SECTION 5.4
5.4
INDEFINITE INTEGRALS AND THE NET CHANGE THEOREM
S Click here for solutions.
1– 2 Verify by differentiation that the formula is correct.
x
sin 2x
1. y sin x dx 苷 ⫺
⫹C
2
4
13.
y ⱍ x ⫺ x ⱍ dx
14.
y ⱍx
15.
y
16.
y
4
17.
y
18.
y
0
19.
y
x4 ⫺ 1
dx
x2 ⫹ 1
20.
y␲
21.
y␲
csc x cot x dx
22.
y (x
2
2
⫺1
3
⫺2
2
yx
2
4.
y sx 共x
2
⫺ 1兾x兲 dx
⫺1
1
sin x dx 苷 ⫺x 2 cos x ⫹ 2 y x cos x dx
3 – 4 Find the general indefinite integral.
3.
1
INDEFINITE INTEGRALS AND THE NET CHANGE THEOREM
A Click here for answers.
2.
■
8
1
共x ⫺ 1兲共3x ⫹ 2兲 dx
冉
⫺2
⫺5
␲兾2
兾3
3
r⫹
s
1
3
r
s
冊
dr
1
st ⫺
兾6
2
st
冊
dt
共x ⫹ 1兲 3 dx
␲兾3
2
ⱍ
⫺ 1 dx
冉
⫺1
0
2
csc 2␪ d␪
2
ⱍ
ⱍ
⫺ x ⫺ 1 ) dx
y 共2x ⫹ sec x tan x兲 dx
23. Water leaked from a tank at a rate of r 共t兲 liters per hour, where
5–22 Evaluate the integral.
5.
7.
9.
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
11.
y
1
y
1
y
2
0
0
1
y
2
0
共1 ⫺ 2x ⫺ 3x 2 兲 dx
共 y ⫺ 2y ⫹ 3y兲 dy
9
5
6.
8.
t6 ⫺ t2
dt
t4
10.
共x 3 ⫺ 1兲2 dx
12.
y
2
y
3
y
2
1
1
1
1
共5x 2 ⫺ 4x ⫹ 3兲 dx
冉
1
1
2 ⫺ 4
t
t
冊
dt
x2 ⫹ 1
dx
sx
y u (su ⫹ su ) du
0
the graph of r is as shown. Express the total amount of water
that leaked out during the first four hours as a definite integral.
Then use the Midpoint Rule to estimate that amount.
r
6
4
2
3
0
1
2
3
4 t
■
SECTION 5.4
5.4
INDEFINITE INTEGRALS AND THE NET CHANGE THEOREM
ANSWERS
E Click here for exercises.
3. 7 x7兾2 ⫺ 2x1兾2 ⫹ C
2
6.
26
3
7.
12.
29
35
18.
1
4
23.
y
19
15
13.
8.
11
6
19. 36
4
0
4.
28
21
14.
9.
28
3
2
20. 3 s3
S Click here for solutions.
x2 ⫹ sec x ⫹ C
11
6
10.
15. 2
6
5
5. ⫺1
(3 s2 ⫺ 2)
16.
2
3
21. ⫺1 ⫹ 3 s3
2
11.
17.
63
4
22.
5
3
86
7
r共t兲 dt ⬇ 19.6 L
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
2
SECTION 5.5
5.5
S Click here for solutions.
1– 6 Evaluate the integral by making the given substitution.
2.
3.
4.
5.
6.
■
T H E S U B S T I T U T I O N RU L E
A Click here for answers.
1.
THE SUBSTITUTION RULE
y x共x
y
2
1 兲 99 dx,
x2
dx,
s2 x 3
dx
y 共2x 1兲
y 共x
2
x3
dx,
6x兲 2
2
y sec a tan a d,
7.
y 共2x 1兲共x
9.
y sx 1 dx
y
26.
yt
sin共e x 兲 dx
28.
y cos x sin x dx
x1
dx
2
2x
30.
ye
32.
y
34.
y cos共7 3x兲 dx
y t sin共t
25.
y sec x tan x s1 sec x dx
27.
ye
29.
yx
31.
y x 共1 x
33.
y sin共2x 3兲 dx
35.
y 共sin 3 sin 3x兲 dx
3
2
x
2
兲 dt
u 苷 a
x 1兲 dx
3
8.
y x 共1 x 兲
10.
y s3 1 x dx
3
4 5
3
兲
2 3兾2
dx
2
(1 sx ) 9
sx
dx
cos共1 t 3 兲 dt
2
4
u 苷 x 2 6x
7–35 Evaluate the indefinite integral.
2
24.
23.
u 苷 2 x3
u 苷 2x 1
,
y tan sec d
y sin x cos x dx
u 苷 x2 1
u 苷 4x
y sin 4x dx,
22.
21.
ex
dx
1
2x
cos sx
dx
sx
dx
36– 43 Evaluate the definite integral, if it exists.
11.
13.
15.
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
17.
19.
y x s2 x
3
2
y 共t 1兲
4
dx
14.
dt
6
y 共1 2y兲
1.3
dy
y cos 2 d
y
12.
3x 1
dx
共3x 2 2x 1兲4
16.
18.
20.
y x共x
2
1兲3兾2 dx
1
y 共1 3t兲
4
36.
y
1
38.
y
4
40.
y
1
42.
y
1
dt
y s3 5y dy
0
1
0
cos t dt
兾4
37.
y
39.
y
3
共2x 1兲100 dx
41.
y
4
共x 4 x兲5共4x 3 1兲 dx
43.
y
3
1
x2
冑
1
1
dx
x
0
0
0
5
y sec
y
2
0
2
sin 4t dt
dx
2x 3
s1 2x dx
3x 2 1
dx
共x 3 x兲 2
3 d
x
dx
sx 2 1
44. Show that the area under the graph of y 苷 sin sx from 0 to 4
is the same as the area under the graph of y 苷 2x sin x from
0 to 2.
1
■
SECTION 5.5
5.5
THE SUBSTITUTION RULE
ANSWERS
E Click here for exercises.
1
1. 200
100
x2 − 1
+C
3.
− 14 cos 4x + C
5.
−
7.
1
4
9. 23
11.
1
6
13.
−
15.
−
17. 12
19.
−
1
21. 4
1
+C
2 (x2 + 6x)
2
4
x +x+1 +C
(x − 1)3/2 + C
3/2
2 + x4
+C
2
2
2. 3
√
2 + x3 + C
4. −
1
+C
2 (2x + 1)
sec aθ
+C
a
6
1
1 − x4 + C
8. − 24
6.
(1 − x)4/3 + C
2
5/2
x +1
+C
10. − 34
12.
1
5
2
+C
5 (t + 1)5
14.
1
+C
9 (1 − 3t)3
(1 − 2y)2.3
+C
4.6
16. − 16
sin 2θ + C
18. 13
6 (3x2
1
+C
− 2x + 1)3
sin4 x + C
2
23. − 12 cos t + C
25. 3
S Click here for solutions.
(1 + sec x)3/2 + C
20.
(3 − 5y)6/5 + C
tan 3θ + C
√
x2 + 1 + C
1
tan3 θ + C
√
(1 + x)10
+C
24.
5
1
3
26. − 3 sin 1 − t + C
22. 3
27.
− cos (ex ) + C
− 15 cos5 x + C
2
1
29. 2 ln x + 2x
+ C
28.
tan−1 (e x) + C
2 7/2
31. 17 1 − x
−
√
32. 2 sin x + C
30.
1
5
1 − x2
33.
− 12 cos (2x + 3) + C
34.
− 13 sin (7 − 3x) + C
35.
(sin 3α) x +
36.
0
1
3
5/2
+C
cos 3x + C
1
37. 2
38.
√
4
1
39. 2
3
2
−
√
5 5
12
ln 3
40.
1
101
41.
− 26
3
42. 32
3
43. 18
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
2
SECTION 6.1
6.1
■
1
A R E A S B E T W E E N C U RV E S
A Click here for answers.
S Click here for solutions.
1– 4 Find the area of the shaded region.
1.
2.
y
y
y=≈+3
(6, 12)
18. y 苷 x 4,
y 苷 ⫺x ⫺ 1,
19. y 2 苷 x,
y 苷 x ⫹ 5,
21. y 苷 x 2 ⫺ 4x,
x
1
x
y=2x
y=≈-4x
3.
y 苷 x ⫹ 2,
y
1
y=x+5
26. y 苷 x,
x
x
_1
y=_1
¥=x
29.
y 苷 cos 2x,
whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the
area of the region.
5. y 苷 4x ,
y苷x ⫹3
2
2
6. y 苷 x ⫹ 1,
2
y 苷 共x ⫺ 1兲 ,
x 苷 ⫺1,
x苷2
7. y 苷 x ⫹ 1,
y苷3⫺x ,
x 苷 ⫺2,
x苷2
2
8. y 2 苷 x,
2
9. y 苷 1兾x,
x 苷 0,
10. y 苷 cos x,
y 苷 1,
30. x 苷 3y,
x ⫹ y 苷 0,
area of the region.
11. y 苷 x,
12. y 苷 sx,
y苷x
y 苷 1兾x 2,
x ⫺ 3y ⫹ 1 苷 0
14. y 苷 x 4 ⫺ x 2,
y 苷 1 ⫺ x2
15. y 苷 x 2 ⫹ 2,
y 苷 2x ⫹ 5,
16. x ⫹ y 苷 2,
x⫹y苷0
17. y 苷 x 2 ⫹ 3,
y 苷 x,
x 苷 1,
34. y 苷 2 x,
y 苷 5 x,
x 苷 ⫺1,
35. y 苷 e x,
y 苷 e 3x,
x苷1
36. y 苷 e x,
y 苷 e ⫺x,
x 苷 ⫺2,
␲
冟
sin x ⫺
x苷2
x苷1
x苷1
冟
2
x dx
␲
and interpret it as the area of a region. Sketch the region.
38– 39 Use the Midpoint Rule with n 苷 4 to approximate the area
of the region bounded by the given curves.
y 苷 1 ⫺ x,
x苷2
y苷x
; 40– 41 Use a graph to find approximate x-coordinates of the
x 苷 0,
x 苷 ⫺1,
7x ⫹ 3y 苷 24
y 苷 2兾共x 2 ⫹ 1兲
39. y 苷 x tan x,
13. y 苷 sx ⫺ 1,
x 苷 ␲兾4
x 苷 0,
33. y 苷 x 2,
38. y 苷 s1 ⫹ x 3,
3
y 苷 x兾2
2
32. y 苷 1兾x,
0
11– 36 Sketch the region bounded by the given curves and find the
x 苷 ␲兾2
y 苷 x ⫺ x3
y
x 苷 ␲兾4
x苷2
2
y苷2
x 苷 ⫺␲兾4,
y 苷 sec2x,
x 苷 ⫺3,
2
37. Evaluate
x ⫺ 2y 苷 3
x苷0
ⱍ ⱍ y 苷 共x ⫹ 1兲 ⫺ 7, x 苷 ⫺4
y 苷 ⱍ x ⫺ 1 ⱍ, y 苷 x ⫺ 3, x 苷 0
31. y 苷 x s1 ⫺ x 2,
5– 10 Sketch the region enclosed by the given curves. Decide
x 苷 ⫺3,
x 苷 ⫺␲兾4,
28. y 苷 x ,
0
y苷3
y 苷 x2 ⫺ x
y 苷 sin x,
27. y 苷 sin x,
y=2
x=1-y$
y 苷 0,
y 苷 x ⫹ 4,
25. y 苷 x 3 ⫺ 4x 2 ⫹ 3x,
y
0
23. y 苷 4 ⫺ x 2,
y苷2
x⫹y⫹2苷0
24. y 苷 x 2 ⫹ 2x ⫹ 2,
4.
x=y#-y
y 苷 ⫺1,
x苷0
y 苷 2x
22. x 2 ⫹ 2x ⫹ y 苷 0,
0
_1
x 苷 ⫺2,
x 苷 y 2 ⫹ 1,
20. x ⫹ y 2 苷 0,
y=x
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
AREAS BETWEEN CURVES
x苷6
points of intersection of the given curves. Then use the Midpoint
Rule with n 苷 4 to approximate the area of the region bounded
by the curves.
40. y 苷 1 ⫹ 3x ⫺ 2x 2,
x苷1
41. y 苷 x 2 ⫺ x,
y 苷 s1 ⫹ x 4
y 苷 sin共x 2 兲
■
SECTION 6.1
AREAS BETWEEN CURVES
42– 43 Find the area of the region bounded by the given
; 46– 48 Use a graph to find approximate x-coordinates of the
curves by two methods: (a) integrating with respect to x, and
(b) integrating with respect to y.
points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves.
42. 4x ⫹ y 2 苷 0,
46. y 苷 sx ⫹ 1,
y 苷 2x ⫹ 4
43. x ⫹ 1 苷 2共 y ⫺ 2兲2,
x ⫹ 6y 苷 7
44– 45 Use calculus to find the area of the triangle with the
47. y 苷 x 4 ⫺ 1,
48. y 苷 x 2,
y 苷 x2
y 苷 x sin共x 2 兲
y 苷 e ⫺x
2
given vertices.
44. 共0, 0兲, 共1, 8兲, 共4, 3兲
45. 共⫺2, 5兲, 共0, ⫺3兲, 共5, 2兲
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
2
SECTION 6.1
6.1
8
5
1.
20
3
2. 36
3.
8.
32
3
9. ln 2
10. 2 ⫺ s2
8
5
15. 36
20. 21
26.
5
32
21. 36
1
␲ 2 ⫹ s2
⫺2
29.
13
3
33.
16
1
⫺
5 ln 5
2 ln 2
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
■
3
ANSWERS
E Click here for exercises.
14.
AREAS BETWEEN CURVES
30. 12
4.
9
2
16.
22.
1
2
5. 4
20
3
31
6
12.
18.
24.
32
5
49
6
(3s3 ⫺ s2 ⫺ 3)
32. ln 2 ⫺
1
6
35. 3 e3 ⫺ e ⫹
1
2
3
31
6
6.
1
2
11.
23.
27.
31.
33
2
17.
9
2
S Click here for solutions.
1
2
36. e 2 ⫹ e ⫹ e⫺1 ⫹ e⫺2 ⫺ 4
7. 8
4
3
13.
19.
25.
1
6
37.
␲
2
33
2
71
6
28. 34
33. ␲ ⫺
2
3
38. 3.22
43.
1
3
39. 0.13
44.
29
2
40. 0.83
45. 25
41. 0.81
46. 1.38
42. 9
47. 1.78
48. 0.98
SECTION 6.2
6.2
VOLUMES
■
1
VO L U M E S
A Click here for answers.
S Click here for solutions.
1– 5 Find the volume of the solid obtained by rotating the
region bounded by the given curves about the specified line.
Sketch the region, the solid, and a typical disk or washer.
1. y 2 苷 x 3, x 苷 4, y 苷 0;
26. y 苷 ln x, y 苷 1, x 苷 1;
about the x-axis
2. x y 苷 1, x 苷 0, y 苷 0;
3. y 苷 x , y 苷 4, x 苷 0, x 苷 2;
4. y 苷 x 2 1, y 苷 3 x 2;
about the x-axis
5. y 苷 2x x , y 苷 0, x 苷 0, x 苷 1;
about the y-axis
6– 13 Find the volume of the solid obtained by rotating the
region bounded by the given curves about the x-axis.
6. y 苷 x 2 1,
y 苷 0,
7. y 苷 1兾x,
8. y 苷 e ,
x
y 苷 0,
y 苷 0,
x 苷 0,
x 苷 1,
x 苷 0,
9. y 苷 1兾sx 1,
y 苷 0,
10. y 苷 sec x,
y 苷 1,
11. y 苷 cos x,
y 苷 sin x,
ⱍ
ⱍ
12. y 苷 x 2 ,
13. y 苷 冀 x 冁,
x苷3
x苷1
x 苷 1,
y 苷 0,
x 苷 1,
x苷2
about y 苷 1
30. y 苷 cos x, y 苷 0, x 苷 0, x 苷 兾2;
about y 苷 1
; 31– 32 Use a graph to find approximate x-coordinates of the points
of intersection of the given curves. Then find (approximately) the
volume of the solid obtained by rotating about the x-axis the region
bounded by these curves.
31. y 苷 x 2,
y 苷 sx 1
32. y 苷 x ,
y 苷 3x x 3
x苷1
33– 34 Sketch and find the volume of the solid obtained by rotating
x苷1
x 苷 0,
x 苷 兾4
x 苷 3,
x苷0
x 苷 6,
about y 苷 7
29. y 苷 cos x, y 苷 0, x 苷 0, x 苷 兾2;
4
x 苷 0,
about the y-axis
28. x y 苷 1, y 苷 共x 4兲 1;
2
about the y-axis
2
about the x-axis
27. y 苷 sx 1, y 苷 0, x 苷 5;
about the x-axis
2
26–30 Set up, but do not evaluate, an integral for the volume of
the solid obtained by rotating the region bounded by the given
curves about the specified line.
the region under the graph of f about the x-axis.
33. f 共x兲 苷
y苷0
14– 25 Refer to the figure and find the volume generated by
34. f 共x兲 苷
再
3 if 0 x 1
1 if 1 x 4
3 if 4 x 5
再
1
2
x 2 2x 2
if 0 x 1
if 1 x 2
rotating the given region about the given line.
35– 40 Each integral represents the volume of a solid. Describe the
y
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
x
y=Œ„
solid.
C(0, 2)
B(8, 2)
35. y
᏾£
兾4
0
᏾™
᏾¡
0
x=4y
1
tan2 x dx
37. y 共 y y 2 兲 dy
A(8, 0)
x
0
2
36. y y 6 dy
1
4
38. y 关16 共x 2兲4 兴 dx
0
1
39. y 关共5 2x 2 兲2 共5 2x兲2 兴 dx
14. ᏾1 about OA
15. ᏾1 about OC
16. ᏾1 about AB
17. ᏾1 about BC
18. ᏾2 about OA
19. ᏾2 about OC
20. ᏾2 about BC
21. ᏾2 about AB
22. ᏾3 about OA
23. ᏾3 about OC
24. ᏾3 about BC
25. ᏾3 about AB
0
40. y
兾2
兾4
关共2 sin x兲2 共2 cos x兲2 兴 dx
41. The base of S is the triangular region with vertices 共0, 0兲, 共2, 0兲,
and 共0, 1兲. Cross-sections perpendicular to the x-axis are semicircles. Find the volume of S.
