval of conver- gence. Then identify the values of xfor which the series converges (b) absolutely and (c)conditionally.

41. 42.

43. 44.

45. 46. a

q n=1

xⁿ 2n a

q n=1

xⁿ nⁿ

a

q n=0

sn + 1ds2x + 1dⁿ s2n + 1d2ⁿ a

q n=1

s-1d^n-1s3x - 1dⁿ n²

a

q n=1

sx - 1d^2n-2 s2n - 1d! a

q n=1

sx + 4dⁿ n3ⁿ

a

q n=2

1 n2n² - 1 a

q n=1

1

2nsn + 1dsn + 2d

a

q n=1

2ⁿ3ⁿ nⁿ a

q n=1

s-3dⁿ n!

a

q n=1

s-1dⁿsn² + 1d 2n² + n - 1 a

q n=1

n + 1 n!

a

q n=1

s-1dⁿ3n² n³ + 1 a

q n=1

s-1dⁿ n2n² + 1

a

q n=3

lnn lnslnnd a

q n=1

lnn n³

a

q n=2

1 nslnnd² a

q n=1

s-1dⁿ lnsn + 1d a

q n=1

1 2n³

a

q n=1

s-1dⁿ 2n a

q n=1

-5 a n

q n=1

1 2n

Chapter 11 Practice Exercises

841

47. 48.

49. 50.

Maclaurin Series

Each of the series in Exercises 51–56 is the value of the Taylor series at of a function ƒ(x) at a particular point. What function and what point? What is the sum of the series?

51.

52.

53.

54.

55.

56.

Find Taylor series at for the functions in Exercises 57–64.

57. 58.

59. 60.

61. 62.

63. 64.

Taylor Series

In Exercises 65–68, find the first four nonzero terms of the Taylor series generated by ƒat

65.

66.

67.

68.

Initial Value Problems

Use power series to solve the initial value problems in Exercises 69–76.

69. 70.

71. 72.

73. 74.

75. y¿ - y = x, ys0d = 1 76. y¿ - y = -x, ys0d = 2 y¿ + y = x, ys0d = 0 y¿ - y = 3x, ys0d = -1

y¿ + y = 1, ys0d = 0 y¿ + 2y = 0, ys0d = 3

y¿ - y = 0, ys0d = -3 y¿ + y = 0, ys0d = -1

ƒsxd = 1>x at x = a 7 0

ƒsxd = 1>sx + 1d at x = 3

ƒsxd = 1>s1 - xd at x = 2

ƒsxd = 23 + x² at x = -1 x = a.

e^-x2 e^spx>2d

cos25x cossx^5>2d

sin2x sinpx 3

1 1 + x³ 1

1 - 2x

x = 0 + s-1d^n-1 1

s2n - 1d

A

23

B

^{2n-1 + Á}

1 23 - 1

923 + 1 4523 - Á 1 + ln2 + sln2d²

2! + Á + sln2dⁿ n! + Á 1 - p²

9

#

2! + p⁴

81

#

4! - Á + s-1dⁿ p²ⁿ 3²ⁿs2nd! + Á p - p³

3! + p⁵

5! - Á + s-1dⁿ p²ⁿ⁺¹ s2n + 1d! + Á 2

3 - 4 18 + 8

81 - Á + s-1d^n-1 2ⁿ n3ⁿ + Á 1 - 1

4 + 1

16 - Á + s-1dⁿ1 4ⁿ + Á x = 0

a

q

n=1scoth ndxⁿ a

q

n=1scsch ndxⁿ

a

q n=0

s-1dⁿsx - 1d²ⁿ⁺¹ 2n + 1 a

q n=0

sn + 1dx^2n-1 3ⁿ

Nonelementary Integrals

Use series to approximate the values of the integrals in Exercises 77–80 with an error of magnitude less than (The answer section gives the integrals’ values rounded to 10 decimal places.)

77. 78.

79. 80.

Indeterminate Forms

In Exercises 81–86:

a. Use power series to evaluate the limit.

b. Then use a grapher to support your calculation.

81. 82.

83. 84.

85. 86.

87. Use a series representation of sin 3xto find values of rand sfor which

88. a. Show that the approximation in Section 11.10, Example 9, leads to the approximation

b. Compare the accuracies of the approximations

and by comparing the graphs of

and Describe what you find.

Theory and Examples

89. a. Show that the series

converges.

b. Estimate the magnitude of the error involved in using the sum of the sines through to approximate the sum of the series. Is the approximation too large, or too small? Give reasons for your answer.

90. a. Show that the series converges.

b. Estimate the magnitude of the error in using the sum of the tangents through to approximate the sum of the series. Is the approximation too large, or too small? Give reasons for your answer.

-tans1>41d a

q n=1 atan 1

2n - tan 1 2n + 1b n = 20

a

q n=1 asin 1

2n - sin 1 2n + 1b

gsxd = sinx- s6x>s6 + x²dd.

