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

256 Chapter 3 Differentiation Rules
NOTE Although the possible error in Example 4 may appear to be rather large, a
better picture of the error is given by the relative error, which is computed by dividing
the error by the total volume:
DV
V
< dV V − 4r 2 dr
4
3 r − 3 dr
3 r
Thus the relative error in the volume is about three times the relative error in the radius.
In Example 4 the relative error in the radius is approximately dryr − 0.05y21 < 0.0024
and it produces a relative error of about 0.007 in the volume. The errors could also be
expressed as percentage errors of 0.24% in the radius and 0.7% in the volume.
3.10 Exercises
;
;
;
1–4 Find the linearization Lsxd of the function at a.
1. f sxd − x 3 2 x 2 1 3, a − 22
2. f sxd − sin x, a − y6
3. f sxd − sx , a − 4
4. f sxd − 2 x , a − 0
5. Find the linear approximation of the function
f sxd − s1 2 x at a − 0 and use it to approximate the
numbers s0.9 and s0.99 . Illustrate by graphing f and the
tangent line.
6. Find the linear approximation of the function
tsxd − s 3 1 1 x at a − 0 and use it to approximate the
numbers s 3 0.95 and s 3 1.1 . Illustrate by graphing t and the
tangent line.
7–10 Verify the given linear approximation at a − 0. Then
determine the values of x for which the linear approximation is
accurate to within 0.1.
7. lns1 1 xd < x 8. s1 1 xd 23 < 1 2 3x
9. s 4 1 1 2x < 1 1 1 2 x 10. e x cos x < 1 1 x
11–14 Find the differential of each function.
11. (a) y − xe 24x (b) y − s1 2 t 4
12. (a) y − 1 1 2u
1 1 3u
(b) y − 2 sin 2
13. (a) y − tan st (b) y − 1 2 v 2
1 1 v 2
14. (a) y − lnssin d (b) y − e x
1 2 e x
15–18 (a) Find the differential dy and (b) evaluate dy for the
given values of x and dx.
15. y − e x y10 , x − 0, dx − 0.1
16. y − cos x, x − 1 3 , dx − 20.02
17. y − s3 1 x 2 , x − 1, dx − 20.1
18. y − x 1 1 , x − 2, dx − 0.05
x 2 1
19–22 Compute Dy and dy for the given values of x and
dx − Dx. Then sketch a diagram like Figure 5 showing the line
segments with lengths dx, dy, and Dy.
19. y − x 2 2 4x, x − 3, Dx − 0.5
20. y − x 2 x 3 , x − 0, Dx − 20.3
21. y − sx 2 2 , x − 3, Dx − 0.8
22. y − e x , x − 0, Dx − 0.5
23–28 Use a linear approximation (or differentials) to estimate
the given number.
23. s1.999d 4 24. 1y4.002
25. s 3 1001 26. s100.5
27. e 0.1 28. cos 29°
29–31 Explain, in terms of linear approximations or differentials,
why the approximation is reasonable.
29. sec 0.08 < 1 30. s4.02 < 2.005
31.
1
9.98 < 0.1002
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