42. Solve Example 9 if the planes intersect at an angle of 45.
2
■
SECTION 6.2
6.2
VOLUMES
ANSWERS
E Click here for exercises.
S Click here for solutions.
1. 64π
x
16. 128
π
3
17. 64
π
3
18. 128
π
15
19. 512
π
21
20. 112
π
15
21. 832
π
21
22. 64
π
5
23. 128
π
7
24. 16
π
5
25. 320
π
7
e
2.
2
2
26. V = π 1 1 − (ln x)
π
3
2
27. V = π 0
6
dx
24 − y − 2y dy
4
2
x4 − 16x3 + 83x2 − 144x + 36 dx
π/2 29. V = π 0
2 cos x − cos2 x dx
π/2 30. V = π 0
2 cos x + cos2 x dx
28. V = π 3
31. 5.80
3. 8π
33. 21π
32. 6.74
34. 127π
60
4. 32
π
3
35. Solid obtained by rotating the region under the curve
y = tan x, from x = 0 to x =
π
,
4
about the x-axis
36. Solid obtained by rotating the region bounded by the curve
37. Solid obtained by rotating the region between the curves
x = y and x =
5. 65 π
√
y about the y-axis
38. Solid obtained by rotating the region bounded by the curve
y = (x − 2)2 and the line y = 4 about the x-axis
39. Solid obtained by rotating the region between the curves
x
6.
46
π
15
8.
π
2
2
e −1
7.
2
π
3
9. π ln 2
π
2
y = 5 − 2x2 and y = 5 − 2x about the x-axis. Or: Solid
obtained by rotating the region bounded by the curves
y = 2x and y = 2x2 about the line y = 5
40. Solid obtained by rotating the region bounded by the curves
y = 2 + cos x and y = 2 + sin x and the line x =
the x-axis
10. 2π (tan 1 − 1)
11.
12. 3π
13. 55π
π
41. 12
14. 32
π
3
15. 256
π
3
42. 128
3
π
2
about
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
x = y 3 and the lines y = 1, y = 2, and x = 0 about the
y-axis
SECTION 6.3
6.3
■
1
VO L U M E S B Y C Y L I N D R I C A L S H E L L S
A Click here for answers.
S Click here for solutions.
1– 7 Use the method of cylindrical shells to find the volume
15. y 苷 e x, x 苷 0, y 苷 ␲ ;
generated by rotating the region bounded by the given curves
about the y-axis.
16. y 苷 e ⫺x, y 苷 0, x 苷 ⫺1, x 苷 0;
1. y 苷 x ⫺ 6x ⫹ 10,
y 苷 ⫺x ⫹ 6x ⫺ 6
2
2. y 苷 x,
x 苷 2y
3. y 苷 x ,
y 苷 4,
2
2
4. y 苷 x 2 ⫺ x 3,
2
y 苷 0,
6. y 苷 ⫺x 2 ⫹ 4x ⫺ 3,
7. y 苷 x ⫺ 2,
x 苷 0,
x苷4
y苷0
of the solid obtained by rotating the region bounded by the
given curves about the x-axis.
9. x 苷 y ,
2
x 苷 0,
x 苷 0,
10. y 苷 x,
x 苷 0,
11. y 苷 x ,
y苷9
2
y 苷 16
y 苷 2,
y苷5
x⫹y苷2
x苷0
13. y 苷 sx,
x⫹y苷2
y 苷 0,
14– 22 Set up, but do not evaluate, an integral for the volume
of the solid obtained by rotating the region bounded by the
given curves about the specified axis.
14. y 苷 e x, y 苷 e ⫺x, x 苷 1;
about y 苷 3
19. y 苷 sin x, y 苷 0, x 苷 2␲, x 苷 3␲ ;
about the y-axis
20. x 苷 cos y, x 苷 0, y 苷 0, y 苷 ␲兾4;
about the x-axis
21. y 苷 ⫺x 2 ⫹ 7x ⫺ 10, y 苷 x ⫺ 2;
about the y-axis
about the x-axis
about y 苷 5
23. The integral x0␲ 2␲ 共4 ⫺ x兲 sin4 x dx represents the volume of a
solid. Describe the solid.
; 24– 25 Use a graph to estimate the x-coordinates of the points of
intersection of the given curves. Then use this information to estimate the volume of the solid obtained by rotating about the y-axis
the region enclosed by these curves.
24. y 苷 0,
25. y 苷 x 4,
12. y ⫺ 6y ⫹ x 苷 0,
2
about x 苷 1
about y 苷 1
22. x 苷 4 ⫺ y 2, x 苷 8 ⫺ 2y 2;
y 苷 sx ⫺ 2
8 – 13 Use the method of cylindrical shells to find the volume
4
8. x 苷 s
y,
17. y 苷 e x, x 苷 0, y 苷 2;
about the x-axis
18. y 苷 ln x, y 苷 0, x 苷 e;
x苷0
y苷0
5. y 苷 s4 ⫹ x 2,
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
VOLUMES BY CYLINDRICAL SHELLS
y 苷 x ⫹ x2 ⫺ x4
y 苷 3x ⫺ x 3
26– 27 The region bounded by the given curves is rotated about
the specified axis. Find the volume of the resulting solid by any
method.
26. y 苷 x 2 ⫹ x ⫺ 2, y 苷 0;
27. y 苷 x 2 ⫺ 3x ⫹ 2, y 苷 0;
about the x-axis
about the y-axis
2
■
SECTION 6.3
6.3
VOLUMES BY CYLINDRICAL SHELLS
ANSWERS
E Click here for exercises.
S Click here for solutions.
2
1. 16π
17. V = 2π 1 (y ln y − ln y) dy
2. 64
π
15
18. V = 2π 0 (3e − ey − 3e + ye ) dy
1
3. 8π
4.
3π
π/4
20. V = 0
2πy cos y dy
4 4
3
2
21. V = π 2 x − 14x + 68x − 136x + 96 dx
2
2
22. V = −2 2π (5 − y) 4 − y dy
6. 16π
3
7. 45 π
23. Solid obtained by rotating the region under the curve
8. 4096
π
9
y = sin4 x, above y = 0, from x = 0 to x = π, about the
line x = 4
9. 609
π
2
10. 23 π
24. 4.05
1944
π
5
25. 4.62
12. 216π
13. 56 π
y
19. V = 2π 2πx sin x dx
1
π
10
√
5. 16
π 5 5−1
3
11.
y
26. 81
π
10
1
x
−x
14. V = 0 2πx e − e
π
27. 12 π
dx
15. V = 1 2πy · ln y dy
−x
dx
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
0
16. V = −1 2π (1 − x) e
SECTION 6.4
6.4
WO R K
A Click here for answers.
1. A particle is moved along the x-axis by a force that measures
5x 2 ⫹ 1 pounds at a point x feet from the origin. Find the
work done in moving the particle from the origin to a distance
of 10 ft.
2. A uniform cable hanging over the edge of a tall building is
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
40 ft long and weighs 60 lb. How much work is required to
pull 10 ft of the cable to the top?
S Click here for solutions.
WORK
■
1
2
■
SECTION 6.4
6.4
WORK
ANSWERS
E Click here for exercises.
S Click here for solutions.
1. 5030
ft-lb
3
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
2. 525 ft-lb
SECTION 7.1
7.1
INTEGRATION BY PARTS
■
1
I N T E G R AT I O N B Y PA RT S
A Click here for answers.
S Click here for solutions.
16. First make a substitution and then use integration by parts to
1–15 Evaluate the integral.
evaluate x x 5 cos共x 3兲 dx.
1.
y xe 2x dx
2.
y x cos x dx
3.
y x sin 4x dx
4.
yx
5.
yx
7.
yt
9.
y
1
0
2
cos 3x dx
2
sin ax dx
6.
y sin cos d
2
ln t dt
8.
ye
te 1 dt
cos 3 d
y
4
12.
y
1
14.
y sin共ln x兲 dx
10.
1
ln sx dx
; 17. Evaluate x sx ln x dx. Illustrate, and check that your answer is
reasonable, by graphing both the function and its antiderivative
(take C 苷 0).
18. Find the area of the region bounded by y 苷 sin 1 x, y 苷 0,
and x 苷 0.5.
; 19–20 Use a graph to find approximate x-coordinates of the points
of intersection of the given curves. Then find (approximately) the
area of the region bounded by the curves.
19. y 苷 x 2 ,
y 苷 xe x兾2
20. y 苷 x 2 5,
兾2
11.
y
13.
y
15.
y x tan
0
x cos 2x dx
0
2 x
x e
y 苷 ln x
dx
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
21. Use the method of cylindrical shells to find the volume
2
x 3e x dx
1
x dx
generated by rotating the region bounded by y 苷 sin x, y 苷 0,
x 苷 2, and x 苷 3 about the y-axis.
22. Find the average value of f 共x兲 苷 x cos 2x on the interval
关0, 兾2兴.
■
SECTION 7.1
7.1
INTEGRATION BY PARTS
ANSWERS
E Click here for exercises.
S Click here for solutions.
12. 2 − 5/e
2x
2x
1. 12 xe − 14 e + C
2
x2
13. 12 e
x −1 +C
2. x sin x + cos x + C
1
3. − 14 x cos 4x + 16
sin 4x + C
4.
1 2
x
3
5. −
sin 3x +
2
x cos 3x
9
−
2
27
14. 12 x [sin (ln x) − cos (ln x)] + C
sin 3x + C
x2
2
2x
cos ax + 2 sin ax + 3 cos ax + C
a
a
a
6. 18 (sin 2θ − 2θ cos 2θ) + C
3
7. 19 t (3 ln t − 1) + C
1 −θ
8. 10
e (3 sin 3θ − cos 3θ) + C
9. 1 − 2/e
10. 2 ln 4 − 32
11. − 12
2
−1
15. 12 x tan
x + tan−1 x − x + C
3
3
3
16. 13 x sin x + 13 cos x + C
3/2
17. 23 x
ln x − 49 x3/2 + C
√
1
18. 12
π + 6 3 − 12
19. 0.080
20. 7.10
21. 10π
22. −
2
1
π
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
2
SECTION 7.2
7.2
S Click here for solutions.
1–36 Evaluate the integral.
3.
5.
7.
9.
11.
13.
15.
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
17.
y sin
y sin
y
兾2
0
3
4
2.
x dx
x cos 3 x dx
sin 2 3x dx
y cos
4
y x sin 共x
3
5
8.
y
2
2
dx
兲 dx
4
sin 2x cos 2x dx
y sin
4
4.
6.
t dt
y 共1 sin 2x兲
y
■
TRIGONOMETRIC INTEGRALS
A Click here for answers.
1.
TRIGONOMETRIC INTEGRALS
x cos 4 x dx
cos 2 (sx )
dx
sx
10.
12.
14.
16.
18.
y sin
3
y cos
y
兾2
0
20.
y tan
22.
y
24.
y
26.
y
28.
y cot
30.
ye
32.
y sin
y cos 3x cos 4x dx
34.
y sin 3x sin 6x dx
y sin x cos 5x dx
36.
y cos x cos 2x cos 3x dx
19.
y tan x sec
21.
y sec
23.
y
25.
y tan
27.
y
29.
y cot
31.
y
33.
35.
6
x dx
x cos 4 x dx
5
x sin 5 x dx
兾4
0
4
x dx
tan 4 t sec 2 t dt
y sin x dx
x sec 3 x dx
兾4
0
cos 2 x dx
3
兾4
0
兾3
0
3
x sec 6 x dx
sec 6 x dx
tan 2 x sec 4 x dx
tan 5 x sec x dx
6
冉 冊
y sin
y cos
6
6
sec 2 x
dx
cot x
cos d
x dx
3
x csc 4 x dx
cos 2 x
dx
sin x
x
2
w csc 4 w dw
cos 7共e x 兲 dx
dx
4
x
y sin x dx
5
y
cos 3 x
dx
ssin x
y
dx
1 sin x
1
■
SECTION 7.2
7.2
TRIGONOMETRIC INTEGRALS
ANSWERS
E Click here for exercises.
S Click here for solutions.
3
1. − cos x + 13 cos x + C
6
2
4
6
19. 16 sec x + C or 12 tan x + 12 tan x + 12 tan x + C
7
5
2. 17 cos x − 15 cos x + C
8
6
20. 18 sec x − 16 sec x + C or
1
8
5
7
3. 15 sin x − 17 sin x + C
tan2 x +
1
3
tan6 x +
1
4. 10
sin10 x − 14 sin8 x + 16 sin6 x + C
3
21. 13 tan x + tan x + C
5. π
4
22. 28
15
6. π
4
1
4
23. 15
8
24. 15
1
7. 38 t + 14 sin 2t + 32
sin 4t + C
5
3
25. 15 sec x − 13 sec x + C
5
1
3
1
8. 16
x − 4π
sin 2πx + 64π
sin 4πx + 48π
sin3 2πx + C
26. 38
15
9. 32 x + cos 2x − 18 sin 4x + C
2
27. 12 tan x + C
√
10. − 83 cos 2θ + 14 θ + 18 sin 2θ + C
2
11. − 12 cos x
+
1
6
cos3 x2 + C
3
12. 18 52 x + 2 sin 2x + 38 sin 4x − 16 sin 2x + C
13. − 12
1
9
cos9 2x −
2
7
cos7 2x +
1
5
cos5 2x + C
5
3
14. − 15 cos x + 23 cos x − cos x + C
3
1
1
15. 128
x − 128
sin 4x + 1024
sin 8x + C
2
16. 2 1 − 15 sin x
17.
√
x+
1
2
√
sin x + C
√
sin (2 x) + C
18. tan x + sec x + C
tan4 x + C
3
5
28. − 13 cot w − 15 cot w + C
6
4
29. − 16 cot x − 14 cot x + C
x
3
x
5
x
7
x
30. sin e − sin e + 35 sin e − 17 sin e + C
31. ln |csc x − cot x| + cos x + C
3
32. − 13 cot x − cot x + C
1
33. 12 sin x + 14
sin 7x + C
1
34. 16 sin 3x − 18
sin 9x + C
1
35. 18 cos 4x − 12
cos 6x + C
1
1
36. 14 x + 18 sin 2x + 16
sin 4x + 24
sin 6x + C
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
2
SECTION 7.3
7.3
TRIGONOMETRIC SUBSTITUTION
A Click here for answers.
S Click here for solutions.
1–21 Evaluate the integral.
1.
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
y
s3兾2
y
5.
y
2
0
1
dx
x 2 s1 x 2
x
dx
s1 x 2
4.
y x s4 x
x3
dx
sx 2 4
6.
y
dx
x 3 sx 2 16
y
9.
y x sx
dx
2 3
s9x 2 4
dx
x
8.
10.
12.
0
3
0
2
dx
s9 x 2
y x s25 x
y
3
0
y 5x s1 x
15.
y sx
17.
y e s9 e
t
2t
19.
yx
dx
sx 2 2
dx
x
dx
共x 2 4兲 5兾2
y
13.
2
dx
4
2
14.
y
dx
4x 8
16.
y s2x x
18.
y se
20.
y
dt
x s9 x dx
2t
2
dx
9 dt
dx
共1 x 2兲 2
21. Find the average value of f 共x兲 苷 共4 x 2 兲 3兾2 on the
interval 关0, 2兴.
2
dx
共4x 2 25兲 3兾 2
dx
2
x 3 s4 x 2 dx
y
7.
y
2
2.
1兾2
3.
11.
TRIGONOMETRIC SUBSTITUTION
■
1
■
SECTION 7.3
7.3
TRIGONOMETRIC SUBSTITUTION
ANSWERS
E Click here for exercises.
1. √2
S Click here for solutions.
12. 9
3
2. 64
15
2 3/2
13. 53 1 + x
+C
3. −
14. −
4.
5.
6.
7.
8.
9.
10.
√
1 − x2 + C
3/2
− 13 4 − x2
+C
√ 8
2− 2
3
√
ln 2 + 1
√
4 x2 −16
−1 x
1
sec
+C
+
128
x2
4
−3/2
− 13 x2 + 4
+C
√
√
x2 +3− 3 1
+C
√
ln x
3
1
3
3/2
+C
x + 25
2
√
11.
9x2 − 4 − 2 sec−1 3x
+C
2
15.
16.
17.
18.
19.
20.
x
√
+C
25 4x2 − 25
√
ln x2 + 4x + 8 + x + 2 + C
√
−1
1
sin (x − 1) + (x − 1) 2x − x2 + C
2
√
9
sin−1 13 et + 12 et 9 − e2t + C
2
√
e2t − 9 − 3 sec−1 13 et + C
√
3/2 2
x −2
1
x2 − 2
+C
−
4
x
3x3
x
−1
1
x
+
+C
tan
2
x2 + 1
21. 3π
2
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
2
SECTION 7.4
7.4
S Click here for solutions.
1–18 Write out the form of the partial fraction decomposition of
the function (as in Example 7). Do not determine the numerical
values of the coefficients.
25.
y
7
3
1
dx
共x ⫹ 1兲共x ⫺ 2兲
1
6x ⫺ 5
26.
y 2x ⫹ 3 dx
3
6x 2 ⫹ 5x ⫺ 3
dx
x 3 ⫹ 2x 2 ⫺ 3x
3
1.
共2x ⫹ 3兲共x ⫺ 1兲
5
2.
2x 2 ⫺ 3x ⫺ 2
27.
y x共x ⫹ 1兲共2x ⫹ 3兲 dx
28.
y
x 2 ⫹ 9x ⫺ 12
3.
共3x ⫺ 1兲共x ⫹ 6兲2
z 2 ⫺ 4z
4.