ƒsxd = sinx - x

sinx L 6x>s6+ x²d

sinx L x

6x>s6 + x²d.

sinx L cscxL 1>x + x>6

xlim:0 asin3x x³ + r

x² + sb = 0 .

ylim:0

y² cosy - cosh y

zlim:0

1 - cos²z lns1- zd + sinz

h:0lim

ssinhd>h - cosh h²

t:0lim a 1

2 - 2cost - 1 t²b

u:lim0

e^u - e^-u - 2u u - sinu lim

x:0

7sinx e^2x - 1

L

1>64 0

tan^-1x 2x L dx

1>2 0

tan^-1x x dx

L

1 0

xsinsx³d dx L

1>2 0

e^-x3 dx

10^-8.

T

T

T

T

91.Find the radius of convergence of the series

92.Find the radius of convergence of the series

93.Find a closed-form formula for the nth partial sum of the series and use it to determine the convergence or divergence of the series.

94.Evaluate by finding the limits as of

the series’nth partial sum.

95. a. Find the interval of convergence of the series

b. Show that the function defined by the series satisfies a differ- ential equation of the form

and find the values of the constants aand b.

96. a. Find the Maclaurin series for the function b. Does the series converge at Explain.

97.If and are convergent series of nonnegative numbers, can anything be said about Give reasons for your answer.

98.If and are divergent series of nonnegative numbers, can anything be said about Give reasons for your answer.

99.Prove that the sequence and the series both converge or both diverge.

100.Prove that converges if for all nand

converges.

101.(Continuation of Section 4.7, Exercise 27.) If you did Exercise 27 in Section 4.7, you saw that in practice Newton’s method stopped too far from the root of to give a use- ful estimate of its value, Prove that nevertheless, for any

starting value the sequence of ap-

proximations generated by Newton’s method really does con- verge to 1.

102.Suppose that are positive numbers satisfying the following conditions:

i.

ii. the series a2 + a4 + a8 + a16 + Ádiverges.

a1 Ú a2 Ú a3 Ú Á; a₁, a₂, a₃,Á, a_n

x₀, x₁, x₂,Á, x_n,Á x₀ Z 1 ,

x = 1 .

ƒsxd = sx - 1d⁴⁰ g^q_n=1a_n

an 7 0

g^q_n=1san>s1 + andd

g^q_k=1sx_k+1 - x_kd

5x_n6

g^q_n=1anbn? g^q_n=1b_n

g^q_n=1a_n

g^q_n=1anbn? g^q_n=1b_n

g^q_n=1a_n

x = 1 ?

x²>s1 + xd.

d²y

dx² = x^ay + b + 1

#

4

#

7

#

Á

#

s3n - 2d

s3nd! x³ⁿ + Á.

y = 1 + 1 6x³ + 1

180x⁶ + Á

n:q

g^q_k=2s1>sk²- 1dd

g^q_n=2lns1 - s1>n²dd a

q n=1

3

#

5

#

7

#

Á

#

s2n + 1d

4

#

9

#

14

#

Á

#

s5n- 1dsx - 1dⁿ. a

q n=1

2

#

5

#

8

#

^Á

#

s3n - 1d 2

#

4

#

6

#

^Á

#

s2nd xⁿ.

Show that the series

diverges.

103.Use the result in Exercise 102 to show that

diverges.

104.Suppose you wish to obtain a quick estimate for the value of There are several ways to do this.

a. Use the Trapezoidal Rule with to estimate

b. Write out the first three nonzero terms of the Taylor series at for to obtain the fourth Taylor polynomial P(x) for Use to obtain another estimate for

c.The second derivative of is positive for all Explain why this enables you to conclude that the Trapezoidal Rule estimate obtained in part (a) is too large.

(Hint: What does the second derivative tell you about the graph of a function? How does this relate to the trapezoidal approximation of the area under this graph?)

d. All the derivatives of are positive for Ex- plain why this enables you to conclude that all Maclaurin polynomial approximations to ƒ(x) for xin [0, 1] will be too

small. (Hint: )

e.Use integration by parts to evaluate

Fourier Series

Find the Fourier series for the functions in Exercises 105–108. Sketch each function.

105.

106.

107.

108.ƒsxd = ƒsinxƒ, 0 … x … 2p ƒsxd = ep - x, 0 … x … p

x- 2p, p 6 x … 2p ƒsxd = ex, 0 … x … p

1, p 6 x… 2p ƒsxd = e0, 0 … x … p

1, p 6 x… 2p

10¹x²e^x dx.

ƒsxd = P_nsxd + R_nsxd.

x 7 0 . ƒsxd = x²e^x

x 7 0 .

ƒsxd = x²e^x 10¹x²e^x dx.

10¹Psxd dx x²e^x.

x²e^x x = 0

10¹x²e^x dx.

n = 2 10¹x²e^x dx.

1 + a

q n=2

1 nlnn a₁

1 + a₂ 2 + a₃

3 + Á

842

Chapter 11: Infinite Sequences and Series

960

Chapter 13: Vector-Valued Functions and Motion in Space

Motion in a Cartesian Plane

In Exercises 1 and 2, graph the curves and sketch their velocity and acceleration vectors at the given values of t. Then write ain the form without finding Tand N, and find the value of at the given values of t.

1. and

2.

