James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Appendix I Answers to Odd-Numbered Exercises A93
7. n − 2: upper − 3 < 9.42, lower − 2 < 6.28
y
3
2
1
0
n − 4: upper − s10 1 s2 dsy4d < 8.96,
π
2
lower − s8 1 s2 dsy4d < 7.39
y
3
2
1
0
π
4
π
2
n − 8: upper < 8.65, lower < 7.86
y
3
2
1
0
π
4
9. 0.2533, 0.2170, 0.2101, 0.2050; 0.2
11. (a) Left: 0.8100, 0.7937, 0.7904;
right: 0.7600, 0.7770, 0.7804
13. 34.7 ft, 44.8 ft 15. 63.2 L, 70 L 17. 155 ft
19. 7840 21. lim
n l ` o n
i−1
π
2
3π
4
3π
4
π
π
π
2s1 1 2iynd
s1 1 2iynd 2 1 1 ? 2 n
23. lim
n l ` on ssinsiynd ?
i−1 n
25. The region under the graph of y − tan x from 0 to y4
27. (a) L n , A , R n
29. (a) lim
n l `
(c) 32 3
31. sin b, 1
64
o n
i 5
n 6 i−1
(b) n 2 sn 1 1d 2 s2n 2 1 2n 2 1d
12
x
x
x
3. 2 49
16 y
The Riemann sum represents the 4
sum of the areas of the two rectangles
above the x-axis minus the 1
3
2
0.5 1 1.5
sum of the areas of the four
2.5 3 x
rectangles below the x-axis.
_1
_2
_3
f(x)=≈-4
5. (a) 6 (b) 4 (c) 2
7. Lower, L 5 − 264; upper, R 5 − 16
9. 6.1820 11. 0.9071 13. 0.9029, 0.9018
15.
n
e x
17. y 1
dx 19. y7
0
2
1 1 x s5x 3 2 4xd dx
21. 29 23. 2 3 25. 23 4
29. lim
nl` o n
s4 1 s1 1 2iynd ? 2
i−1 n
31. lim
n l ` o n
i−1Ssin 5i
n
D n − 2 5
The values of R n appear to be
approaching 2.
33. (a) 4 (b) 10 (c) 23 (d) 2
35. 3 2 37. 3 1 9 4 39. 25 4 41. 0 43. 3
45. e 5 2 e 3 47. y 5 f sxd dx 49. 122
21
51. B , E , A , D , C 53. 15
59. 0 < y 1 x
3
0
dx < 1 61.
12 < y y3
y4
63. 0 < y 2 0 xe2x dx < 2ye 67. y 2 1
arctan x dx
73. y 1 0 x 4 dx 75. 1 2
R n
5 1.933766
10 1.983524
50 1.999342
100 1.999836
tan x dx < 12 s3
Exercises 5.3 • page 399
1. One process undoes what the other one does. See the Fundamental
Theorem of Calculus, page 398.
3. (a) 0, 2, 5, 7, 3 (d) y
(b) (0, 3)
g
(c) x − 3
Exercises 5.2 • page 388
1. 210 y
The Riemann sum represents
3
the sum of the areas of the two
_6 _4 _2 1
rectangles above the x-axis minus
0 2 4 x
the sum of the areas of the three
rectangles below the x-axis; that is,
_3
the net area of the rectangles with
_5
respect to the x-axis.
f(x)=x-1
5.
y
0 1
y=t@
x
t
1
0 1
x
(a), (b) x 2
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