共3z ⫹ 5兲3共z ⫹ 2兲
29.
y
x
dx
x 2 ⫹ 4x ⫹ 4
30.
yx
31.
y 共x ⫺ 1兲共x ⫺ 2兲共x ⫺ 3兲
32.
y x共x ⫹ 1兲共2x ⫹ 3兲
33.
y 2x
3x 2 ⫺ 6x ⫹ 2
dx
3
⫺ 3x 2 ⫹ x
34.
y
35.
y
x2 ⫹ 3
dx
x 3 ⫹ 2x
36.
y 共x ⫺ 1兲共x
37.
y
38.
yx
39.
yx
2x 3 ⫺ x
dx
4
⫺ x2 ⫹ 1
40.
y
1
41.
y
x
dx
x ⫹x⫹1
42.
y
1兾2
43.
y
3x 3 ⫺ x 2 ⫹ 6x ⫺ 4
dx
共x 2 ⫹ 1兲共x 2 ⫹ 2兲
44.
y sin x ⫺ 3 sin x ⫹ 2 dx
45.
y
sin x cos 2 x
dx
5 ⫹ cos 2 x
46.
y 共x ⫹ 1兲共x
47.
y 共x ⫺ 3兲共x
48.
y 共x
49.
y
50.
y 共x
5.
1
x4 ⫺ x3
6.
x2 ⫹ 1
7. 2
x ⫺1
7
10.
2x 2 ⫹ 5x ⫺ 12
12.
x ⫺x
13.
共x ⫺ 6兲共5x ⫹ 3兲3
15.
x ⫹ 3x ⫺ 4
共2x ⫺ 1兲 2共2x ⫹ 3兲
2
1
共x ⫺ 1兲共x ⫹ 2兲
3
x4 ⫹ x3 ⫺ x2 ⫺ x ⫹ 1
x3 ⫺ x
x 3 ⫺ 4x 2 ⫹ 2
8.
共x 2 ⫹ 1兲共x 2 ⫹ 2兲
x⫹1
9. 2
x ⫹ 2x
11.
2
19x
共x ⫺ 1兲3共4x 2 ⫹ 5x ⫹ 3兲2
1
14. 6
x ⫺ x3
x ⫹x ⫹1
x 4 ⫹ x 3 ⫹ 2x 2
3
16.
2
3 ⫺ 11x
17.
3 2
共x ⫺ 2兲 共x ⫹ 1兲共2x 2 ⫹ 5x ⫹ 7兲2
4
18.
x
共x 2 ⫹ 9兲3
1
0
dx
2
1
2t 3 ⫺ t 2 ⫹ 3t ⫺ 1
dt
共t 2 ⫹ 1兲共t 2 ⫹ 2兲
1
0
2
21.
y
x2
dx
x⫹1
y
x2 ⫹ 1
dx
x2 ⫺ x
20.
y
3
2
y
22.
y
2
0
23.
y
dy
y⫹2
x 3 ⫹ x 2 ⫺ 12x ⫹ 1
dx
x 2 ⫹ x ⫺ 12
1
dx
x 3 ⫹ x 2 ⫺ 2x
24.
y
4
2
4x ⫺ 1
dx
共x ⫺ 1兲共x ⫹ 2兲
2
18 ⫺ 2x ⫺ 4x 2
dx
3
⫹ 4x 2 ⫹ x ⫺ 6
dx
1
0
x3
dx
x ⫹1
2
3x 2 ⫺ 4x ⫹ 5
dx
2
⫹ 1兲
0
4
x4
dx
⫺1
x⫺1
dx
x 2 ⫹ 2x ⫹ 2
⫺1兾2
4x 2 ⫹ 5x ⫹ 7
dx
4x 2 ⫹ 4x ⫹ 5
共2 sin x ⫺ 3兲 cos x
2
x 2 ⫹ 7x ⫺ 6
dx
2
⫺ 4x ⫹ 7兲
4x ⫹ 1
dx
2
⫹ 6x ⫹ 12兲
19 – 51 Evaluate the integral.
19.
■
I N T E G R AT I O N O F R AT I O N A L F U N C T I O N S B Y PA RT I A L F R AC T I O N S
A Click here for answers.
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
2
x⫹1
dx
⫹ x ⫹ 2兲
3x 4 ⫺ 2x 3 ⫹ 20x 2 ⫺ 5x ⫹ 34
dx
共x ⫺ 1兲共x 2 ⫹ 4兲2
2
8x
dx
⫹ 4兲 3
51.
x2 ⫹ 1
dx
3
⫹ 3x兲2
y 共x
1
■
SECTION 7.4
7.4
INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
ANSWERS
E Click here for exercises.
1.
A
B
+
2x + 3
x−1
2.
A
B
+
2x + 1
x−2
18.
20. y − 2 ln |y + 2| + C
A
B
C
+
+
3x − 1
x+6
(x + 6)2
4.
A
C
D
B
+
+
+
3z + 5
z+2
(3z + 5)2
(3z + 5)3
A
C
D
B
+ 2 + 3 +
x
x
x
x−1
6. x + 1 +
7. 1 +
8.
Ex + F
Ax + B
Cx + D
+
+
x2 + 9
(x2 + 9)2
(x2 + 9)3
2
19. 12 x − x + ln |x + 1| + C
3.
5.
S Click here for solutions.
21. x + ln
(x − 1)2
+C
|x|
22. 2 + 17 ln 29
23. 12 ln 2 − 12 ln 3 + 16 ln 5
24. ln 81
8
A
B
C
+
+
x
x+1
x−1
25. 13 ln 52
26. 3x − 7 ln |2x + 3| + C
27. 13 ln |x| − ln |x + 1| + 23 ln |2x + 3| + C
A
B
+
x−1
x+1
28. 4 ln 6 − 3 ln 5
Ax + B
Cx + D
+ 2
x2 + 1
x +2
29. ln 32 − 13
30. ln |x − 1| − 2 ln |x + 2| − 3 ln |x + 3| + C
A
B
9.
+
x
x+2
31.
1 |(x − 1) (x − 3)|
+C
ln
2
(x − 2)2
10.
A
B
+
2x − 3
x+4
11.
A
B
+
x−1
x+2
12.
A
C
B
+
+
2x − 1
2x + 3
(2x − 1)2
35. 54 ln 2
A
D
B
C
+
+
2 +
x−6
5x + 3
(5x + 3)
(5x + 3)3
2
2
−1
37. 12 ln t + 1 + 12 ln t + 2 − √1 tan
A
C
D
B
Ex + F
+ 2 + 3 +
+ 2
x
x
x
x−1
x +x+1
x − 1 1
38. x + 14 ln − tan−1 x + C
x + 1 2
13.
14.
32. 13 ln |x| − ln |x + 1| + 23 ln |2x + 3| + C
33. 2 ln |x| − ln |x − 1| + 12 ln |2x − 1| + C
34. 12 (1 − ln 2)
A
C
B
15.
+
+
x−1
(x − 1)2
(x − 1)3
+
Dx + E
Fx + G
+
4x2 + 5x + 3
(4x2 + 5x + 3)2
√
x2 + 1 − 3 tan−1 x + C
2
A
C
Dx + E
B
17.
+
2 +
3 +
x−2
x2 + 1
(x − 2)
(x − 2)
Hx + I
Fx + G
+
2x2 + 5x + 7
(2x2 + 5x + 7)2
1
√
t
2
+C
4
2
39. 12 ln x − x + 1 + C
−1
40. 12 ln 52 − 2 tan
2 + π2
41. ln
√
3−
π
√
6 3
42. 1 + 18 ln 2 + 3π
32
A
Cx + D
B
16.
+ 2 + 2
x
x
x +x+2
+
2
36. (x − 1) + ln
√
2
−1
43. 32 ln x + 1 − 3 tan
x + 2 tan−1
44. ln sin x − 3 sin x + 2 + C
2
1
√
x
2
+C
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
2
SECTION 7.4
45. − cos x +
√
5 tan−1 √15 cos x + C
2
5
3
−1
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
47.
ln |x − 3| −
1
6
x−2
√
+C
3
2
ln x + 6x + 12 + √ tan−1
3
2
x+3
√
3
+C
■
3
√
2x + 1
x−3
2 7
−1
√
+C
+
tan
7 (x2 + x + 2)
49
7
2
−1 1
49. 2 ln |x − 1| + 12 ln x + 4 − 12 tan
x
2
1
2x
1
−1 1
+
x
+
+
tan
+C
2
2 (x2 + 4)
8
x2 + 4
48.
46. − ln |x + 1| + ln x − 4x + 7 + √ tan
1
3
INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
50. −
2
+C
(x2 + 4)2
51. −
1
+C
3 (x3 + 3x)
SECTION 7.5
7.5
S T R AT E G Y F O R I N T E G R AT I O N
A Click here for answers.
S Click here for solutions.
1–45 Evaluate the integral.
1.
3.
5.
7.
9.
2x 5
dx
x3
y
yx
2
4.
x
dx
2x 2
6.
1 cos x
dx
sin x
8.
y sin x cos x dx
10.
y
2
3
11.
y ln共1 x
13.
y ⱍx
15.
yx
17.
y s9 cos
19.
yx
2
21.
yx
4
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
2.
sx 2
dx
x2
y
STRATEGY FOR INTEGRATION
兲 dx
cos 5x dx
24.
y cos 3x cos 5x dx
dx
x2 x 1
26.
y s9x
y sx (1 sx) dx
28.
yx
29.
y sin x cos x dx
30.
y s5 4x x
31.
ye
32.
y
s9 x 2
dx
x
33.
y
34.
y tan
2
4x dx
3
z ) dz
sz (z s
35.
y sin x cos x dx
36.
y cot
3
2x csc 3 2x dx
37.
y 共x
x
dx
1兲共x 2 4兲
38.
y 4 5 sin x
39.
y 共x
4x 3兲 sin 2x dx
40.
y s16 x
tan 3x sec 4x dx
41.
y ⱍ ln共x兾2兲 ⱍ dx
42.
yx
cosh x dx
43.
y x sec x tan x dx
44.
y 1 cos x dx
45.
y xs2x 25
x
y 共x 2兲
2
cos x
dx
1 sin 2x
y
y
2
1
y
1
x 3 ln x dx
14.
y s1 x x
16.
ye
dx
18.
y
x
dx
3x 2
20.
yx
x3 x 1
dx
2x 2 4x
22.
y
3
3
3
ⱍ
x 2 2x dx
1
dx
8
sin 2x
4
x
25.
yx
27.
dx
y
3
ye
3x
sin x cos x
y sin x cos x dx
12.
2
23.
0
2x
兾4
0
1
0
2
2
dx
sin 3x dx
cos x tan x dx
3
3
2
4
ex
dx
1
2x
兾4
0
cos5 d
2
2
2
4
3
1
dx
4
2
1
dx
12x 5
x
dx
2x 2 10
1
1
1
2
dx
x 5 cosh x dx
dx
x
2
4
dx
x2
dx
x2
1 cos 2 x
2
■
1
■
SECTION 7.5
7.5
STRATEGY FOR INTEGRATION
ANSWERS
E Click here for exercises.
1. 2x + 11 ln |x − 3| + C
2. − ln |sin x + cos x| + C
√
√
−1 1
3. 2 x − 2 − 4 tan
x−2 +C
2
2
+C
x+2
2
−1
5. 12 ln x − 2x + 2 + tan
(x − 1) + C
4. ln |x + 2| +
−1
6. tan
(sin x) + C
7. ln |1 − cos x| + C
8. 4 ln 2 −
S Click here for solutions.
1
24. 16
sin 8x + 14 sin 2x + C
2
−1
25. 12 ln |x + 1| − 14 ln x + 1 + 12 tan
x+C
√
26. 13 ln 3x + 2 + 9x2 + 12x − 5 + C
4/3
6 11/6
27. 34 x
− 11
x
+C
28.
1
x2 + 1
tan−1
+C
6
3
1
1
1
29. 16
x − 64
sin 4x + 48
sin3 2x + C
x+2
+C
3
1 ex − 1 31.
ln x
+C
2
e + 1
−1
30. sin
15
16
3
5
9. 13 sin x − 15 sin x + C
√
3 − 9 − x2 √
+C
9 − x2 + 3 ln x
2
−1
11. x ln 1 + x − 2x + 2 tan
x+C
10.
32. 0
√
2
33. 43
120
12. 52
55
34. 14 tan 4x − x + C
13. 86
3
1
1
1
35. 16
x − 64π
sin 4πx + 48π
sin3 2πx + C
√
(2x − 1) + 14 (2x − 1) 1 + x − x2 + C
1
1
15. 12
ln |x − 2| − 24
ln x2 + 2x + 4
1
− 4√
tan−1 √13 (x + 1) + C
3
−1
14. 58 sin
1
√
5
1 2x
16. 13
e (2 sin 3x − 3 cos 3x) + C
−1
17. − sin
5
18. 12
1
3
cos2 x + C
(x + 2)2
19. ln
+C
|x + 1|
2
20. x + 2 sinh x − 2x cosh x + C
4
2
21. 14 ln x + 2x + 4x + C
22.
2
π
1 3x
23. 34
e (5 sin 5x + 3 cos 5x) + C
3
1
36. 16 csc 2x − 10
csc5 2x + C
1 x2 + 1
ln
+C
6 x2 + 4
1 tan (x/2) − 2 38.
ln +C
3
2 tan (x/2) − 1 2
39. 12 (x + 2) sin 2x − 14 2x + 8x − 7 cos 2x + C
−1 1 2
40. 12 sin
x +C
4
37.
41. ln 27
16
42. ln
√
x2 + x + 2 +
3
√
7
tan−1
1
√
7
43. x sec x − ln |sec x + tan x| + C
44. −2 cot x − x + C
√
−1 1
45. 25 tan
2x − 25 + C
5
(2x + 1) + C
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
2
SECTION 7.6
7.6
S Click here for solutions.
1–13 Use the Table of Integrals [on Reference Pages 6–10]
to evaluate the integral.
⫺1
共x 2 兲 dx
1.
y x sin
3.
y s1 ⫹ sin x dx
5.
ye
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
7.
sin x cos x
y
■
I N T E G R AT I O N U S I N G TA B L E S A N D C O M P U T E R A L G E B R A S Y S T E M S
A Click here for answers.
x
INTEGRATION USING TABLES AND COMPUTER ALGEBRA SYSTEMS
ln共1 ⫹ e x 兲 dx
s4 ⫺ 3x 2
dx
x
s5 ⫺ 4x ⫺ x 2 dx
y sx
cos共3x ⫹ 4兲 dx
13.
yx
4.
ye
6.
y
s9x 2 ⫺ 1
dx
x2
y
sin ␪
d␪
1 ⫹ 2 cos ␪
8.
1
⫺2
11.
yx
x
y
sin⫺1共x 2 兲 dx
2.
3
9.
2
2
x
dx
⫺ 4x
10.
y s2 ⫹ 3 cos x
12.
y
␲兾2
0
tan x dx
cos 5x dx
tan⫺1x dx
14. Find the volume of the solid obtained when the region under
the curve y 苷 1兾共1 ⫹ 5x兲2 from 0 to 1 is rotated about the
y-axis.
1
■
SECTION 7.6
7.6
INTEGRATION USING TABLES AND COMPUTER ALGEBRA SYSTEMS
ANSWERS
E Click here for exercises.
2 −1 2 √
1. 12 x sin
x + 1 − x4 + C
√
x2 1 − x4
2x4 − 1
2.
sin−1 x2 +
+C
8
8
√
3. − 23 (2 − sin x) 1 + sin x + C
3 x 1
4. 10
e 3 cos (3x + 4) + sin (3x + 4) + C
x
x
x
5. (1 + e ) ln (1 + e ) − e + C
√
√
9x2 − 1
6. −
+ 3 ln 3x + 9x2 − 1 + C
x
√
2 + 4 − 3x2 √
+C
7.
4 − 3x2 − 2 ln x
S Click here for solutions.
8. − 12 ln |1 + 2 cos θ| + C
9. 9π
4
√ √
2 + 3 cos x − 2 √
√
√ +C
2 + 3 cos x − 2 ln √
2 + 3 cos x + 2 √
√
11.
x2 − 4x + 2 ln x − 2 + x2 − 4x + C
10. −2
8
12. 15
3
−1
13. 13 x tan
x − 16 x2 + 16 ln 1 + x2 + C
14. 2π
ln 6 − 56
25
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
2
SECTION 7.7
7.7
■
A P P ROX I M AT E I N T E G R AT I O N
A Click here for answers.
S Click here for solutions.
1– 3 Use (a) the Trapezoidal Rule and (b) Simpson’s Rule to
approximate the given integral with the specified value of n.
(Round your answers to six decimal places.)
1.
y
3.
y
1
⫺1
s1 ⫹ x 3 dx,
␲兾4
0
n苷8
2.
sin x
dx, n 苷 6
x
␲
y␲
兾2
x tan x dx, n 苷 6
4 –10 Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and
(c) Simpson’s Rule to approximate the given integral with the
specified value of n. (Round your answers to six decimal
places.)
1
1
2
2
4. y e⫺x dx, n 苷 10
5. y
dx, n 苷 10
0
0 s1 ⫹ x 3
6.
y
3
8.
y
1
10.
y
3
2
0
0
1
dx, n 苷 10
ln x
7.
y
1
n 苷 10
9.
y
1兾2
x 5e x dx,
0
0
x 共m兲
A 共m 2 兲
x 共m兲
A 共m 2 兲
0
1
2
3
4
5
0.68
0.65
0.64
0.61
0.58
0.59
6
7
8
9
10
0.53
0.55
0.52
0.50
0.48
14. (a) Use the Trapezoidal Rule and the following data to esti-
mate the value of the integral x13.2 y dx.
ln共1 ⫹ e x 兲 dx, n 苷 8
cos共e x 兲 dx,
n苷8
1
dx, n 苷 6
1 ⫹ x4
x
y
x
y
1.0
1.2
1.4
1.6
1.8
2.0
4.9
5.4
5.8
6.2
6.7
7.0
2.2
2.4
2.6
2.8
3.0
3.2
7.3
7.5
8.0
8.2
8.3
8.3
(b) If it is known that ⫺1 艋 f ⬙共 x兲 艋 3 for all x, estimate the
error involved in the approximation in part (a).