3. The position of a particle in the plane at time tis

Find the particle’s highest speed.

4. Suppose Show that the angle be-

tween rand anever changes. What isthe angle?

5. Finding curvature At point P, the velocity and acceleration of

a particle moving in the plane are and

Find the curvature of the particle’s path at P.

6. Find the point on the curve where the curvature is greatest.

7. A particle moves around the unit circle in the xy-plane. Its posi- tion at time t is where x and y are differentiable functions of t. Find dy dtif Is the motion clockwise, or counterclockwise?

8. You send a message through a pneumatic tube that follows the curve (distance in meters). At the point (3, 3), and Find the values of and at (3, 3).

9. Characterizing circular motion A particle moves in the plane so that its velocity and position vectors are always orthogonal.

Show that the particle moves in a circle centered at the origin.

10. Speed along a cycloid A circular wheel with radius 1 ft and center Crolls to the right along the x-axis at a half-turn per sec- ond. (See the accompanying figure.) At time tseconds, the posi- tion vector of the point Pon the wheel’s circumference is

r = spt - sinptdi + s1 - cosptdj.

a

#

j v

#

j a

#

i= -2 .

v

#

i = 4 9y = x³

v

#

i = y. r =>xi+ yj,

y = e^x a = 5i+ 15j.

v= 3i + 4j rstd = se^tcostdi + se^tsintdj.

r = 1

21 + t²

i + t

21 + t² j.

rstd =

A

23sect

B

ⁱ+

A

23tant

B

^{j, t} = 0 p>4 rstd = s4costdi +

A

22sint

B

^{j, t} = 0

k a = aTT + aNN

Chapter 13 Practice Exercises

a. Sketch the curve traced by Pduring the interval

b. Find vand aat and 3 and add these vectors to your sketch.

c. At any given time, what is the forward speed of the topmost point of the wheel? Of C?

Projectile Motion and Motion in a Plane

11. Shot put A shot leaves the thrower’s hand 6.5 ft above the ground at a 45° angle at 44 ft sec. Where is it 3 sec later?

12. Javelin A javelin leaves the thrower’s hand 7 ft above the ground at a 45° angle at 80 ft sec. How high does it go?

13. A golf ball is hit with an initial speed at an angle to the hori- zontal from a point that lies at the foot of a straight-sided hill that is inclined at an angle to the horizontal, where

Show that the ball lands at a distance

measured up the face of the hill. Hence, show that the greatest range that can be achieved for a given occurs when i.e., when the initial velocity vector bisects the angle between the vertical and the hill.

a = sf>2d + sp>4d,

y0

2y02cosa

gcos²f sinsa - fd, 0 6 f 6 a 6 p

2. f

a y0

>

>

x y

1 C

P

␲t r

0

t = 0, 1, 2 ,

0… t … 3 .

25.

26.

In Exercises 27 and 28, write ain the form at without finding Tand N.

27.

28.

29. Find T, N, B, and as functions of t if

30. At what times in the interval are the velocity and ac- celeration vectors of the motion

orthogonal?

31. The position of a particle moving in space at time is

Find the first time ris orthogonal to the vector

32. Find equations for the osculating, normal, and rectifying planes of the curve at the point (1, 1, 1).

33. Find parametric equations for the line that is tangent to the curve at

34. Find parametric equations for the line tangent to the helix at the point where t = p>4 .

A

^2cost

B

ⁱ +

A

^2sint

B

^j + tk

r(t) = t = 0 .

rstd = e^ti + ssintdj + lns1 - tdk rstd = ti + t²j + t³k

i - j.

rstd = 2i + a4sint

2bj + a3 - t pbk.

t Ú 0

s3sintdk rstd = i+ s5costdj +

0 … t … p

A

22cost

B

^j + ssintdk.

rstd = ssintdi+ t

k,

rstd = s2 + tdi + st + 2t²dj + s1+ t²dk rstd = s2 + 3t + 3t²di+ s4t + 4t²dj - s6costdk

t = 0 a = a_TT + a_NN rstd = s3cosh2tdi + s3sinh2tdj + 6tk, t = ln2 rstd = ti + 1

2e^2tj, t= ln2 35. The view from Skylab 4 What percentage of Earth’s surface area could the astronauts see when Skylab 4was at its apogee height, 437 km above the surface? To find out, model the visible surface as the surface generated by revolving the circular arc GT, shown here, about the y-axis. Then carry out these steps:

1. Use similar triangles in the figure to show that Solve for

2. To four significant digits, calculate the visible area as

3. Express the result as a percentage of Earth’s surface area.

y

x

437 G

6380 0

T

⎧⎨

⎩

S (Skylab)

y₀

x 兹^(6380)2 y² VA =

L

6380 y0

2px

C^{1 + a} dx dyb² dy.

y₀.

6380>s6380 + 437d.

y₀>6380 =

962

Chapter 13: Vector-Valued Functions and Motion in Space

for If we hold constant and let the limit is 0. On the

path however, we have and

In Exercises 51–56, find the limit of ƒas or show that the limit does not exist.

51. 52.

53. 54.

55.

56.