15. Water leaked from a tank at a rate of r共t兲 liters per hour,
11. Use Simpson’s Rule and the following data to estimate the
value of the integral x26 y dx.
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
APPROXIMATE INTEGRATION
where the graph of r is as shown. Use Simpson’s Rule to estimate the total amount of water that leaked out during the first
four hours.
r
6
x
y
x
y
2.0
2.5
3.0
3.5
4.0
9.22
9.01
8.76
8.30
7.52
4.5
5.0
5.5
6.0
6.83
7.32
7.69
7.91
4
2
0
12. The speedometer reading 共v兲 on a car was observed at
1-minute intervals and recorded in the following chart. Use
Simpson’s Rule to estimate the distance traveled by the car.
t 共min兲
v 共mi兾h兲
t 共min兲
v 共mi兾h兲
0
1
2
3
4
5
40
42
45
49
52
54
6
7
8
9
10
56
57
57
55
56
13. A log 10 meters long is cut at 1-meter intervals and its
cross-sectional areas A (at a distance x from the end of the
log) are listed in the following table. Use Simpson’s Rule to
estimate the volume of the log.
2
1
3
4 t
16. The table (supplied by Pacific Gas and Electric) gives the
power consumption in megawatts in the San Francisco Bay
Area from midnight to noon on September 19, 1996. Use
Simpson’s Rule to estimate the energy used during that time
period. (Use the fact that power is the derivative of energy.)
t
P
t
P
0
1
2
3
4
5
6
4182
3856
3640
3558
3547
3679
4112
7
8
9
10
11
12
4699
5151
5514
5751
6044
6206
1
2
■
SECTION 7.7
7.7
APPROXIMATE INTEGRATION
ANSWERS
E Click here for exercises.
S Click here for solutions.
1. (a) 1.913972
(b) 1.934766
2. (a) 0.481672
(b) 0.481172
3. (a) 0.189445
(b) 0.185822
4. (a) 0.746211
(b) 0.747131
(c) 0.746825
5. (a) 1.401435
(b) 1.402558
(c) 1.402206
6. (a) 1.119061
(b) 1.118107
(c) 1.118428
7. (a) 0.984120
(b) 0.983669
(c) 0.983819
8. (a) 0.409140
(b) 0.388849
(c) 0.395802
9. (a) 0.132465
(b) 0.132857
(c) 0.132727
10. (a) 1.098004
(b) 1.098709
(c) 1.109031
11. 31.94
12. 8.6 mi
3
13. 5.8 m
14. (a) 15.4
(b) 0.022
15. 19.46 L
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
16. 54,730 megawatt-hours
SECTION 7.8
7.8
I M P RO P E R I N T E G R A L S
A Click here for answers.
S Click here for solutions.
1–33 Determine whether each integral is convergent or
divergent. Evaluate those that are convergent.
1.
3.
5.
7.
y
y
y
21.
23.
⫺1
⬁
⬁
⫺⬁
⬁
1
y
y
⬁
⫺⬁
y
0
dx
x 2 ⫹ 4x ⫹ 6
1
dx
4x ⫺ 5
y
y
y
24.
27.
y
1
dw
3
w⫺5
s
29.
y
2
1
dx
共2x ⫺ 3兲2
31.
y
9
x dx
33.
y
e
⬁
1
⬁
⫺⬁
y
22.
y␲
e⫺x dx
⫺⬁
0
⫺⬁
0
y
1
dx
x共ln x兲 2
⬁
e
y
x
dx
共x ⫹ 2兲共x ⫹ 3兲
⬁
0
y
␲兾2
⫺⬁
y
⬁
0
y
⬁
0
y
e 3x dx
5
dx
2x ⫹ 3
⬁
5
4
␲兾2
25.
⫺⬁
12.
20.
⬁
0
y
18.
xe 2x dx
y
10.
cos x dx
1
2
共2x 2 ⫺ x ⫹ 3兲 dx
8.
16.
⫺⬁
y
1
dx
3
x⫺1
s
6.
1
dx
共x ⫹ 2兲共x ⫹ 3兲
⬁
0
4.
14.
0
y
x 3 dx
1
dx
x2 ⫹ 9
3
⬁
2.
sin ␲ x dx
⫺⬁
y
1
dx
共x ⫹ 3兲3兾2
1
dx
sx ⫹ 3
⬁
2
y
19.
⬁
⫺⬁
11.
17.
⬁
2
y
15.
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
y
9.
13.
IMPROPER INTEGRALS
sin 2␪ d␪
1
dx
共5 ⫺ x兲 2兾5
␲兾4
0
⫺2
0
1
␲兾2
sec 2x dx
26.
y␲
csc 2t dt
28.
y
1
dx
x2 ⫺ 1
30.
y
32.
y␲
dx
共x ⫹ 9兲sx
兾4
tan 2x dx
␲兾4
0
2
0
cos x
dx
ssin x
x
dx
s4 ⫺ x 2
3␲兾4
兾4
tan x dx
1
dx
4
xs
ln x
; 34–37 Sketch the region and find its area (if the area is finite).
ⱍ
S 苷 {共x, y兲 ⱍ x 艌 0, 0 艋 y 艋 1兾sx ⫹ 1 }
S 苷 兵共x, y兲 ⱍ 0 艋 x 艋 ␲, 0 艋 y 艋 tan x sec x 其
S 苷 {共x, y兲 ⱍ 3 ⬍ x 艋 7, 0 艋 y 艋 1兾sx ⫺ 3 }
34. S 苷 兵共x, y兲 x 艌 1, 0 艋 y 艋 共ln x兲兾x 2 其
35.
36.
37.
38– 40 Use the Comparison Theorem to determine whether the
integral is convergent or divergent.
38.
y
39.
y
40.
y
xe⫺x dx
1
dx
2x
兾4
⬁
1
⬁
1
⬁
1
sin2 x
dx
x2
1
dx
sx ⫹ 1
3
s1 ⫹ sx
sx
dx
■
1
■
SECTION 7.8
7.8
IMPROPER INTEGRALS
ANSWERS
E Click here for exercises.
S Click here for solutions.
1. √2
2. 1
31. π
6
3. Divergent
4. Divergent
33. 43
5. Divergent
6. 12
34.
7. Divergent
8. Divergent
5
9. Divergent
12. Divergent
13. π
4
14. 1
15. − ln
17. Divergent
19.
1 2
e
4
π
2
35.
10. 13
11. Divergent
2
3
32. Divergent
16. Divergent
Area = 1
36.
Area is infinite
37.
18. Divergent
20. 1
1
ln 2
21. √
22.
23. Divergent
24. 53
25. Divergent
26. Divergent
27. Divergent
28. 2
29. Divergent
30. 2
3/4
Area is infinite
38. Convergent
Area = 4
39. Convergent
40. Divergent
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
2
SECTION 8.1
8.1
■
ARC LENGTH
A Click here for answers.
S Click here for solutions.
1– 4 Find the length of the arc of the given curve from point A
10–12 Set up, but do not evaluate, an integral for the length of
to point B.
the curve.
1. y 苷 1 ⫺ x 2兾3;
A共1, 0兲,
2. 9y 苷 x共x ⫺ 3兲 ;
2
2
3. y 2 苷 共x ⫺ 1兲3;
B共8, ⫺3兲
A共0, 0兲,
B(4,
5. y 苷 3 共x 2 ⫹ 2兲3兾2,
1
0艋x艋1
4
1
x
6. y 苷
,
⫹
4
8x 2
8. y 苷 ln共1 ⫺ x 2 兲,
9. y 苷 ln共cos x兲,
10. y 苷 tan x,
)
A( 127 , 1), B( 67
24 , 2)
5–9 Find the length of the curve.
7. y 苷 ln共sin x兲,
2
3
A共1, 0兲, B共2, 1兲
4. 12xy 苷 4y 4 ⫹ 3;
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
ARC LENGTH
1艋x艋3
␲兾6 艋 x 艋 ␲兾3
0 艋 x 艋 12
0 艋 x 艋 ␲兾4
11. y 苷 x 3,
0 艋 x 艋 ␲兾4
0艋x艋1
12. y 苷 e x cos x,
0 艋 x 艋 ␲兾2
13–16 Use Simpson’s Rule with n 苷 10 to estimate the arc length
of the curve.
13. y 苷 x 3,
0艋x艋1
14. y 苷 1兾x,
1艋x艋2
15. y 苷 sin x,
0艋x艋␲
16. y 苷 tan x,
0 艋 x 艋 ␲兾4
1
■
SECTION 8.1
8.1
ARC LENGTH
ANSWERS
E Click here for exercises.
√ √
1
1. 27
80 10 − 13 13
2. 14
3
√
13−8
3. 13 27
4. 59
24
5. 43
6.
S Click here for solutions.
√
2+1
π/4 √
10. 0
1 + sec4 x dx
1√
11. 0
1 + 9x4 dx
π/2 12. L = 0
1 + e2x (1 − sin 2x) dx
9. ln
13. 1.548
181
9
7. ln 1 + √2
3
8. ln 3 − 12
14. 1.132
15. 3.820
16. 1.278
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
2
SECTION 8.3
8.3
APPLICATIONS TO PHYSICS AND ENGINEERING
■
A P P L I C AT I O N S TO P H Y S I C S A N D E N G I N E E R I N G
A Click here for answers.
S Click here for solutions.
1–3 The end of a tank containing water is vertical and has the
indicated shape. Explain how to approximate the hydrostatic
force against the end of the tank by a Riemann sum. Then
express the force as an integral and evaluate it.
4. The masses m 1 苷 4 and m 2 苷 8 are located at the points
P1共⫺1, 2兲 and P2共2, 4兲. Find the moments Mx and My and the
center of mass of the system.
5–6 Sketch the region bounded by the curves, and visually
1.
2.
10 m
estimate the location of the centroid. Then find the exact coordinates of the centroid.
water
level
6 ft
5m
water
level
4 ft
5. y 苷 x 2 ,
6. y 苷 sx ,
y 苷 0,
y 苷 0,
x苷2
x苷9
4 ft
7–8 Find the centroid of the region bounded by the curves.
3.
7. y 苷 sin 2x,
y 苷 0,
x 苷 0,
h
8. y 苷 ln x,
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
b
y 苷 0,
x苷e
x 苷 ␲兾2
1
2
■
SECTION 8.3
8.3
APPLICATIONS TO PHYSICS AND ENGINEERING
ANSWERS
E Click here for exercises.
S Click here for solutions.
6
1. 1.23 × 10 N
3
2. 1.56 × 10 lb
2
3. 13 ρgbh
4. 40, 12, 1, 10
3
5. (1.5, 1.2)
7.
8.
27
5
π
4
,
, 98
π
2
8
e +1 e−2
,
4
2
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
6.
SECTION 8.5
8.5
P RO B A B I L I T Y
A Click here for answers.
S Click here for solutions.
1. If f 共t兲 is the probability density function for the lifetime of
a type of battery, where t is measured in hours, what is the
meaning of each integral?
(a)
y
200
100
f 共t兲 dt
(b)
y
⬁
200
f 共t兲 dt
2. If f 共x兲 is the probability density function for the blood cho-
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
lesterol level of men over the age of 40, where x is measured
in milligrams per deciliter, express the following probabilities as integrals.
(a) The probability that the cholesterol level of such a man
lies between 180 and 240
(b) The probability that the cholesterol level of such a man
is less than 200
PROBABILITY
■
1
2
■
SECTION 8.5
8.5
PROBABILITY
ANSWERS
E Click here for exercises.
S Click here for solutions.
1. (a) The probability that a randomly chosen battery will have
a lifetime of between 100 and 200 hours
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
(b) The probability that a randomly chosen battery will have
a lifetime of at least 200 hours
240
2. (a) 180 f (x) dx
200
(b) 0 f (x) dx
SECTION 9.1
9.1
M O D E L I N G W I T H D I F F E R E N T I A L E Q UAT I O N S
S Click here for solutions.
3
1. Show that y 苷 2 ⫹ e⫺x is a solution of the differential
equation y⬘ ⫹ 3x y 苷 6x 2.
2
2. Verify that y 苷 共2 ⫹ ln x兲兾x is a solution of the initial-value
problem
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
x 2 y⬘ ⫹ xy 苷 1
y共1兲 苷 2
MODELING WITH DIFFERENTIAL EQUATIONS
■
1
2
■
SECTION 9.1
9.1
MODELING WITH DIFFERENTIAL EQUATIONS
SOLUTIONS
E Click here for exercises.
−x3
1. y = 2 + e
3
⇒ y = −3x2 e−x .
LHS = y + 3x2 y
3
3
= −3x2 e−x + 3x2 2 + e−x
3
3
= −3x2 e−x + 6x2 + 3x2 e−x
= 6x2
= RHS
2 + ln x
⇒
x
x (1/x) − (2 + ln x) (1)
−1 − ln x
=
and
y =
x2
(x)2
2 + ln 1
y (1) =
= 2.
1
2. y =
LHS = x2 y + xy
−1 − ln x
2 + ln x
= x2
+
x
x2
x
= (−1 − ln x) + (2 + ln x)
= 1
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
= RHS
SECTION 9.2
9.2
DIRECTION FIELDS AND EULER’S METHOD
■
1
DIRECTION FIELDS AND EULER’S METHOD
A Click here for answers.
S Click here for solutions.
1. A direction field for the differential equation y⬘ 苷 y ⫺ e⫺x is
shown. Sketch the graphs of the solutions that satisfy the
given initial conditions.
(a) y共0兲 苷 0
(b) y共0兲 苷 1
(c) y共0兲 苷 ⫺1
3–4 Sketch a direction field for the differential equation. Then use
it to sketch three solution curves.
3. y⬘ 苷 x ⫺ y
4. y⬘ 苷 xy ⫹ y 2
y
2
5– 8 Sketch the direction field of the given differential equation.
Then use it to sketch a solution curve that passes through the given
point.
1
5. y⬘ 苷 y 2,
_2
_1
0
1
2
x
共0, 1兲
7. y⬘ 苷 x 2 ⫹ y 2,
6. y⬘ 苷 x 2 ⫹ y,
共0, 0兲
8. y⬘ 苷 y共4 ⫺ y兲,
共1, 1兲
共0, 1兲
_1
9. Use Euler’s method with step size 0.5 to compute the approxi-
mate y-values y1 , y2 , y3 , and y4 of the solution of the initialvalue problem y⬘ 苷 1 ⫹ 3x ⫺ 2y, y共1兲 苷 2.
_2
2. (a) A direction field for the differential equation
y⬘ 苷 2y共 y ⫺ 2兲 is shown. Sketch the graphs of the solutions that satisfy the given initial conditions.
(i) y共0兲 苷 1
(ii) y共0兲 苷 2.5
(iii) y共0兲 苷 ⫺1
(b) Suppose the initial condition is y共0兲 苷 c. For what values of c is lim t l ⬁ y共t兲 finite? What are the equilibrium
solutions?
2
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
1
0
_1
11. Use Euler’s method with step size 0.1 to estimate y共0.5兲, where
y共x兲 is the solution of the initial-value problem y⬘ 苷 x 2 ⫹ y 2,
y共0兲 苷 1.
where y共x兲 is the solution of the initial-value problem
y⬘ 苷 2xy 2, y共0兲 苷 1.
(b) Repeat part (a) with step size 0.1.
3
_1
y共x兲 is the solution of the initial-value problem y⬘ 苷 x ⫹ y 2,
y共0兲 苷 0.
12. (a) Use Euler’s method with step size 0.2 to estimate y共0.4兲,
y
_2
10. Use Euler’s method with step size 0.2 to estimate y共1兲, where
1
2
x
■
SECTION 9.2
9.2
DIRECTION FIELDS AND EULER’S METHOD
ANSWERS
E Click here for exercises.
1. (a)
S Click here for solutions.
(b)
(c)
3.
4.
5.
6.
7.
2. (a) (i)
8.
(ii)
9. 2, 2.75, 3.5, 4.25
10. 0.4150
(iii)
11. 1.8371
12. (a) 1.08
(b) c ≤ 2; y = 0, y = 2
(b) 1.1292
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
2
SECTION 9.3
9.3
SEPARABLE EQUATIONS
S E PA R A B L E E Q UAT I O N S
A Click here for answers.
S Click here for solutions.
1– 8 Solve the differential equation.
11. x ⫹ 2y sx 2 ⫹ 1
dy
1.
苷 y2
dx
2. yy⬘ 苷 x
3. y⬘ 苷 xy
dy
x ⫹ sin x
苷
4.
dx
3y 2
5. x 2 y⬘ ⫹ y 苷 0
6. y⬘ 苷
du
苷 e u⫹2t
7.
dt
ln x
xy ⫹ xy 3
dx
苷 1 ⫹ t ⫺ x ⫺ tx
8.
dt
12. e y y⬘ 苷
3x 2
,
1⫹y
dy
苷 0,
dx
y共0兲 苷 1
y共2兲 苷 0
13.
du
2t ⫹ 1
, u共0兲 苷 ⫺1
苷
dt
2共u ⫺ 1兲
14.
ty ⫹ 3t
dy
,
苷 2
dt
t ⫹1
y共2兲 苷 2
; 15. Solve the initial-value problem y⬘ 苷 y sin x, y共0兲 苷 1, and
9–14 Find the solution of the differential equation that satisfies
the given initial condition.
dy
1⫹x
9.
苷
,
dx
xy
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
10. xe⫺t
dx
苷 t,
dt
x ⬎ 0,
graph the solution.
16. Find a function f such that f ⬘共x兲 苷 x 3 f 共x兲 and f 共0兲 苷 1.
y共1兲 苷 ⫺4
17. Find a function t such that t⬘共x兲 苷 t共x兲共1 ⫹ t共x兲兲 and
t共0兲 苷 1.
x共0兲 苷 1
■
1
2
■
SECTION 9.3
9.3
SEPARABLE EQUATIONS
ANSWERS
E Click here for exercises.
1. y =
−1
,y=0
x+C
2
y
√
t2 + t + 4
√
14. y = −3 + 5t2 + 5
13. u = 1 −
2
x2 /2
3. y = Ce
3
1 2
x
2
3
12. ye = x − 8
2. x − y = C
4. y =
S Click here for solutions.
− cos x + C
1−cos x
15. y = e
1/x
2
6. y + 1 =
2 (ln x)2 + C
2t
7. u = − ln C − 12 e
− t2 /2+t)
8. x = 1 + Ce (
2
9. y = 2 ln x + 2x + 14
2 (t −
+3
√
2
11. y = 2 − x2 + 1
10. x =
1) et
x4 /4
16. f (x) = e
17. g (x) =
ex
2 − ex
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
5. y = Ce
SECTION 3.8
3.8
■
E X P O N E N T I A L G ROW T H A N D D E C AY
A Click here for answers.
1. A bacteria culture starts with 4000 bacteria and the population
triples every half-hour.
(a) Find an expression for the number of bacteria after t hours.
(b) Find the number of bacteria after 20 min.
(c) When will the population reach 20,000?
2. A bacteria culture grows with constant relative growth rate.
The count was 400 after 2 hours and 25,600 after 6 hours.
(a) What was the initial population of the culture?
(b) Find an expression for the population after t hours.
(c) In what period of time does the population double?
(d) When will the population reach 100,000?
3. Polonium-210 has a half-life of 140 days.
(a) If a sample has a mass of 200 mg, find a formula for the
mass that remains after t days.
(b) Find the mass after 100 days.
(c) When will the mass be reduced to 10 mg?
(d) Sketch the graph of the mass function.
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
EXPONENTIAL GROWTH AND DECAY
S Click here for solutions.
4. Polonium-214 has a very short half-life of 1.4 ⫻ 10 ⫺4 s.
(a) If a sample has a mass of 50 mg, find a formula for the
mass that remains after t seconds.
(b) Find the mass that remains after a hundredth of a second.
(c) How long would it take for the mass to decay to 40 mg?
5. On a hot day a thermometer is taken outside from an air-
conditioned room where the temperature is 21⬚C. After one
minute it reads 27⬚C and after 2 minutes it reads 30⬚C.
(a) What is the outdoor temperature?
(b) Sketch the graph of the temperature function.
1
2
■
SECTION 3.8
3.8
EXPONENTIAL GROWTH AND DECAY
ANSWERS
E Click here for exercises.
1. (a) y 共t兲 苷 4000 ⴢ 9 t
2. (a) 50
(b) 8320
(b) y 共t兲 苷 50 ⴢ 8
⫺t兾140
3. (a) y 共t兲 苷 200 ⴢ 2
t兾2
S Click here for solutions.
(c) ⬇44 min
(c) 40 min
(b) ⬇121.9 mg
(c) ⬇605 days
4. (a) y 共t兲 苷 50 ⴢ 2⫺t兾0.00014
(c) ⬇ 4.5 ⫻ 10
5. (a) 33⬚
⫺5
(b) ⬇1.57 ⫻ 10⫺20 mg
s
(b)
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
(d)
SECTION 11.1
11.1
SEQUENCES
A Click here for answers.
S Click here for solutions.
1– 8 List the first five terms of the sequence.
n
1. a n 苷
2n ⫹ 1
共⫺1兲 n
3. a n 苷
2n
n⫺1
5.
再 冎
4n ⫺ 3
2. a n 苷
3n ⫹ 4
冉 冊
2
4. a n 苷 ⫺
3
n
n␲
sin
2
1 ⴢ 3 ⴢ 5 ⴢ ⭈ ⭈ ⭈ ⴢ 共2n ⫺ 1兲
7. a n 苷
n!
再
共⫺7兲 n⫹1
n!
23. a n 苷 sin
25.
27.
1
6. a 1 苷 1, a n⫹1 苷
1 ⫹ an
8.
SEQUENCES
n␲
2
再 冎
再 冎
3 ⫹ 共⫺1兲 n
n2
26.
ln共n 2 兲
n
28.
n!
共n ⫹ 2兲!
冎
共⫺1兲 n sin
冎
冎
1
n
ln共2 ⫹ e n兲
3n
29. 兵sn ⫹ 2 ⫺ sn 其
30.
31. a n 苷 n2 ⫺n
32. a n 苷 共1 ⫹ 3n兲1兾n
34. a n 苷 (sn ⫹ 1 ⫺ sn ) sn ⫹
35. a n 苷 共⫺1兲 n⫺1
9–14 Find a formula for the general term a n of the sequence,
assuming that the pattern of the first few terms continues.
10.
再
再
再
33. a n 苷 n ⫺1兾n
冎
9. 兵1, 4, 7, 10, . . .其
24. a n 苷 2 ⫹ cos n␲
{
, , , , . . .}
3 4 5 6
16 25 36 49
11.
{32 , ⫺ 94 , 278 , ⫺ 8116 , . . .}
12. 兵⫺1, 2, ⫺6, 24, . . .其
13.
{23 , ⫺ 35 , 47 , ⫺ 59 , . . .}
14. 兵0, 2, 0, 2, 0, 2, . . .其
36.
37.
再 冉
再 冎
arctan
1
2
n4
1 ⫹ n2 ⫹ n3
2n
2n ⫹ 1
冊冎
sin n
sn
38. a n 苷
1
2
n
2 ⫹
2 ⫹ ⭈⭈⭈ ⫹
n
n
n2
39. a n 苷
n cos n
n2 ⫹ 1
15–39 Determine whether the sequence converges or diverges.
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
If it converges, find the limit.
15. a n 苷
1
4n 2
16. a n 苷 4 sn
17. a n 苷
n2 ⫺ 1
n2 ⫹ 1
18. a n 苷
n2
19. a n 苷
n⫹1
n2
21. a n 苷 共⫺1兲
1 ⫹ n3
n
4n ⫺ 3
3n ⫹ 4
3
4
n⫹s
n
s
20. a n 苷
5
sn ⫹ sn
22.
再 冎
␲n
3n
40–43 Determine whether the sequence is increasing,
decreasing, or not monotonic. Is the sequence bounded?
40. a n 苷
1
3n ⫹ 5
41. a n 苷
n⫺2
n⫹2
42. a n 苷
3n ⫹ 4
2n ⫹ 5
43. a n 苷
sn
n⫹2
■
1
■
SECTION 11.1
11.1
SEQUENCES
ANSWERS
E Click here for exercises.
1.
2.
3.
4.
5.
1 2 3 4 5
, , , ,
3 5 7 9 11
1 1 9 13 17
, , , ,
7 2 13 16 19
1
1 3
1 5
,− , ,− ,
2
2 8
4 32
2 4 −8 16
32
− , ,
, ,−
3 9 27 81
243
1, 0, −1, 0, 1
1
,
2
3
7. 1, ,
2
2 3 5
, ,
3 5 8
5 35 63
, ,
2 8 8
343 2401
16,807 117,649
8. 49, −
,
,−
,
2
6
24
120
9. an = 3n − 2
6. 1,
10. an =
n+2
(n + 3)2
n+1
11. an = (−1)
n
12. an = (−1)
S Click here for solutions.
20. 0
21. 0
22. Diverges
23. Diverges
24. Diverges
25. 0
26. 0
27. 0
28. 0
29. 0
1
3
31. 0
30.
32. 1
n
3
2
n!
33. 1
34.
35.
n+1
2n + 1
36.
n−1
37.
n+1
13. an = (−1)
14. an = 1 − (−1)
or an = 1 + (−1)n
1
2
Diverges
π
4
0
1
2
0
15. 0
38.
16. Diverges
39.
17. 1
40. Decreasing; yes
4
18.
3
19. Diverges
41. Increasing; yes
42. Increasing; yes
43. Not monotonic; yes
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
2
SECTION 11.2
11.2
S Click here for solutions.
; 1–5 Find at least 10 partial sums of the series. Graph both the
sequence of terms and the sequence of partial sums on the same
screen. Does it appear that the series is convergent or divergent?
If it is convergent, find the sum. If it is divergent, explain why.
⬁
兺
n苷1
⬁
3.
兺
n苷1
⬁
5.
兺
n苷1
⬁
24.
n苷1
⬁
26.
⬁
10
3n
2.
4.
冉 冊
⫺
兺
n苷4
28.
3
n共n ⫺ 1兲
1
2
n⫺1
3
n⫺1
冊
⬁
25.
⬁
27.
1
⫹ 2n
n
1
n
sin
⬁
⫺ sin
6–33 Determine whether the series is convergent or divergent.
If it is convergent, find its sum.
n
6. 4 ⫹ 5 ⫹
8
32
⫹ 125
⫹ ⭈⭈⭈
16
25
8. 1 ⫺ 2 ⫹ 4 ⫺ 8 ⫹ ⭈ ⭈ ⭈
1
10. ⫺
81
100
1
⫹
1
⫺1⫹
9
10
10
9
7. 1 ⫺ 2 ⫹ 4 ⫺
3
9.
2
3
1
1
1
1
⫹ 8 ⫹ 10 ⫹ 12 ⫹ ⭈ ⭈ ⭈
26
2
2
2
12.
1
36
6
⫹ 301 ⫹ 251 ⫹ 125
⫹ ⭈⭈⭈
n苷1
⬁
29.
兺3
n苷1
⬁
16.
兺
n苷1
⬁
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
18.
兺
n苷0
⬁
20.
兺
n苷1
8
n苷1
⬁
32.
⬁
13.
兺
⬁
21.
兺
n苷1
n
⫹ 共0.2兲n 兴
1
n共n ⫹ 1兲共n ⫹ 2兲
n2 ⫺ 1
n2
34–38 Express the number as a ratio of integers.
37. 1.123
38. 4.1570
冉冊
兺 共⫺1兲
n苷1
1
2n
n苷1
4 n⫹1
5n
e
17. 兺 5
3
n苷1
19.
兺 关2共0.1兲
n苷1
1
e 2n
⬁
23.
兺
n苷1
39–43 Find the values of x for which the series converges. Find
n
⬁
⬁
22.
兺
⬁
n
5
8n
兺
36. 0.307 苷 0.307307307307 . . .
n苷0
冉 冊
1
4n 2 ⫺ 1
35. 0.15 苷 0.15151515 . . .
15.
n⫺1
3
⫺
␲
兺
1
共3n ⫺ 2兲共3n ⫹ 1兲
34. 0.5 苷 0.5555 . . .
⬁
n⫹1
n苷2
⫹ ⭈⭈⭈
⫺ 29 ⫹ 272 ⫺ 812 ⫹ ⭈ ⭈ ⭈
n苷1
⫺n
27
8
⫺ ⭈⭈⭈
11.
⬁
9
兺 ln
兺
1
5 ⫹ 2⫺n
1
n⫹1
兺 ln n ⫹ 1
⬁
33.
兺
n苷1
冉 冊
兺冋 冉 冊
冉 冊册
兺
n苷1
31.
2
n
3共n ⫹ 1兲共n ⫹ 2兲
⬁
30.
⫹
2
n苷1
n⫺1
2
7
冉
⬁
n苷1
n
n⫹1
兺
n苷1
兺 sin n
⬁
兺
n苷1
14.
■
SERIES
A Click here for answers.
1.
SERIES
the sum of the series for those values of x.
⬁
2n
n⫺1
3
2 3n⫹1
1
n共n ⫹ 2兲
n
s1 ⫹ n 2
39.
兺3x
n n
⬁
40.
n苷0
兺2
n
sin n x
n苷0
兺 tan x
n
n苷0
42.
⬁
1
xn
兺
n苷0
⬁
43.
xn
5n
n苷2
⬁
41.
兺
1
■
SECTION 11.2
11.2
SERIES
ANSWERS
E Click here for exercises.
1. 3.33333, 4.44444,
4.81481, 4.93827,
4.97942, 4.99314,
4.99771, 4.99924,
4.99975, 4.99992
S Click here for solutions.
6.
20
3
9.
1
2
7. Divergent
8.
10. Divergent
11.
13.
1
e2 − 1
14. Divergent
15. 20
16.
π
π+3
17.
2. 0.8415, 1.7508,
approach 0)
18.
8
3
19. Divergent
21.
3
4
22.
1.91667, 2.71667,
3.55000, 4.40714,
5.28214, 6.17103,
7.07103, 7.98012
Divergent (terms do not
approach 0)
4. 0.25000, 0.40000,
0.50000, 0.57143,
0.62500, 0.66667,
0.70000, 0.72727,
0.75000, 0.76923
Convergent, sum = 1
17
36
5e
3−e
20. Divergent
23. Divergent
24. 5
25. Divergent
26. Divergent
1
3
28. Divergent
29.
30. sin 1
31. Divergent
27.
3. 0.50000, 1.16667,
1
48
12. Divergent
Convergent, sum = 5
1.8919, 1.1351,
0.1762, −0.1033,
0.5537, 1.5431,
1.9552, 1.4112
Divergent (terms do not
2
3
5
9
307
556
36.
37.
999
495
1
1
1
39. − 3 < x < 3 ;
1 − 3x
33. ln 12
34.
40. −5 < x < 5;
x2
25 − 5x
41. nπ − π
< x < nπ + π6 (n any integer);
6
42. |x| > 1;
1
2
1
32.
4
5
35.
33
41,566
38.
9999
x
x−1
1
1 − 2 sin x
1
5. 1.000000, 0.714286,
0.795918, 0.772595,
0.779259, 0.777355,
0.777899, 0.777743,
0.777788, 0.777775
Convergent, sum =
7
9
43. nπ − π
< x < nπ + π4 (n any integer);
4
1 − tan x
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
2
SECTION 11.3
11.3
T H E I N T E G R A L T E S T A N D E S T I M AT E S O F S U M S
A Click here for answers.
S Click here for solutions.
1. Use the Integral Test to determine whether the series
⬁
8.
⬁
10.
is convergent or divergent.
兺
n苷5
⬁
4.
兺
n苷1
⬁
6.
兺
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
n苷5
1
n 1.0001
1
sn ⫹ 1
兺 ne
⫺n 2
⬁
9.
⬁
3.
⬁
5.
1
共n ⫺ 4兲2
7.
兺
n苷1
⬁
兺
n苷1
⫺0.99
冉
2
3
⫹ 3
n sn
n
1
2n ⫹ 3
兺
n苷1
兺n
n苷1
2
3
n
s
⬁
12.
冊
⬁
14.
兺
n苷1
兺
n苷2
⬁
11.
n苷1
2–15 Determine whether the series is convergent or divergent.
2.
兺
n苷1
1
1
1
1
⫹ ⫹
⫹
⫹ ⭈⭈⭈
3
7
11
15
⬁
THE INTEGRAL TEST AND ESTIMATES OF SUMS
兺
n苷1
⬁
1
4n ⫹ 1
13.
ln n
n2
15.
2
兺
n苷1
⬁
兺
n苷1
1
n2 ⫺ 1
n
2n
arctan n
1 ⫹ n2
1
n 2 ⫹ 2n ⫹ 2
■
1
2
■
SECTION 11.3
11.3
THE INTEGRAL TEST AND ESTIMATES OF SUMS
ANSWERS
E Click here for exercises.
1. Divergent
S Click here for solutions.
9. Convergent
2. Convergent
10. Convergent
3. Divergent
11. Convergent
4. Divergent
12. Convergent
5. Convergent
13. Convergent
6. Convergent
14. Convergent
7. Divergent
15. Convergent
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
8. Divergent
SECTION 11.4
11.4
T H E C O M PA R I S O N T E S T S
A Click here for answers.
S Click here for solutions.
1–18 Determine whether the series converges or diverges.
⬁
1.
兺
n苷1
⬁
3.
兺
n苷1
1
n3 ⫹ n 2
3
n2n
⬁
2.
兺
n苷1
⬁
4.
兺
n苷2
⬁
11.
n苷1
3
4n ⫹ 5
1
sn ⫺ 1
兺
⬁
13.
兺
n苷1
⬁
⬁
5.
兺
n苷0
⬁
7.
兺
n苷1
⬁
9.
兺
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
n苷1
THE COMPARISON TESTS
⬁
1 ⫹ 5n
4n
6.
3
n共n ⫹ 3兲
8.
1
3
sn共n ⫹ 1兲共n ⫹ 2兲
兺
n苷1
⬁
兺
n苷1
⬁
10.
兺
n苷1
sin 2 n
n sn
15.
1
sn共n ⫹ 1兲共n ⫹ 2兲
17.
兺
n苷3
⬁
n
共n ⫹ 1兲2 n
兺
n苷1
⬁
3 ⫹ cos n
3n
12.
n
sn 5 ⫹ 4
14.
1
n ⫺4
16.
n⫹1
n2 n
18.
2
兺
5n
2n 2 ⫺ 5
兺
⬁
arctan n
n4
⬁
n2 ⫹ 1
n4 ⫹ 1
n苷1
n苷1
兺
n苷1
⬁
兺
n苷1
n 2 ⫺ 3n
sn 10 ⫺ 4n 2
3
■
1
■
SECTION 11.4
11.4
THE COMPARISON TESTS
ANSWERS
E Click here for exercises.
S Click here for solutions.
1. Converges
10. Converges
2. Converges
11. Converges
3. Converges
12. Diverges
4. Diverges
13. Converges
5. Diverges
14. Converges
6. Converges
15. Converges
7. Converges
16. Converges
8. Converges
17. Converges
9. Diverges
18. Converges
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
2
SECTION 11.5
11.5
A LT E R N AT I N G S E R I E S
A Click here for answers.
S Click here for solutions.
1–14 Test the series for convergence or divergence.
1.
3
5
⬁
12.
⫺ ⫹ ⫺ ⫹ ⫺ ⭈⭈⭈
3
6
3
7
3
8
3
9
兺 共⫺1兲
2. ⫺5 ⫺ ⫹ ⫺ ⫹
5
5
5
8
5
11
⫺
5
14
14.
⫹ ⭈⭈⭈
⬁
4.
兺
n苷1
3
4
5
⬁
兺 共⫺1兲
n⫹1
5.
兺
n苷1
n苷1
⬁
8.
兺 共⫺1兲
n
n苷1
⬁
10.