In Exercises 57 and 58, define ƒ(0, 0) in a way that extends ƒto be continuous at the origin.

57.

58. ƒsx, yd = 3x²y x²+ y²

ƒsx, yd = lna3x² - x²y²+ 3y² x² + y² b ƒsx, yd = x²- y²

x²+ y² ƒsx, yd = tan^-1aƒxƒ + ƒyƒ

x²+ y²b

ƒsx, yd = 2x x² + x + y² ƒsx, yd = y²

x²+ y²

ƒsx, yd = cosax³ - y³ x² + y²b ƒsx, yd = x³- xy²

x²+ y²

sx, yd:s0, 0d

= 2rcos²usinu

2r²cos⁴u = rsinu r²cos²u = 1.

ƒsrcosu, rsinud = rcosusin2u r²cos⁴u + srcos²ud²

rsinu = r²cos²u y = x²,

r:0, u

r Z 0.

Using the Definition

Each of Exercises 59–62 gives a function ƒ(x,y) and a positive num- ber In each exercise, show that there exists a such that for all (x,y),

59.

60.

61.

62.

Each of Exercises 63–66 gives a function ƒ(x,y,z) and a positive num- ber In each exercise, show that there exists a such that for all (x,y,z),

63.

64.

65.

66.

67. Show that is continuous at every point

68. Show that ƒsx, y, zd = x² + y² + z²is continuous at the origin.

sx₀, y₀, z₀d.

ƒsx, y, zd = x + y - z

ƒsx, y, zd = tan²x + tan²y + tan²z, P = 0.03 ƒsx, y, zd = x + y + z

x²+ y²+ z² + 1, P = 0.015 ƒsx, y, zd = xyz, P =0.008

ƒsx, y, zd = x² + y² + z², P = 0.015

2x² + y² + z² 6 d Q ƒƒsx, y, zd - ƒs0, 0, 0dƒ 6 P. d 7 0

P.

ƒsx, yd = sx + yd>s2+ cosxd, P = 0.02 ƒsx, yd = sx + yd>sx² + 1d, P =0.01

ƒsx, yd = y>sx² + 1d, P = 0.05

ƒsx, yd = x² + y², P = 0.01

2x² + y² 6 d Q ƒƒsx, yd - ƒs0, 0dƒ 6 P. d 7 0 P.

d - P

984

Chapter 14: Partial Derivatives

1060

Chapter 14: Partial Derivatives

Chapter 14 Practice Exercises

Domain, Range, and Level Curves

In Exercises 1–4, find the domain and range of the given function and identify its level curves. Sketch a typical level curve.

1. 2.

3. 4.

In Exercises 5–8, find the domain and range of the given function and identify its level surfaces. Sketch a typical level surface.

5. 6.

7.

8.

Evaluating Limits

Find the limits in Exercises 9–14.

9. 10.

11. 12.

13. 14.

By considering different paths of approach, show that the limits in Ex- ercises 15 and 16 do not exist.

15. 16.

17. Continuous extension Let for

Is it possible to define ƒ(0, 0) in a way that makes ƒcontinuous at the origin? Why?

18. Continuous extension Let

Is ƒcontinuous at the origin? Why?

Partial Derivatives

In Exercises 19–24, find the partial derivative of the function with respect to each variable.

19.

20.

21.

22. hsx, y, zd = sins2px + y - 3zd ƒsR₁, R₂, R₃d = 1

R₁ + 1 R₂ + 1

R₃ ƒsx, yd = 1

2lnsx²+ y²d + tan^-1 y x gsr, ud = rcosu + rsinu

ƒsx, yd = L

sinsx - yd

ƒxƒ + ƒyƒ , ƒxƒ + ƒyƒ Z 0 0, sx, yd = s0, 0d. sx, yd Z s0, 0d.

ƒsx, yd = sx²- y²d>sx² + y²d

sx,yd:s0,0dlim

xyZ0

x² + y² lim xy

sx,yd:s0,0d yZx²

y x²- y

P:s1,-1,-1dlim tan^-1sx + y + zd

P:s1, -1, edlim lnƒx + y + zƒ

sx,yd:s1,1dlim

x³y³ - 1 xy - 1

sx,ydlim:s1,1d

x - y x² - y²

sx,yd:slim0,0d 2+ y x + cosy

sx,yd:splim, ln2de^ycosx ksx, y, zd = 1

x²+ y² + z² + 1 hsx, y, zd = 1

x²+ y² + z²

gsx, y, zd = x²+ 4y²+ 9z² ƒsx, y, zd = x² + y² - z

gsx, yd = 2x² - y gsx, yd = 1>xy

ƒsx, yd = e^x+y ƒsx, yd = 9x² + y²

23. (the ideal gas law)

24.

Second-Order Partials

Find the second-order partial derivatives of the functions in Exercises 25–28.

25. 26.

27.

28.

Chain Rule Calculations

29. Find dw dt at if and

30. Find dw dt at if

and

31. Find and when and if

32. Find and when if

and

33. Find the value of the derivative of with

respect to ton the curve at

34. Show that if is any differentiable function of sand if then

Implicit Differentiation

Assuming that the equations in Exercises 35 and 36 define yas a dif- ferentiable function of x, find the value of dy dxat point P.