兺
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
n苷1
共⫺1兲 n
n
5n ⫹ 1
⬁
7.
兺
n苷2
共⫺1兲 n
sn ⫹ 3
⬁
15.
兺
共⫺1兲 n⫺1
n ln n
⬁
16.
兺
n苷0
⬁
n
n ⫹1
9.
2n
4n ⫹ 1
11.
2
⬁
13.
兺 共⫺1兲
n苷1
n⫹1
n
2n
共⫺1兲n⫺1
3
ln n
s
15–18 Approximate the sum of the series to the indicated
accuracy.
n苷1
6.
sn
n⫹4
6
⬁
共⫺1兲 n⫺1
n2
兺
n苷2
3. ⫺ 2 ⫹ 3 ⫺ 4 ⫹ 5 ⫺ 6 ⫹ 7 ⫺ ⭈ ⭈ ⭈
2
n⫺1
n苷1
⬁
5
2
1
ALTERNATING SERIES
兺 共⫺1兲
n苷1
⬁
兺
n苷1
n
n2
n ⫹1
共⫺1兲 n⫺1
2
2n 2
4n 2 ⫹ 1
⬁
17.
兺
n苷0
⬁
18.
兺
n苷1
共⫺1兲n⫺1
共2n ⫺ 1兲!
(four decimal places)
共⫺1兲 n
共2n兲!
(four decimal places)
共⫺1兲 n
2 n n!
(four decimal places)
共⫺1兲 n⫺1
n6
(five decimal places)
■
1
2
■
SECTION 11.5
11.5
ALTERNATING SERIES
ANSWERS
E Click here for exercises.
S Click here for solutions.
1. Convergent
10. Divergent
2. Convergent
11. Divergent
3. Divergent
12. Convergent
4. Convergent
13. Convergent
5. Convergent
14. Convergent
6. Divergent
15. 0.8415
7. Convergent
16. 0.5403
8. Convergent
17. 0.6065
9. Divergent
18. 0.98555
SECTION 11.6
11.6
A B S O L U T E C O N V E R G E N C E A N D T H E R AT I O A N D RO OT T E S T S
A Click here for answers.
S Click here for solutions.
1–28 Determine whether the series is absolutely convergent,
conditionally convergent, or divergent.
⬁
1.
兺
n苷1
ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS
⬁
共⫺1兲
n sn
n⫺1
2.
兺
n苷1
共⫺1兲
sn
⬁
17.
兺
n苷1
3.
兺
n苷1
⬁
共⫺3兲 n
n3
4.
兺
n苷0
⬁
19.
兺
共⫺3兲 n
n!
⬁
21.
兺
n苷1
⬁
5.
兺
n苷1
⬁
共⫺1兲 n
5⫹n
6.
兺
n苷1
共⫺1兲 n⫺1
n!
⬁
23.
兺
n苷1
⬁
7.
兺
n苷1
⬁
共⫺1兲
2n ⫹ 1
n⫹1
8.
兺
n苷1
9.
兺
n苷1
共⫺1兲 n⫺1
共2n ⫺ 1兲!
⬁
11.
兺 共⫺1兲
n⫺1
n苷1
⬁
13.
兺 共⫺1兲
n苷1
⬁
15.
兺
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
n苷1
n
sn
n⫹1
2n
n ⫹1
2
共⫺1兲 n⫹1 5 n⫺1
共n ⫹ 1兲 2 4 n⫹2
cos共n␲兾6兲
n sn
20.
8 ⫺ n3
n!
22.
共⫺n兲 n
5 2n⫹3
24.
共⫺2兲 n n 2
共n ⫹ 2兲!
26.
兺
n苷1
共⫺1兲n arctan n
n3
共⫺1兲
n2 ⫹ 1
⬁
兺
n苷1
⬁
兺
n苷2
⬁
兺
n苷1
n!
共⫺10兲 n
共⫺1兲 n
ln n
冉
1 ⫺ 3n
3 ⫹ 4n
冊
n
n⫺1
⬁
25.
兺
n苷1
⬁
18.
n
n苷1
⬁
⬁
sin 2n
n2
⬁
兺
n苷1
共n ⫹ 2兲!
n! 10 n
⬁
n
10. 兺 共⫺1兲 n 2
n ⫹4
n苷1
⬁
12.
兺 共⫺1兲
n
n苷1
⬁
14.
兺
n苷1
⬁
16.
兺
n苷1
27. 1 ⫺
2n
3n ⫺ 4
共⫺2兲 n
n 3 n⫹1
共n ⫹ 1兲5 n
n 3 2n
2!
3!
4!
⫹
⫺
⫹ ⭈⭈⭈
1ⴢ3
1ⴢ3ⴢ5
1ⴢ3ⴢ5ⴢ7
⫹
28.
共⫺1兲n⫺1 n!
⫹ ⭈⭈⭈
1 ⴢ 3 ⴢ 5 ⴢ ⭈ ⭈ ⭈ ⴢ 共2n ⫺ 1兲
1
1ⴢ4
1ⴢ4ⴢ7
1 ⴢ 4 ⴢ 7 ⴢ 10
⫹
⫹
⫹
⫹ ⭈⭈⭈
3
3ⴢ5
3ⴢ5ⴢ7
3ⴢ5ⴢ7ⴢ9
⫹
1 ⴢ 4 ⴢ 7 ⴢ ⭈ ⭈ ⭈ ⴢ 共3n ⫺ 2兲
⫹ ⭈⭈⭈
3 ⴢ 5 ⴢ 7 ⴢ ⭈ ⭈ ⭈ ⴢ 共2n ⫹ 1兲
■
1
2
■
SECTION 11.6
11.5
11.6
ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS
ANSWERS
E Click here for exercises.
S Click here for solutions.
1. Absolutely convergent
15. Divergent
2. Conditionally convergent
16. Absolutely convergent
3. Divergent
17. Absolutely convergent
4. Absolutely convergent
18. Absolutely convergent
5. Conditionally convergent
19. Absolutely convergent
6. Absolutely convergent
20. Divergent
7. Conditionally convergent
21. Absolutely convergent
8. Absolutely convergent
22. Conditionally convergent
9. Absolutely convergent
23. Divergent
10. Conditionally convergent
24. Absolutely convergent
11. Conditionally convergent
25. Absolutely convergent
12. Divergent
26. Absolutely convergent
13. Divergent
27. Absolutely convergent
14. Absolutely convergent
28. Divergent
SECTION 11.7
11.7
S T R AT E G Y F O R T E S T I N G S E R I E S
A Click here for answers.
S Click here for solutions.
1–23 Test the series for convergence or divergence.
⬁
1.
兺
n苷1
sn
n2 ⫹ 1
⬁
3.
兺
n苷1
⬁
5.
兺
n苷2
⬁
7.
兺
n苷0
⬁
9.
兺
n苷2
⬁
11.
兺
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
m苷1
STRATEGY FOR TESTING SERIES
4n
3
2n⫺1
共⫺1兲 n
共ln n兲 2
10 n
n!
2
n共ln n兲3
2m
8m ⫺ 5
⬁
2.
⬁
13.
兺 cos n
n苷1
⬁
n苷1
⬁
4.
兺
i苷1
⬁
6.
15.
i4
4i
⬁
n苷1
⬁
10.
17.
12.
19.
n
en
n
⬁
n ⫹1
n4 ⫺ 1
n苷2
冊
n
⬁
14.
21.
⬁
16.
23.
⬁
n
18.
n2
sn 5 ⫹ n 2 ⫹ 2
3
n⫹1
s
兺 n (sn ⫹ 1)
⬁
兺
⬁
1
si共i ⫹ 1兲
22.
⬁
2n
共2n ⫹ 1兲!
n苷1
sn
e sn
n苷1
3n⫺1
20.
兺
兺
n苷1
2
n2 ⫹ 1
i苷1
兺
n苷1
兺
n苷1
兺
兺
3
4n ⫺ 5
兺 共⫺␲兲
⬁
3
n2 ⫹ 1
2n 2 ⫹ 1
n苷0
⫺1.7
3
n
5 ⫹n
n苷1
冉
⬁
兺k
兺
兺
n苷1
k苷1
8.
兺
兺
n苷1
⬁
兺
n苷1
共⫺1兲 n n
共n ⫹ 1兲共n ⫹ 2兲
tan共1兾n兲
n
■
1
2
■
SECTION 11.7
11.5
11.7
STRATEGY FOR TESTING SERIES
ANSWERS
E Click here for exercises.
S Click here for solutions.
1. Convergent
13. Convergent
2. Divergent
14. Convergent
3. Convergent
15. Divergent
4. Convergent
16. Divergent
5. Convergent
17. Divergent
6. Convergent
18. Convergent
7. Convergent
19. Divergent
8. Convergent
20. Convergent
9. Convergent
21. Divergent
10. Convergent
22. Convergent
11. Divergent
23. Convergent
12. Divergent
SECTION 11.8
11.8
P OW E R S E R I E S
A Click here for answers.
S Click here for solutions.
1–19 Find the radius of convergence and interval of
convergence of the series.
⬁
1.
兺
n苷0
⬁
3.
兺
n苷1
⬁
n
x
n⫹2
2.
共⫺1兲 n x n
n2n
4.
兺 nx
7.
兺
n苷0
⬁
9.
兺
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
n苷2
兺 n5
3nx n
共n ⫹ 1兲2
xn
ln n
n
xn
6.
兺
n苷1
⬁
8.
兺
n苷0
⬁
10.
兺
n苷1
兺
n苷0
n
n苷1
⬁
n
共⫺1兲 x
3
n
s
n
⬁
11.
兺 sn 共3x ⫹ 2兲
n 2x n
10 n
共⫺1兲n x 2n⫺1
共2n ⫺ 1兲!
12.
⬁
15.
⬁
n
14.
⬁
17.
兺
n苷0
⬁
19.
兺
n苷1
⬁
16.
共⫺3兲 n 共x ⫺ 1兲 n
sn ⫹ 1
18.
n
兺
n苷0
共x ⫺ 1兲 n
sn
兺 共⫺1兲
兺
n苷1
n苷0
n苷1
xn
n2
⬁
2 n共x ⫺ 3兲n
n⫹3
⬁
13.
⬁
n苷0
⬁
兺
n苷1
⬁
5.
POWER SERIES
n
nx
1 ⴢ 3 ⴢ 5 ⴢ ⭈ ⭈ ⭈ ⴢ 共2n ⫺ 1兲
兺
n苷1
⬁
兺
n苷1
共x ⫹ 1兲n
n共n ⫹ 1兲
n
共2x ⫺ 1兲 n
4n
共x ⫺ 4兲 n
n5n
共2x ⫺ 1兲 n
n3
■
1
2
■
SECTION 11.8
11.5
11.8
POWER SERIES
ANSWERS
E Click here for exercises.
S Click here for solutions.
1. 1, [−1, 1)
11. 12 , 52 , 72
2. 1, (−1, 1]
12. 1, [−2, 0]
3. 2, (−2, 2]
1
4. 15 , − 15 , 5
5. 1, (−1, 1)
6. 1, [−1, 1]
7. 13 , − 13 , 13
8. 10, (−10, 10)
9. 1, [−1, 1)
10. ∞, (−∞, ∞)
13. 13 , −1, − 13
14. 2, − 32 , 52
15. 1, (0, 2]
16. 5, [−1, 9)
17. 13 , 23 , 43
18. 12 , [0, 1]
19. ∞, (−∞, ∞)
SECTION 11.9
11.9
■
1
R E P R E S E N TAT I O N S O F F U N C T I O N S A S P OW E R S E R I E S
A Click here for answers.
S Click here for solutions.
1–7 Find a power series representation for the function and
determine the interval of convergence.
1. f 共x兲 苷
x
1⫺x
2. f 共x兲 苷
1
4 ⫹ x2
3. f 共x兲 苷
1 ⫹ x2
1 ⫺ x2
4. f 共x兲 苷
1
1 ⫹ 4x 2
5. f 共x兲 苷
1
x 4 ⫹ 16
6. f 共x兲 苷
x
x⫺3
7. f 共x兲 苷
2
3x ⫹ 4
8 –9 Express the function as the sum of a power series by first
using partial fractions. Find the interval of convergence.
8. f 共x兲 苷
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
REPRESENTATIONS OF FUNCTIONS AS POWER SERIES
3x ⫺ 2
2x 2 ⫺ 3x ⫹ 1
9. f 共x兲 苷
x
x 2 ⫺ 3x ⫹ 2
10–11 Find a power series representation for the function and
determine the radius of convergence.
10. f 共x兲 苷 tan⫺1共2x兲
冉 冊
11. f 共x兲 苷 ln
1⫹x
1⫺x
12–14 Evaluate the indefinite integral as a power series.
1
12.
y 1⫹x
14.
y
4
dx
13.
x
dx
1 ⫹ x5
y
arctan x
dx
x
15–16 Use a power series to approximate the definite integral to
six decimal places.
15.
y
0.2
0
1
dx
1 ⫹ x4
16.
y
1兾2
0
tan⫺1共x 2 兲 dx
2
■
SECTION 11.9
11.5
11.9
REPRESENTATIONS OF FUNCTIONS AS POWER SERIES
ANSWERS
E Click here for exercises.
1.
∞
xn , (−1, 1)
S Click here for solutions.
9.
n=1
2.
n
2n
(−1) x
4n+1
n=0
∞
, (−2, 2)
2x2n , (−1, 1)
n=1
4.
∞
n=0
5.
(−1)n 4n x2n , − 12 , 12
∞
(−1)n x4n
, (−2, 2)
24n+4
n=0
6. −
∞ x n
n=1
7.
3
, (−3, 3)
n=0
n
∞
(−1)n 22n+1 x2n+1 1
,
2n + 1
2
n=0
11.
∞
2x2n+1
,1
2n + 1
n=0
12. C +
∞
(−1)n x4n+1
4n + 1
n=0
13. C +
∞
(−1)n x5n+2
5n + 2
n=0
∞
(−1)n
n=0
(−1)n 3n xn 4 4 , −3, 3
22n+1
n=0
∞
10.
14. C +
∞
8. −
1 − 2−n xn , (−1, 1)
n=0
∞
3. 1 +
∞
n
(2 + 1) x ,
− 12 , 12
15. 0.199936
16. 0.041303
x2n+1
(2n + 1)2
SECTION 11.10
11.10
TAYLOR AND MACLAURIN SERIES
■
1
TAY L O R A N D M AC L A U R I N S E R I E S
A Click here for answers.
S Click here for solutions.
1–2 Find the Maclaurin series for f 共x兲 using the definition of
a Maclaurin series. [Assume that f has a power series expansion.
Do not show that Rn共x兲 l 0.] Also find the associated radius of
convergence.
1
共1 ⫹ x兲2
1. f 共x兲 苷
2. f 共x兲 苷
x
1⫺x
19–20 Use series to approximate the definite integral correct to
three decimal places.
19.
y
1
0
sin共x 2 兲 dx
20.
y
0.5
cos共x 2 兲 dx
0
21. Use multiplication or division of power series to find the
first three nonzero terms in the Maclaurin series for
3–6 Find the Taylor series for f 共x兲 centered at the given value
of a. [Assume that f has a power series expansion. Do not show
that Rn共x兲 l 0.]
3. f 共x兲 苷 1兾x,
a苷1
4. f 共x兲 苷 sx,
a苷4
y苷
22–24 Find the sum of the series.
⬁
5. f 共x兲 苷 sin x,
a 苷 ␲兾4
6. f 共x兲 苷 cos x,
a 苷 ⫺␲兾4
22.
兺
n苷2
⬁
24.
兺
n苷0
7–13 Use a Maclaurin series derived in this section to obtain the
Maclaurin series for the given function.
7. f 共x兲 苷 e
3x
9. f 共x兲 苷 x cos x
2
11. f 共x兲 苷 x sin共x兾2兲
13. f 共x兲 苷
再
1 ⫺ cos x
x2
1
2
8. f 共x兲 苷 sin 2x
10. f 共x兲 苷 cos共x 3兲
12. f 共x兲 苷 xe ⫺x
if x 苷 0
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
late ln 1.1 correct to five decimal places.
17 –18 Evaluate the indefinite integral as an infinite series.
2
兲 dx
n苷0
18.
ye
x3
dx
x n⫹1
共n ⫹ 1兲!
x
2 共n ⫹ 1兲!
n
26. Show that cosh x 艌 1 ⫹ 2 x 2 for all x.
1
27–32 Use the binomial series to expand the function as a
power series. State the radius of convergence.
29.
15. f 共x兲 苷 共1 ⫹ x兲⫺3
兺
n
28.
x
s1 ⫺ x
30.
x2
s1 ⫺ x 3
if x 苷 0
16. Find the Maclaurin series for ln共1 ⫹ x兲 and use it to calcu-
y sin共x
23.
3
27. s
1 ⫹ x2
radius of convergence. Graph f and its first few Taylor polynomials on the same screen. What do you notice about the relationship between these polynomials and f ?
17.
⬁
x 3n⫹1
n!
25. Show that e x ⬎ 1 ⫹ x for all x ⬎ 0.
; 14– 15 Find the Maclaurin series of f (by any method) and its
14. f 共x兲 苷 1兾s1 ⫹ 2x
ln共1 ⫺ x兲
ex
31.
1
s2 ⫹ x
冉 冊
x
1⫺x
5
5
32. s
x⫺1
33. (a) Expand 1兾s1 ⫹ x as a power series.
(b) Use part (a) to estimate 1兾s1.1 correct to three decimal
places.
3
34. (a) Expand s
8 ⫹ x as a power series.
3
(b) Use part (a) to estimate s
8.2 correct to four decimal
places.
2
■
SECTION 11.10
11.10
TAYLOR AND MACLAURIN SERIES
ANSWERS
E Click here for exercises.
1.
∞
(−1)n (n + 1) xn , R = 1
S Click here for solutions.
15.
n=0
2.
∞
∞
(−1)n (n + 1) (n + 2) xn
,R=1
2
n=0
xn , R = 1
n=1
3.