35.

36.

Directional Derivatives

In Exercises 37–40, find the directions in which ƒincreases and de- creases most rapidly at and find the derivative of ƒin each direc- tion. Also, find the derivative of ƒat in the direction of the vector v.

37.

38.

39.

v = 2i+ 3j + 6k

ƒsx, y, zd = lns2x + 3y + 6zd, P₀s-1, -1, 1d, ƒsx, yd = x²e^-2y, P₀s1, 0d, v = i + j

ƒsx, yd = cosxcosy, P₀sp>4, p>4d, v = 3i + 4j P₀

P0

2xy + e^x+y - 2 = 0, Ps0, ln2d 1 - x - y² - sinxy = 0, Ps0, 1d

>

0w 0x - 50w

0y = 0.

s = y + 5x,

w = ƒssd

t = 1 . x = cost, y = sint, z = cos2t

ƒsx, y, zd = xy + yz + xz x = 2e^ucosy.

ln21 + x² - tan^-1x

w = u= y = 0

0w>0y

0w>0u

x = r + sins, y = rs.

w = sins2x - yd, s = 0

r = p

0w>0s

0w>0r

z = pt.

y = t - 1 + lnt,

w = xe^y + ysinz - cosz, x = 22t, t = 1

>

lnst+ 1d.

y = w = sinsxy + pd, x = e^t,

t= 0

>

ƒsx, yd = y² - 3xy + cosy + 7e^y ƒsx, yd = x + xy - 5x³ + lnsx² + 1d

gsx, yd = e^x + ysinx gsx, yd = y + x

y ƒsr, l, T, wd = 1

2rlA pTw Psn, R, T, Vd = nRT

V

40.

41. Derivative in velocity direction Find the derivative of in the direction of the velocity vector of the helix

at

42. Maximum directional derivative What is the largest value that the directional derivative of can have at the point (1, 1, 1)?

43. Directional derivatives with given values At the point (1, 2), the function ƒ(x,y) has a derivative of 2 in the direction toward (2, 2) and a derivative of in the direction toward (1, 1).

a. Find and

b. Find the derivative of ƒat (1, 2) in the direction toward the point (4, 6).

44. Which of the following statements are true if ƒ(x,y) is differen- tiable at Give reasons for your answers.

a. If uis a unit vector, the derivative of ƒat in the direction of uis

b. The derivative of ƒat in the direction of uis a vector.

c. The directional derivative of ƒat has its greatest value in the direction of

d. At vector is normal to the curve

Gradients, Tangent Planes, and Normal Lines

In Exercises 45 and 46, sketch the surface together with at the given points.

45.

46.

In Exercises 47 and 48, find an equation for the plane tangent to the level surface at the point Also, find parametric equations for the line that is normal to the surface at

47.

48.

In Exercises 49 and 50, find an equation for the plane tangent to the surface at the given point.

49.

50.

In Exercises 51 and 52, find equations for the lines that are tangent and normal to the level curve at the point Then sketch the lines and level curve together with at

51. 52. y²

2 - x² 2 = 3

2, P₀s1, 2d y - sinx = 1, P₀sp, 1d

P₀.

§ƒ

P₀. ƒsx, yd = c

z = 1>sx²+ y²d, s1, 1, 1>2d

z = lnsx² + y²d, s0, 1, 0d z = ƒsx, yd

x²+ y² + z = 4, P₀s1, 1, 2d x²- y - 5z = 0, P₀s2, -1, 1d

P₀. P₀.

ƒsx, y, zd = c

y² + z²= 4; s2, ;2, 0d, s2, 0, ;2d x²+ y + z² = 0; s0, -1, ;1d, s0, 0, 0d

§ƒ

ƒsx, y, zd = c ƒsx, yd = ƒsx₀, y₀d.

§ƒ sx0, y₀d,

§ƒ .

sx₀, y₀d sx₀, y₀d

sƒ_xsx₀, y₀di + ƒ_ysx₀, y₀djd

#

u.

sx0, y₀d sx0, y₀d?

ƒ_ys1, 2d.

ƒ_xs1, 2d -2

ƒsx, y, zd = xyz t = p>3.

rstd = scos3tdi + ssin3tdj + 3tk ƒsx, y, zd = xyz

v = i + j + k

ƒsx, y, zd = x² + 3xy - z² + 2y + z + 4, P₀s0, 0, 0d,

Tangent Lines to Curves

In Exercises 53 and 54, find parametric equations for the line that is tangent to the curve of intersection of the surfaces at the given point.

53. Surfaces:

Point: (1, 1, 1 2) 54. Surfaces:

Point: (1 2, 1, 1 2)

Linearizations

In Exercises 55 and 56, find the linearization L(x,y) of the function ƒ(x,y) at the point Then find an upper bound for the magnitude of the error Ein the approximation over the rectangle R.

55.

56.

Find the linearizations of the functions in Exercises 57 and 58 at the given points.

57. at (1, 0, 0) and (1, 1, 0)

58. at and

Estimates and Sensitivity to Change

59. Measuring the volume of a pipeline You plan to calculate the volume inside a stretch of pipeline that is about 36 in. in diameter and 1 mile long. With which measurement should you be more careful, the length or the diameter? Why?

60. Sensitivity to change Near the point (1, 2), is

more sensitive to changes in xor to changes in y? How do you know?

61. Change in an electrical circuit Suppose that the current I(am- peres) in an electrical circuit is related to the voltage V(volts) and the resistance R(ohms) by the equation If the voltage drops from 24 to 23 volts and the resistance drops from 100 to 80 ohms, will Iincrease or decrease? By about how much? Is the change in Imore sensitive to change in the voltage or to change in the resistance? How do you know?

62. Maximum error in estimating the area of an ellipse If and to the nearest millimeter, what should you expect the maximum percentage error to be in the calculated area of the ellipse

63. Error in estimating a product Let and where uand yare positive independent variables.

a. If uis measured with an error of 2% and ywith an error of 3%, about what is the percentage error in the calculated value of y?

z = u + y, y = uy

x²>a² + y²>b² = 1 ? A = pab

b= 16 cm a = 10 cm

I = V>R.

x²- xy + y² - 3

ƒsx, yd = p>4, 0d sp>4, s0, 0, p>4d

ƒsx, y, zd = 22cosxsinsy + zd ƒsx, y, zd = xy + 2yz - 3xz R: ƒx - 1ƒ … 0.1, ƒy - 1ƒ … 0.2 ƒsx, yd = xy - 3y² + 2, P0s1, 1d R: `x - p

4` … 0.1, `y - p 4` … 0.1 ƒsx, yd = sinxcosy, P0sp>4, p>4d

ƒsx, yd L Lsx, yd P₀.

>

>

x + y² + z = 2, y = 1

>

x²+ 2y + 2z = 4, y = 1

Chapter 14 Practice Exercises

1061

b. Show that the percentage error in the calculated value of zis less than the percentage error in the value of y.

64. Cardiac index To make different people comparable in studies of cardiac output (Section 3.7, Exercise 25), researchers divide the measured cardiac output by the body surface area to find the cardiac index C:

The body surface area Bof a person with weight wand height his approximated by the formula

which gives Bin square centimeters when wis measured in kilo- grams and hin centimeters. You are about to calculate the cardiac index of a person with the following measurements:

Which will have a greater effect on the calculation, a 1-kg error in measuring the weight or a 1-cm error in measuring the height?

Local Extrema

Test the functions in Exercises 65–70 for local maxima and minima and saddle points. Find each function’s value at these points.

65.

66.

67.

68.

69.

70.

Absolute Extrema

In Exercises 71–78, find the absolute maximum and minimum values of ƒon the region R.

71.

R: The triangular region cut from the first quadrant by the line

72.

R: The rectangular region in the first quadrant bounded by the co- ordinate axes and the lines and

73.

R: The square region enclosed by the lines and 74.

R: The square region bounded by the coordinate axes and the lines in the first quadrant

x = 2, y = 2

ƒsx, yd = 2x + 2y - x² - y²

y = ;2 x = ;2

ƒsx, yd = y² - xy - 3y + 2x

y = 2 x = 4

ƒsx, yd = x² - y² - 2x + 4y + 1 x + y = 4

ƒsx, yd = x² + xy + y² - 3x + 3y ƒsx, yd = x⁴ - 8x²+ 3y²- 6y ƒsx, yd = x³ + y³ + 3x² - 3y² ƒsx, yd = x³ + y³ - 3xy + 15 ƒsx, yd = 2x³ + 3xy + 2y³

ƒsx, yd = 5x² + 4xy - 2y²+ 4x - 4y ƒsx, yd = x² - xy + y² + 2x + 2y - 4

Cardiac output: 7 L>min

Weight: 70 kg

Height: 180 cm

B = 71.84w^0.425h^0.725, C = cardiac output

body surface area.

75.

R: The triangular region bounded below by the x-axis, above by the line and on the right by the line

76.

R: The triangular region bounded below by the line above by the line and on the right by the line 77.

R: The square region enclosed by the lines and 78.

R: The square region enclosed by the lines and

Lagrange Multipliers

79. Extrema on a circle Find the extreme values of on the circle

80. Extrema on a circle Find the extreme values of on the circle

81. Extrema in a disk Find the extreme values of on the unit disk

82. Extrema in a disk Find the extreme values of on the disk

83. Extrema on a sphere Find the extreme values of on the unit sphere

84. Minimum distance to origin Find the points on the surface closest to the origin.

85. Minimizing cost of a box A closed rectangular box is to have volume The cost of the material used in the box is for top and bottom, for front and back, and for the remaining sides. What dimensions mini- mize the total cost of materials?

86. Least volume Find the plane that passes through the point (2, 1, 2) and cuts off the least volume from the first octant.

87. Extrema on curve of intersecting surfaces Find the extreme values of on the curve of intersection of the right circular cylinder and the hyperbolic cylinder

88. Minimum distance to origin on curve of intersecting plane and cone Find the point closest to the origin on the curve of in-

tersection of the plane and the cone

Partial Derivatives with Constrained Variables

In Exercises 89 and 90, begin by drawing a diagram that shows the relations among the variables.