∞
(−1)n (x − 1)n , R = 1
n=0
∞
x − 4 (−1)n−1 1 · 3 · 5 · · · · · (2n − 3)
+
(x − 4)n ,
3n−1 n!
4
2
n=2
R=4
√ ∞
2
1 π 2n
5.
(−1)n
x−
2 n=0
(2n)!
4
1
π 2n+1
+
,R=∞
x−
(2n + 1)!
4
n
√ ∞
n(n−1)/2 x + π4
2 (−1)
6.
,R=∞
2 n=0
n!
4. 2 +
16. 0.09531
17. C +
∞
(−1)n x4n+3
(4n + 3) (2n + 1)!
n=0
∞
3n xn
7.
,R=∞
n!
n=0
19. 0.310
∞
(−1)n 22n+1 x2n+1
8.
,R=∞
(2n + 1)!
n=0
22. x e
n
(−1) x2n+2
,R=∞
(2n)!
n=0
∞
(−1)n x6n
10.
,R=∞
(2n)!
n=0
11.
∞
(−1)n x2n+2
,R=∞
(2n + 1)!22n+1
n=0
12.
∞
(−1)n−1 xn
,R=∞
(n − 1)!
n=1
13.
∞
(−1)n x2n
,R=∞
(2n + 2)!
n=0
∞
(−1)n 1 · 3 · 5 · · · · · (2n − 1) n
14.
x , R = 12 .
n!
n=0
x3
− 1 − x3
21. −x +
x
23. e − 1
∞
x3n+1
(3n + 1) n!
n=0
x2
x3
−
+ ···
2
3
24.
2 x/2
−1
e
x
x2 (−1)n−1 · 2 · 5 · 8 · · · · · (3n − 4) x2n
+
,
3
3n n!
n=2
∞
27. 1 +
R=1
28. x +
∞
1 · 3 · 5 · · · · · (2n − 1) n+1
, R=1
x
2n n!
n=1
√ ∞
(−1)n · 1 · 3 · 5 · · · · · (2n − 1) xn
2
29.
1+
, R=2
2
22n · n!
n=1
2
30. x +
31.
∞
1 · 3 · 5 · · · · · (2n − 1) x3n+2
, R=1
2n · n!
n=1
∞
(n + 4)! n+5
, R=1
x
4! · n!
n=0
x 4 · 9 · · · (5n − 6) xn
+
, R=1
5 n=2
5n · n!
∞
32. −1 +
33. (a) 1 +
∞
(−1)n 1 · 3 · 5 · · · · · (2n − 1) n
x
2n · n!
n=1
(b) 0.953
(−1)n−1 · 2 · 5 · · · · · (3n − 4) xn
x
+
34. (a) 2 1 +
24 n=2
24n · n!
(b) 2.0165
∞
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
9.
∞
20. 0.497
18. C +
SECTION 11.11
11.11
■
1
A P P L I C AT I O N S O F TAY L O R P O LY N O M I A L S
A Click here for answers.
S Click here for solutions.
; 1–6 Find the Taylor polynomial Tn共x兲 for the function f at the
number a. Graph f and Tn on the same screen.
1. f 共x兲 苷 sx,
a 苷 9,
3
2. f 共x兲 苷 1兾s
x,
n苷3
a 苷 8,
n苷3
7–10
(a) Approximate f by a Taylor polynomial with degree n at the
number a.
(b) Use Taylor’s Inequality to estimate the accuracy of the approximation f 共x兲 ⬇ Tn共x兲 when x lies in the given interval.
; (c) Check your result in part (b) by graphing Rn共x兲 .
ⱍ
3. f 共x兲 苷 sec x,
a 苷 ␲兾3,
4. f 共x兲 苷 tan x,
a 苷 0,
5. f 共x兲 苷 tan x,
a 苷 ␲兾4,
n苷4
9. f 共x兲 苷 x 3兾4,
a 苷 0,
n苷3
10. f 共x兲 苷 ln x,
6. f 共x兲 苷 e x sin x,
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
APPLICATIONS OF TAYLOR POLYNOMIALS
n苷3
n苷4
7. f 共x兲 苷 sin x,
a 苷 ␲兾4,
3
8. f 共x兲 苷 s
1 ⫹ x 2,
a 苷 0,
a 苷 16,
ⱍ
n 苷 5, 0 艋 x 艋 ␲兾2
n 苷 2,
n 苷 3,
a 苷 4, n 苷 3,
ⱍ x ⱍ 艋 0.5
15 艋 x 艋 17
3艋x艋5
2
■
SECTION 11.11
11.11
APPLICATIONS OF TAYLOR POLYNOMIALS
ANSWERS
E Click here for exercises.
S Click here for solutions.
2
3
4
+ 2 x − π4 + 83 x − π4 + 10
x − π4
5. 1 + 2 x − π
4
3
1
1
1. 3 + 16 (x − 9) − 216
(x − 9)2 + 3888
(x − 9)3
1
1
7
2. 12 − 48
(x − 8) + 576
(x − 8)2 − 41,472
(x − 8)3
2
3
6. x + x + 13 x
√ 2
3 x − π3 + 7 x − π3 +
3. 2 + 2
√
23 3
3
x−
π 3
3
√
√ √2 2
7. (a) 22 + 22 x − π
− 4 x − π4
4
−
√
2
12
x−
π 3
4
+
√
2
48
x−
π 4
4
+
√
2
240
x−
π 5
4
(b) 0.00033
2
8. (a) 1 + 13 x
4. x +
x3
3
3
5
9. (a) 8 + 38 (x − 16) − 1024
(x − 16)2 + 65,536
(x − 16)3
(b) 0.000003
1
1
10. (a) ln 4 + 14 (x − 4) − 32
(x − 4)2 + 192
(x − 4)3
(b) 0.0031
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
(b) 0.014895
SECTION 12.1
12.1
THREE-DIMENSIONAL COORDINATE SYSTEMS
■
T H R E E - D I M E N S I O N A L C O O R D I N AT E S Y S T E M S
A Click here for answers.
S Click here for solutions.
1– 4 Draw a rectangular box that has P and Q as opposite ver-
tices and has its face parallel to the coordinate planes. Then
find (a) the coordinates of the other six vertices of the box and
(b) the length of the diagonal of the box.
14–19 Show that the equation represents a sphere, and find its
center and radius.
14. x 2 ⫹ y 2 ⫹ z 2 ⫹ 2x ⫹ 8y ⫺ 4z 苷 28
1. P共0, 0, 0兲,
Q共2, 3, 5兲
15. 2x 2 ⫹ 2y 2 ⫹ 2z 2 ⫹ 4y ⫺ 2z 苷 1
2. P共0, 0, 0兲,
Q共⫺4, ⫺1, 2兲
16. x 2 ⫹ y 2 ⫹ z 2 苷 6x ⫹ 4y ⫹ 10z
3. P共1, 1, 2兲,
Q共3, 4, 5兲
4. P共4, 3, 0兲,
Q共1, 6, ⫺4兲
17. x 2 ⫹ y 2 ⫹ z 2 ⫹ x ⫺ 2y ⫹ 6z ⫺ 2 苷 0
18. x 2 ⫹ y 2 ⫹ z 2 苷 x
5. Sketch the points 共3, 0, 1兲, 共⫺1, 0, 3兲, 共0, 4, ⫺2兲, and
共1, 1, 0兲 on a single set of coordinate axes.
19. x 2 ⫹ y 2 ⫹ z 2 ⫹ ax ⫹ by ⫹ cz ⫹ d 苷 0,
where a 2 ⫹ b 2 ⫹ c 2 ⬎ 4d
6–9 Find the lengths of the sides of the triangle ABC and
determine whether the triangle is isosceles, a right triangle,
both, or neither.
6. A共3, ⫺4, 1兲,
B共5, ⫺3, 0兲,
C共6, ⫺7, 4兲
7. A共2, 1, 0兲,
B共3, 3, 4兲,
C共5, 4, 3兲
8. A共5, 5, 1兲,
B共3, 3, 2兲,
C共1, 4, 4兲
9. A共⫺2, 6, 1兲,
B共5, 4, ⫺3兲,
C共2, ⫺6, 4兲
10. Find an equation of the sphere with center 共0, 1, ⫺1兲 and
radius 4. What is the intersection of this sphere with the
yz-plane?
11–13 Find the equation of the sphere with center C and radius
r.
11. C共⫺1, 2, 4兲,
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
12. C共⫺6, ⫺1, 2兲,
13. C共1, 2, ⫺3兲,
r 苷 12
r 苷 2 s3
r苷7
20. Find an equation of the sphere that has center 共1, 2, 3兲 and
passes through the point 共⫺1, 1, ⫺2兲.
21–28 Describe in words the region of ⺢ 3 represented by the
equation or inequality.
21. x 苷 9
22. z 苷 ⫺8
23. y ⬎ 2
24. z 艋 0
25.
ⱍzⱍ 艋 2
27. xy 苷 0
26. z 苷 x
28. xy 苷 1
1
2
■
SECTION 12.1
12.1
THREE-DIMENSIONAL COORDINATE SYSTEMS
ANSWERS
E Click here for exercises.
S Click here for solutions.
√
38
1. (a)
(b)
2. (a)
√
(b) 21
√
√
√
6, |BC| = 33, |CA| = 3 3; right triangle
√
√
√
7. |AB| = 21, |BC| = 6, |CA| = 3 3; right triangle
√
8. |AB| = 3, |BC| = 3, |CA| = 26; isosceles
√
√
9. |AB| = 69, |BC| = 158, |CA| = 13; neither
6. |AB| =
2
2
2
2
2
10. x + (y − 1) + (z + 1) = 16; (y − 1) + (z + 1) = 16,
x=0
2
2
2
2
2
2
2
2
2
11. (x + 1) + (y − 2) + (z − 4) = 14
12. (x + 6) + (y + 1) + (z − 2) = 12
13. (x − 1) + (y − 2) + (z + 3) = 49
14. (−1, −4, 2), 7
3. (a)
(b)
√
22
√
0, −1, 12 , 27
√
16. (3, 2, 5), 38
17. − 12 , 1, −3 , 72
18. 12 , 0, 0 , 12
15.
19.
4. (a)
√
(b) 34
1
− 2 a, − 12 b, − 12 c , 14 (a2 + b2 + c2 ) − d
2
2
2
20. (x − 1) + (y − 2) + (z − 3) = 30
21. A plane parallel to the yz-plane and 9 units in front of it
22. A plane parallel to the xy-plane and 8 units below it
23. A half-space containing all points to the right of the plane
y=2
24. A half-space containing all points on and below the xy-plane
25. All points on and between the two horizontal planes z = 2
and z = −2
26. A plane perpendicular to the xz-plane and intersecting the
xz-plane in the line x = z, y = 0
27. The two planes x = 0 and y = 0
28. A hyperbolic cylinder
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
5.
SECTION 12.2
12.2
VECTORS
■
V E C TO R S
A Click here for answers.
S Click here for solutions.
1. Express w in terms of the vectors u and v in the figure.
cally.
v
u
6. 具2, 3典 ,
w
2. Write each combination of vectors as a single vector.
l
l
(a) AB BC
l
l
(c) BC DC
6–9 Find the sum of the given vectors and illustrate geometri-
l
l
(b) CD DA
l
l
l
(d) BC CD DA
具3, 4典
7. 具1, 2典 ,
具5, 3 典
8. 具1, 0, 1典 ,
具0, 0, 1典
9. 具0, 3, 2典 ,
具1, 0, 3典
10–15 Find a unit vector that has the same direction as the
given vector.
D
A
10. 具1, 2典
11. 具3, 5典
12. 具 2, 4, 3典
13. 具1, 4, 8典
14. i j
15. 2 i 4 j 7 k
C
B
16. A quadrilateral has one pair of opposite sides parallel and
3–5 Find a vector a with representation given by the directed
l
l
line segment AB. Draw AB and the equivalent representation
starting at the origin.
3. A共1, 3兲,
B共4, 4兲
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
5. A共1, 2, 0兲,
B共1, 2, 3兲
4. A共4, 1兲,
B共1, 2兲
of equal length. Use vectors to prove that the other pair of
opposite sides is parallel and of equal length.
1
2
■
SECTION 12.2
12.2
VECTORS
ANSWERS
E Click here for exercises.
1. w = v − u
−→
2. (a) AC
3. 3, 1
−→
(b) CA
S Click here for solutions.
−−→
(c) BD
−→
(d) BA
9. 1, 3, −1
4. −3, 3
10.
5. 0, 0, 3
6. 5, −1
11.
1
√
, √25
5
√3 , − √5
34
34
− √229 , √429 , √329
13. 19 , − 49 , 89
12.
14. √1 i + √1 j
2
15.
2
i−
√4
69
j+
√7
69
k
8. 1, 0, 2
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
7. 4, 5
√2
69
SECTION 12.3
12.3
■
1
T H E D OT P RO D U C T
A Click here for answers.
S Click here for solutions.
1–7 Find a ⴢ b.
1. a 苷 具 2, 5典 ,
16–21 Determine whether the given vectors are orthogonal,
parallel, or neither.
b 苷 具⫺3, 1典
2. a 苷 具⫺2, ⫺8 典 ,
b 苷 具6, ⫺4典
3. a 苷 具 4, 7, ⫺1典 ,
b 苷 具 ⫺2, 1, 4典
4. a 苷 具⫺1, ⫺2, ⫺3典 ,
b 苷 具2, 8, ⫺6 典
5. a 苷 2 i ⫹ 3 j ⫺ 4 k,
b 苷 i ⫺ 3j ⫹ k
6. a 苷 i ⫺ k,
ⱍ ⱍ
7. a 苷 2,
b 苷 i ⫹ 2j
ⱍ b ⱍ 苷 3,
the angle between a and b is ␲兾3
8–13 Find the angle between the vectors. (First find an exact
expression and then approximate to the nearest degree.)
8. a 苷 具1, 2, 2 典 ,
b 苷 具 3, 4, 0典
9. a 苷 具6, 0, 2 典 ,
b 苷 具5, 3, ⫺2典
10. a 苷 具1, 2 典 ,
b 苷 具 12, ⫺5典
11. a 苷 具3, 1典 ,
b 苷 具2, 4 典
12. a 苷 6 i ⫺ 2 j ⫺ 3 k,
13. a 苷 i ⫹ j ⫹ 2 k,
b苷i⫹j⫹k
b 苷 2 j ⫺ 3k
14 –15 Find, correct to the nearest degree, the three angles of
the triangle with the given vertices.
14. A共1, 2, 3兲,
15. P共0, ⫺1, 6兲,
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
THE DOT PRODUCT
B共6, 1, 5兲,
C共⫺1, ⫺2, 0兲
Q共2, 1, ⫺3兲,
R共5, 4, 2兲
16. a 苷 具2, ⫺4 典 ,
b 苷 具⫺1, 2典
17. a 苷 具2, ⫺4 典 ,
b 苷 具4, 2 典
18. a 苷 具2, 8, ⫺3典 ,
b 苷 具 ⫺1, 2, 5 典
19. a 苷 具⫺1, 5, 2典 ,
b 苷 具4, 2, ⫺3典
20. a 苷 3 i ⫹ j ⫺ k,
b 苷 i ⫺ j ⫹ 2k
21. a 苷 ⫺i ⫹ 2 j ⫹ 5 k,
b 苷 3i ⫹ 4j ⫺ k
22. Find the values of x such that the vectors 具⫺3x, 2x 典 and
具4, x 典 are orthogonal.
23. For what values of c is the angle between the vectors
具1, 2, 1 典 and 具1, 0, c 典 equal to 60⬚ ?
24–28 Find the direction cosines and direction angles of the
vector. (Give the direction angles correct to the nearest degree.)
24. 具1, 2, 2 典
25. 具⫺4, ⫺1, 2典
26. ⫺8 i ⫹ 3 j ⫹ 2 k
27. 3 i ⫹ 5 j ⫺ 4 k
28. 具2, 1.2, 0.8典
29–30 Find the scalar and vector projections of b onto a.
29. a 苷 具 2, 3典 ,
30. a 苷 具3, ⫺1典 ,
b 苷 具4, 1典
b 苷 具2, 3 典
2
■
SECTION 12.3
12.3
THE DOT PRODUCT
ANSWERS
E Click here for exercises.
S Click here for solutions.
1. −1
16. Parallel
2. 20
17. Orthogonal
3. −5
18. Neither
4. 0
19. Orthogonal
5. −11
20. Orthogonal
6. 1
21. Orthogonal
7. 3
−1
8. cos
9.
10.
11.
12.
13.
22. 0, 6
11 ◦
≈ 43
13
cos−1 2√
≈ 48◦
95
cos−1 132√5 ≈ 86◦
√ cos−1 22 = 45◦
1
cos−1 7√
≈ 85◦
3
cos−1 − √478 ≈ 117◦
15
◦
◦
◦
14. 114 , 33 , 33
◦
◦
√
3
23. 2 ±
◦
◦
◦
24. 13 , 23 , 23 ; 71 , 48 , 48
◦
◦
◦
25. − √4 , − √1 , √2 ; 151 , 103 , 64
21
21
21
◦
◦
◦
26. − √8 , √3 , √2 ; 156 , 70 , 77
77
27.
77
3
√
, √12 ,
5 2
77
4
− 5√
;
2
65◦ , 45◦ , 124◦
◦
◦
◦
28. √5 , √3 , √2 ; 36 , 61 , 71
38
29.
11
√
,
13
30. √3 ,
10
38
22
13
,
38
33
13
9
3
, − 10
10
◦
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
15. 43 , 58 , 79
SECTION 12.4
12.4
THE CROSS PRODUCT
■
T H E C RO S S P RO D U C T
A Click here for answers.
1–9 Find the cross product a ⫻ b.