89. If and find

a. b. c. a0w

0zb

y

a0w .

0zb

a0w x

0yb

z

z = x² - y² w = x²e^yz

2x² + 2y².

z² = x + y + z = 1

xz = 1.

x² + y² = 1 ƒsx, y, zd = xsy + zd

x>a + y>b+ z>c = 1 c cents>cm²

b cents>cm² a cents>cm²

V cm³. z² - xy = 4

x²+ y² + z² = 1.

x - y + z

ƒsx, y, zd = x² + y² … 9.

x²+ y² - 3x - xy

ƒsx, yd = x² + y² … 1.

x²+ 3y² + 2y

ƒsx, yd = x² + y² = 1.

ƒsx, yd = xy x² + y²= 1.

x³+ y²

ƒsx, yd = y = ;1 x = ;1

ƒsx, yd = x³ + 3xy + y³ + 1

y = ;1 x = ;1

ƒsx, yd = x³ + y³ + 3x² - 3y²

x = 2 y = x,

y = -2, ƒsx, yd = 4xy - x⁴ - y⁴ + 16

x = 2 y = x + 2,

ƒsx, yd = x² - y² - 2x + 4y

1062

Chapter 14: Partial Derivatives

1138

Chapter 15: Multiple Integrals

Chapter 15 Practice Exercises

Planar Regions of Integration

In Exercises 1–4, sketch the region of integration and evaluate the double integral.

1. 2.

3. 4.

Reversing the Order of Integration

In Exercises 5–8, sketch the region of integration and write an equiva- lent integral with the order of integration reversed. Then evaluate both integrals.

5. 6.

7. 8.

Evaluating Double Integrals

Evaluate the integrals in Exercises 9–12.

9. 10.

11. 12.

Areas and Volumes

13. Area between line and parabola Find the area of the region enclosed by the line and the parabola in the xy-plane.

14. Area bounded by lines and parabola Find the area of the “tri- angular” region in the xy-plane that is bounded on the right by the parabola on the left by the line and above by the line

15. Volume of the region under a paraboloid Find the volume under the paraboloid above the triangle enclosed by

the lines and in the xy-plane.

16. Volume of the region under parabolic cylinder Find the vol- ume under the parabolic cylinder above the region enclosed by the parabola and the line in the xy-plane.

Average Values

Find the average value of over the regions in Exercises 17 and 18.

17. The square bounded by the lines in the first quadrant 18. The quarter circle x² + y² … 1in the first quadrant

x = 1, y = 1 ƒsx, yd = xy

y = x y = 6 - x²

z = x² x + y = 2 y = x, x = 0 ,

z = x² + y² y = 4 .

x + y = 2 , y = x²,

y = 4 - x² y = 2x + 4

L

1 0L

1 2³y

2psinpx² x² dx dy L

8

0 L

2 2³x

dy dx y⁴+ 1

L

2 0L

1 y>2e^x2 dx dy L

1 0 L

2 2y

4cossx²d dx dy

L

2 0L

4-x² 0

2x dy dx L

3>2

0 L

29-4y² -29-4y²y dx dy

L

1 0L

x

x²2x dy dx L

4 0 L

sy-4d>2 -24-y dx dy

L

1 0L

2-2y 2y xy dx dy L

3>2

0 L

29-4t² -29-4t²t ds dt

L

1 0L

x³ 0

e^y>x dy dx L

10

1 L

1>y 0

ye^xy dx dy

Masses and Moments

19. Centroid Find the centroid of the “triangular” region bounded by the lines and the hyperbola in the xy-plane.

20. Centroid Find the centroid of the region between the parabola and the line in the xy-plane.

21. Polar moment Find the polar moment of inertia about the ori- gin of a thin triangular plate of constant density bounded by the y-axis and the lines and in the xy-plane.

22. Polar moment Find the polar moment of inertia about the cen- ter of a thin rectangular sheet of constant density bounded by the lines

a. in the xy-plane

b. in the xy-plane.

(Hint:Find Then use the formula for to find and add the two to find ).

23. Inertial moment and radius of gyration Find the moment of inertia and radius of gyration about the x-axis of a thin plate of constant density covering the triangle with vertices (0, 0), (3, 0), and (3, 2) in the xy-plane.

24. Plate with variable density Find the center of mass and the moments of inertia and radii of gyration about the coordinate axes of a thin plate bounded by the line and the parabola in the xy-plane if the density is

25. Plate with variable density Find the mass and first moments about the coordinate axes of a thin square plate bounded by the lines in the xy-plane if the density is

26. Triangles with same inertial moment and radius of gyration Find the moment of inertia and radius of gyration about the x-axis of a thin triangular plate of constant density whose base lies along the interval [0, b] on the x-axis and whose vertex lies on the line above the x-axis. As you will see, it does not matter where on the line this vertex lies. All such triangles have the same moment of inertia and radius of gyration about the x-axis.

Polar Coordinates

Evaluate the integrals in Exercises 27 and 28 by changing to polar coordinates.

27.

28.