1. a 苷 具 1, 0, 1 典 ,
b 苷 具0, 1, 0 典
2. a 苷 具2, 4, 0 典 ,
b 苷 具 ⫺3, 1, 6 典
3. a 苷 具 ⫺2, 3, 4典 ,
b 苷 具 3, 0, 1典
4. a 苷 具 1, 2, ⫺3典 ,
b 苷 具5, ⫺1, ⫺2典
5. a 苷 i ⫹ j ⫹ k,
6. a 苷 i ⫹ 2 j ⫺ k,
7. a 苷 2 i ⫺ k,
b苷 i⫹j⫺k
b 苷 3i ⫺ j ⫹ 7k
b 苷 i ⫹ 2j
8. a 苷 具1, ⫺1, 0典 ,
b 苷 具3, 2, 1 典
9. a 苷 具⫺3, 2, 2典 ,
b 苷 具 6, 3, 1典
S Click here for solutions.
14. Find the area of the parallelogram with vertices P共0, 0, 0兲,
Q共5, 0, 0兲, R共2, 6, 6兲, and S共7, 6, 6兲.
15–17 (a) Find a vector orthogonal to the plane through the
points P, Q, and R, and (b) find the area of triangle PQR.
15. P共1, 0, ⫺1兲,
16. P共0, 0, 0兲,
Q共2, 4, 5兲,
R共3, 1, 7兲
Q共1, ⫺1, 1兲,
R共4, 3, 7兲
17. P共⫺4, ⫺4, ⫺4兲,
Q共0, 5, ⫺1兲,
R共3, 1, 2兲
18–19 Find the volume of the parallelepiped determined by the
vectors a, b, and c.
18. a 苷 具1, 0, 6 典 ,
b 苷 具2, 3, ⫺8典 ,
19. a 苷 2 i ⫹ 3 j ⫺ 2 k ,
b 苷 i ⫺ j,
c 苷 具8, ⫺5, 6典
c 苷 2i ⫹ 3k
10. If a 苷 具 0, 1, 2 典 and b 苷 具 3, 1, 0 典 , find a ⫻ b and b ⫻ a.
11. If a 苷 具 ⫺4, 0, 3典 , b 苷 具2, ⫺1, 0典 , and c 苷 具0, 2, 5 典 ,
show that a ⫻ 共b ⫻ c兲 苷 共a ⫻ b兲 ⫻ c.
12. Find two unit vectors orthogonal to both i ⫹ j and .
13. Find the area of the parallelogram with vertices A共0, 1兲,
B共3, 0兲, C共5, ⫺2兲, and D共2, ⫺1兲.
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
1
20. Given the points P共1, 1, 1兲, Q共2, 0, 3兲, R共4, 1, 7兲, and
S共3, ⫺1, ⫺2兲, find the volume of the parallelepiped with
adjacent edges PQ, PR, and PS.
■
SECTION 12.4
12.4
THE CROSS PRODUCT
ANSWERS
E Click here for exercises.
S Click here for solutions.
1. −i + k
12. ± √1 1, −1, −2
2. 24i − 12j + 14k
13. 4
3. 3i + 14j − 9k
14. 30
4. −7i − 13j − 11k
15. (a) 26, 4, −7
5. −2i + 2j
16. (a) −10, −3, 7
6. 13i − 10j − 7k
7. 2i − j + 4k
8. −i − j + 5k
9. −4i + 15j − 21k
10. −2i + 6j − 3k, 2i − 6j + 3k
6
√
2
17. (a) 39, −3, −43
√
741
√
(b) 12 158
√
(b) 12 3379
(b)
1
2
18. 226
19. 19
20. 21
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
2
SECTION 12.5
12.5
EQUATIONS OF LINES AND PLANES
■
1
E Q U AT I O N S O F L I N E S A N D P L A N E S
A Click here for answers.
S Click here for solutions.
1– 4 Find a vector equation and parametric equations for the
19–22 Find an equation of the plane passing through the given
line passing through the given point and parallel to the vector
a.
point and parallel to the specified plane.
1. 共3, ⫺1, 8兲,
a 苷 具2, 3, 5 典
2. 共⫺2, 4, 5兲,
a 苷 具3, ⫺1, 6 典
3. 共0, 1, 2兲,
20. 共3, 0, 8兲,
x⫹y⫺z⫹1苷0
2x ⫹ 5y ⫹ 8z 苷 17
21. 共⫺1, 3, ⫺8兲,
a 苷 6i ⫹ 3j ⫹ 2k
4. 共1, ⫺1, ⫺2兲,
19. 共6, 5, ⫺2兲,
22. 共2, ⫺4, 5兲,
3x ⫺ 4y ⫺ 6z 苷 9
z 苷 2x ⫹ 3y
a 苷 2i ⫺ 7k
23–26 Find an equation of the plane passing through the three
5–10 Find parametric equations and symmetric equations for
the line through the given points.
5. 共2, 1, 8兲,
共6, 0, 3兲
6. 共⫺1, 0, 5兲,
7. 共3, 1, ⫺1兲,
共3, 2, ⫺6兲
8. (3, 1, 2 ),
9. (⫺ 3 , 1, 1),
共0, 5, ⫺8兲
10. 共2, ⫺7, 5兲,
1
1
共4, ⫺3, 3兲
共⫺1, 4, 1兲
共⫺4, 2, 5兲
11. Show that the line through the points 共0, 1, 1兲 and
共1, ⫺1, 6兲
is perpendicular to the line through the points 共⫺4, 2, 1兲
and 共⫺1, 6, 2兲.
12–14 Determine whether the lines L 1 and L 2 are parallel,
skew, or intersecting. If they intersect, find the point of intersection.
12. L 1 :
L 2:
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
13. L 1 :
L 2:
x⫺4
y⫹5
z⫺1
苷
苷
,
2
4
⫺3
y⫹1
z
x⫺2
苷
苷
1
3
2
x⫺1
y
z⫺1
苷 苷
,
2
1
4
x
y⫹2
z⫹2
苷
苷
1
2
3
14. L 1 : x 苷 1 ⫹ t, y 苷 2 ⫺ t, z 苷 3t
L 2 : x 苷 2 ⫺ s, y 苷 1 ⫹ 2s, z 苷 4 ⫹ s
15 –18 Find an equation of the plane passing through the given
given points.
23. 共0, 0, 0兲,
共1, 1, 1兲,
24. 共⫺1, 1, ⫺1兲,
共1, 2, 3兲
共1, ⫺1, 2兲,
共4, 0, 3兲
25. 共1, 0, ⫺3兲,
共0, ⫺2, ⫺4兲,
共4, 1, 6兲
26. 共2, 1, ⫺3兲,
共5, ⫺1, 4兲,
共2, ⫺2, 4兲
27–30 Find an equation of the plane that passes through the
given point and contains the specified line.
27. 共1, 6, ⫺4兲;
x 苷 1 ⫹ 2t, y 苷 2 ⫺ 3t, z 苷 3 ⫺ t
28. 共⫺1, ⫺3, 2兲;
29. 共0, 1, 2兲;
x 苷 ⫺1 ⫺ 2t, y 苷 4t, z 苷 2 ⫹ t
x苷y苷z
30. 共⫺1, 0, 1兲;
x 苷 5t, y 苷 1 ⫹ t, z 苷 ⫺t
31–34 Find the point at which the line intersects the given
plane.
31. x 苷 1 ⫹ t, y 苷 2t, z 苷 3t;
x⫹y⫹z苷1
32. x 苷 5, y 苷 4 ⫺ t, z 苷 2t;
2x ⫺ y ⫹ z 苷 5
33. x 苷 1 ⫹ 2t, y 苷 ⫺1, z 苷 t;
34. x 苷 1 ⫺ t, y 苷 t, z 苷 1 ⫹ t;
2x ⫹ y ⫺ z ⫹ 5 苷 0
z 苷 1 ⫺ 2x ⫹ y
35–40 Determine whether the planes are parallel, perpendicular,
or neither. If neither, find the angle between them.
35. x ⫹ z 苷 1,
y⫹z苷1
36. ⫺8x ⫺ 6y ⫹ 2z 苷 1,
z 苷 4x ⫹ 3y
point and with normal vector n.
37. x ⫹ 4y ⫺ 3z 苷 1,
⫺3x ⫹ 6y ⫹ 7z 苷 0
15. 共1, 4, 5兲,
38. 2x ⫹ 2y ⫺ z 苷 4,
6x ⫺ 3y ⫹ 2z 苷 5
16. 共⫺5, 1, 2兲,
17. 共1, 2, 3兲,
n 苷 具7, 1, 4 典
n 苷 具3, ⫺5, 2 典
n 苷 15 i ⫹ 9 j ⫺ 12 k
18. 共⫺1, ⫺6, ⫺4兲,
n 苷 ⫺5 i ⫹ 2 j ⫺ 2 k
39. 2x ⫹ 4y ⫺ 2z 苷 1,
40. 2x ⫺ 5y ⫹ z 苷 3,
⫺3x ⫺ 6y ⫹ 3z 苷 10
4x ⫹ 2y ⫹ 2z 苷 1
2
■
SECTION 12.5
12.5
EQUATIONS OF LINES AND PLANES
ANSWERS
E Click here for exercises.
S Click here for solutions.
12. Skew
13. Intersecting, (1, 0, 1)
14. Skew
15. 7x + y + 4z = 31
16. 3x − 5y + 2z = −16
17. 5x + 3y − 4z = −1
18. −5x + 2y − 2z = 1
19. x + y − z = 13
x = 6t, y = 1 + 3t, z = 2 + 2t
20. 2x + 5y + 8z = 70
21. 3x − 4y − 6z = 33
4. r = (1 + 2t) i − j + (−2 − 7t) k;
22. 2x + 3y − z = −13
23. x − 2y + z = 0
24. −5x + 7y + 8z = 4
25. −17x + 6y + 5z = −32
26. 7x − 21y − 9z = 20
27. 25x + 14y + 8z = 77
28. x + 2z = 3
29. x − 2y + z = 0
30. y + z = 1
31. (1, 0, 0)
1. r = (3 + 2t) i + (−1 + 3t) j + (8 + 5t) k;
x = 3 + 2t, y = −1 + 3t, z = 8 + 5t
2. r = (−2 + 3t) i + (4 − t) j + (5 + 6t) k;
x = −2 + 3t, y = 4 − t, z = 5 + 6t
3. r = (6t) i + (1 + 3t) j + (2 + 2t) k;
x = 1 + 2t, y = −1, z = − (2 + 7t)
5. x = 2 + 4t, y = 1 − t, z = 8 − 5t;
x−2
y−1
z−8
=
=
4
−1
−5
x+1
y
z−5
6. x = −1 + 5t, y = −3t, z = 5 − 2t;
=
=
5
−3
−2
7. x = 3, y = 1 + t, z = −1 − 5t; x = 3, y − 1 =
z+1
−5
8. x = −1 − 4t, y = 4 + 3t, z = 1 + 12 t;
x+1
y−4
z−1
=
=
−4
3
1/2
32. 5,
2
13
, −3
3
◦
34. (0, 1, 2)
35. Neither, 60
36. Parallel
37. Perpendicular
◦
38. Neither, 79
9. x = − 13 + 13 t, y = 1 + 4t, z = 1 − 9t;
33. (−3, −1, −2)
39. Parallel
40. Perpendicular
x + 1/3
y−1
z−1
=
=
1/3
4
−9
x−2
y+7
=
,z =5
−6
9
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
10. x = 2 − 6t, y = −7 + 9t, z = 5;
SECTION 14.1
14.1
FUNCTIONS OF SEVERAL VARIABLES
F U N C T I O N S O F S E V E R A L VA R I A B L E S
A Click here for answers.
S Click here for solutions.
1. If f 共x, y兲 苷 x 2 ⫺ y 2 ⫹ 4xy ⫺ 7x ⫹ 10, find
(a) f 共2, 1兲
(c) f 共x ⫹ h, y兲
(e) f 共x, x兲
(b) f 共⫺3, 5兲
(d) f 共x, y ⫹ k兲
2. If t共x, y兲 苷 ln共xy ⫹ y ⫺ 1兲, find
(a) t共1, 1兲
(c) t共x, 1兲
(e) t共x, y ⫹ k兲
(b) t共e, 1兲
(d) t共x ⫹ h, y兲
3. If F 共x, y兲 苷 3xy兾共x 2 ⫹ 2y 2 兲, find
(a) F 共1, 1兲
(c) F 共t, 1兲
(e) F 共x, x 2 兲
(b) F 共⫺1, 2兲
(d) F 共⫺1, y兲
4. If G共x, y, z兲 苷 x sin y cos z, find
(a) G共2, ␲兾6, ␲兾3兲
(c) G共t, t, t兲
(e) G共x, x ⫹ y, x兲
(b) G共4, ␲兾4, 0兲
(d) G共u, v, 0兲
5–10 Find the domain and range of the function.
5. f 共x, y兲 苷 x ⫹ 2y ⫺ 5
6. f 共x, y兲 苷 sx ⫺ y
7. f 共x, y兲 苷 2兾共x ⫹ y兲
8. f 共x, y兲 苷 tan⫺1 共 y兾x兲
15–25 Find and sketch the domain of the function.
15. f 共x, y兲 苷 xy sx 2 ⫹ y
16. f 共x, y兲 苷
s9 ⫺ x 2 ⫺ y 2
x ⫹ 2y
17. f 共x, y兲 苷
x2 ⫹ y2
x2 ⫺ y2
18. f 共x, y兲 苷 tan共x ⫺ y兲
19. f 共x, y兲 苷 ln共xy ⫺ 1兲
20. f 共x, y兲 苷 ln共x 2 ⫺ y 2 兲
21. f 共x, y兲 苷 x 2 sec y
22. f 共x, y兲 苷 sin⫺1 共x ⫹ y兲
23. f 共x, y兲 苷 s4 ⫺ 2x 2 ⫺ y 2
24. f 共x, y兲 苷 ln x ⫹ ln sin y
25. f 共x, y兲 苷 sy ⫺ x ln共 y ⫹ x兲
26–33 Sketch the graph of the function.
26. f 共x, y兲 苷 x
27. f 共x, y兲 苷 sin y
28. f 共x, y兲 苷 x 2 ⫹ 9y 2
9. f 共x, y, z兲 苷 x兾共 yz兲
29. f 共x, y兲 苷 y 2
10. f 共x, y, z兲 苷 x sin共 y ⫹ z兲
30. f 共x, y兲 苷 s16 ⫺ x 2 ⫺ 16y 2
11. Let f 共x, y兲 苷 e x
31. f 共x, y兲 苷 y 2 ⫺ x 2
2⫺y
.
(a) Evaluate f 共2, 4兲.
(b) Find the domain of f .
(c) Find the range of f .
32. f 共x, y兲 苷 1 ⫺ x 2
33. f 共x, y兲 苷 x 2 ⫹ y 2 ⫺ 4x ⫺ 2y ⫹ 5
12. Let t共x, y兲 苷 s36 ⫺ 9x 2 ⫺ 4y 2.
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
■
(a) Evaluate t共1, 2兲.
(b) Find and sketch the domain of t.
(c) Find the range of t.
13. Let f 共x, y, z兲 苷 x 2 ln共x ⫺ y ⫹ z兲.
(a) Evaluate f 共3, 6, 4兲.
(b) Find the domain of f .
(c) Find the range of f .
14. Let f 共x, y, z兲 苷 1兾sx ⫹ y ⫹ z ⫺ 1.
2
(a) Evaluate f 共1, 3, ⫺4兲.
(b) Find the domain of f .
(c) Find the range of f .
2
2
34–39 Draw a contour map of the function showing several
level curves.
34. f 共x, y兲 苷
x
y
35. f 共x, y兲 苷
x⫹y
x⫺y
36. f 共x, y兲 苷 y ⫺ cos x
37. f 共x, y兲 苷 e 1兾( x
38. f 共x, y兲 苷 x 2 ⫹ 9y 2
39. f 共x, y兲 苷 e xy
2⫹ y 2 )
1
2
■
SECTION 14.1
14.1
FUNCTIONS OF SEVERAL VARIABLES
ANSWERS
E Click here for exercises.
S Click here for solutions.
1. (a) 7
15.
(x, y) | y ≥ −x2
16.
(x, y) | y = − 12 x and x2 + y 2 ≤ 9
(b) −45
(c) x2 + 2xh + h2 − y 2 + 4xy + 4hy − 7x − 7h + 10
(d) x2 − y 2 − 2ky − k2 + 4xy + 4xk − 7x + 10
(e) 4x2 − 7x + 10
2. (a) 0
(b) ln e = 1
(c) ln x
(d) ln (xy + hy + y − 1)
(e) ln (xy + kx + y + k − 1)
3. (a) 1
(b) − 23
3t
t2 + 2
3y
(d) −
1 + 2y 2
3x
(e)
1 + 2x2
(c)
17. {(x, y) | y = x and y = −x}
4. (a) 12
√
(b) 2 2
(c) t sin t cos t
(d) u sin v
(e) x cos x [sin x cos y + sin y cos x]
2
5. R , R
18.
(x, y) | x − y =
π
2
+ nπ, n an integer
6. {(x, y) | x ≥ y}, {z | z ≥ 0}
8. {(x, y) | x = 0},
z | − π2 < z <
π
2
9. {(x, y, z) | yz = 0}, R
3
10. R , R
11. (a) 1
19. {(x, y) | xy > 1}
(b) R2
(c) {z | z > 0}
√
12. (a) 11
(b) (x, y) | 14 x2 + 19 y 2 ≤ 1
(c) {z | 0 ≤ z ≤ 6}
13. (a) 0
(b) {(x, y, z) | x + z > y}
(c) R
14. (a) 15
(b) (x, y, z) | x2 + y 2 + z 2 > 1
(c) (0, ∞)
20. {(x, y) | |y| < |x|}
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
7. {(x, y) | x + y = 0}, {z | z = 0}
SECTION 14.1
21.
(x, y) | y =
π
2
+ nπ, n an integer
22. {(x, y) | −1 − x ≤ y and y ≤ 1 − x}
23.
(x, y) | 2x2 + y 2 ≤ 4
30.
31.
32.
33.
34.
35.
36.
24. {(x, y) | x > 0 and 2nπ < y < (2n + 1) π, n an integer}
37.
25. {(x, y) | −y < x ≤ y, y > 0}
Stewart: Calculus, Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.
38.
26.
27.
39.
28.
29.
FUNCTIONS OF SEVERAL VARIABLES
■
3