29. Centroid Find the centroid of the region in the polar coordinate plane defined by the inequalities 0 … r … 3, -p>3 … u … p>3.

L

1 -1L

21-y²

-21-y²lnsx² + y² + 1d dx dy L

1 -1L

21-x² -21-x²

2 dy dx s1 + x² + y²d² y = h

d x²+ y² + 1>3.

dsx, yd = x = ;1, y = ;1

dsx, yd = x + 1 .

y = x² y = x

d I0

I_y I_x

I_x. x = ;a, y = ;b x = ;2, y = ;1

d = 1 y = 4

y = 2x

d = 3 x + 2y = 0

x + y² - 2y = 0

xy = 2 x = 2, y = 2

30. Centroid Find the centroid of the region in the first quadrant bounded by the rays and and the circles and

31. a. Centroid Find the centroid of the region in the polar coordi- nate plane that lies inside the cardioid and out- side the circle

b. Sketch the region and show the centroid in your sketch.

32. a. Centroid Find the centroid of the plane region defined by the polar coordinate inequalities

How does the centroid move as

b. Sketch the region for and show the centroid in your sketch.

33. Integrating over lemniscate Integrate the function

over the region enclosed by one loop of the lemniscate

34. Integrate over

a. Triangular region The triangle with vertices (0, 0), (1, 0), b. First quadrant The first quadrant of the xy-plane.

Triple Integrals in Cartesian Coordinates

Evaluate the integrals in Exercises 35–38.

35.

36.

37.

38.

39. Volume Find the volume of the wedge-shaped region enclosed

on the side by the cylinder on

the top by the plane and below by the xy-plane.

40. Volume Find the volume of the solid that is bounded above by the cylinder on the sides by the cylinder

and below by the xy-plane.

y² = 4,

x²+ z = 4- x²,

y x

z

– x ⫽ –cos y

z ⫽ –2x

␲

2 ␲

2

z

y x

x²⫹ y²⫽ 4 z ⫽ 4 ⫺ x² z = -2x,

x = -cosy, -p>2… y … p>2, L

e 1L

x 1L

z 0

2y z³dy dz dx L

1 0 L

x²

0 L

x+y 0

s2x - y - zd dz dy dx L

ln7 ln6 L

ln2

0 L

ln5 ln4

e^sx+y+zd dz dy dx

L

p

0 L

p

0 L

p 0

cossx + y + zd dx dy dz

A

^1, 23

B

^.

ƒsx, yd = 1>s1 + x² + y²d²

sx² + y²d² - sx² - y²d = 0.

1>s1 + x² + y²d²

ƒsx, yd = a = 5p>6

a:p^-? s0 6 a … pd.

-a … u … a 0 … r … a,

r = 1.

r = 1 + cosu r = 3.

r = 1 u = p>2

u = 0

41. Average value Find the average value of

over the rectangular solid in the first octant bounded by the coordinate planes and the planes

42. Average value Find the average value of over the solid sphere (spherical coordinates).

Cylindrical and Spherical Coordinates

43. Cylindrical to rectangular coordinates Convert

to (a) rectangular coordinates with the order of integration dz dx dyand (b)spherical coordinates. Then (c)evaluate one of the integrals.

44. Rectangular to cylindrical coordinates (a)Convert to cylin- drical coordinates. Then (b)evaluate the new integral.

45. Rectangular to spherical coordinates (a)Convert to spherical coordinates. Then (b)evaluate the new integral.

46. Rectangular, cylindrical, and spherical coordinates Write an iterated triple integral for the integral of over the region in the first octant bounded by the cone the cylinder and the coordinate planes in (a)rectangular coordinates, (b)cylindrical coordinates, and (c)spherical coordinates. Then (d)find the integral of ƒby evaluating one of the triple integrals.

47. Cylindrical to rectangular coordinates Set up an integral in rectangular coordinates equivalent to the integral

Arrange the order of integration to be zfirst, then y, then x.

48. Rectangular to cylindrical coordinates The volume of a solid is

a. Describe the solid by giving equations for the surfaces that form its boundary.

b. Convert the integral to cylindrical coordinates but do not evaluate the integral.

49. Spherical versus cylindrical coordinates Triple integrals involving spherical shapes do not always require spherical coordi- nates for convenient evaluation. Some calculations may be accomplished more easily with cylindrical coordinates. As a case in point, find the volume of the region bounded above by the

L

2 0 L

22x-x²

0 L

24-x²-y²

-24-x²-y² dz dy dx.

L

p>2

0 L

23

1 L

24-r² 1

r³ssinucosudz² dz dr du. x² + y² = 1 , z = 2x² + y²,

ƒsx, y, zd = 6 + 4y L

1 -1L

21-x² -21-x²L

1

2x²+y² dz dy dx L

1 0L

21-x² -21-x²L

sx²+y²d -sx²+y²d

21xy² dz dy dx L

2p

0 L

22

0 L

24-r² r

3 dz r dr du, r Ú 0 r … a

r z = 1.

x = 1, y = 3, 30xz 2x² + y

ƒsx, y, zd = Chapter 15 Practice Exercises

1139