9.9
Convergence of Taylor Series 549
Equations (6) and (9) give
f(b)=Pn(b) +
J
f(n+l)(c)
^ ( b -aT*1.
This concludes the proof.
Ixercists 9 3
Finding
Use substitution (as in Example 4) to find the Taylor series at x = 0 of the
functions in Exercises 1-10.
„-5JC
2. e-*'2 5. cos
1. e~
3. 5 sin (—x)
5x2
4. sin
6. cos (x2/3/V2)
8. tan"1 (3x4)
7. ln(l + x ) 1
41 The approximation ex = 1 + x + (x2/2) is used when x is small. Use the
Remainder Estimation Theorem to estimate the error when |x| < 0.1.
10
Use power series operations to find the Taylor series at x = 0 for the functions
in Exercises 11-28.
12. x2 sinx
11. xex
14. sinx - x +
3!
2
2
15. X COS 7TX 16. x cos (x )
2
27.
2L | ln(l + x )
(1 - x ) 2 24.
v2
19. 1 - 2x 2 25. ex + 20. xln(l + 2x) 23. x
(1 - -x)3 1 + x
22.
1
sinx* cosx
29. ex sin x
28.
44. (Continuation of Exercise 43.) Use the identity cos2 x = cos 2x + sin2 x to
obtain a power series for cos2 x.
-1 2
In tan x 26. cosx —
45. Taylor's Theorem and the Mean Value Theorem Explain how the Mean
Value Theorem (Section 4.2, Theorem 4) is a special case of Taylor's
Theorem.
sinx
31. (tan_1x)2
30.
42. (Continuation of Exercise 41.) When x < 0, the series for ex is an alternating
series. Use the Alternating Series Estimation Theorem to estimate the error
that results from replacing ex by 1 + x + (x2/2) when —0.1 < x < 0.
Compare your estimate with the one you obtained in Exercise 41.
Theory and ' I '-....<
43. Use the identity sin2x = (1 - cos 2x)/2 to obtain the Maclaurin series for
sin2x. Then differentiate this series to obtain the Maclaurin series for 2 sin
x cos x. Check that this is the series for sin 2x.
17. cos x (Hint: cos x = (1 +
cos 2x)/2.)
18. sin2x 1
39. How close is the approximation sinx = x when |x| < 10~ 3? For which of
these values of x is x < sin x ?
40 The estimate VT^-Tx = 1 + (x/2) is used when x is small. Estimate the
error when |x| < 0.01.
9. 1
1+ I x 3
2
13. —— 1 + cosx
38. If cos x is replaced by 1 - (x2/2) and |x| < 0.5, what estimate can be made of
the error? Does 1 — (x2/2) tend to be too large, or too small? Give reasons
for your answer.
1-x
Find the first four nonzero terms in the Maclaurin series for the functions in
Exercises 29-34.
46. Linearizations at inflection points Show that if the graph of a
twice-differentiable function f ( x ) has an inflection point at x = a, then the
linearization of / at x = a is also the quadratic approximation of / at x = a.
This explains why tangent lines fit so well at inflection points.
47. The (second) second derivative test
Use the equation
ln(l + x)
(1 + x) - ln(l - x)
32. cos2x• sinx
33. e&
34. sin (tan 1 x)
f(x) = f(a) + f'(a)(x - a)
f"(c2)
(x - a)2
Error Ls^
35. Estimate the error if Ps(x) = x — (x3/6) is used to estimate the value of sin
x at x — 0.1.
36. Estimate the error if P4(x) = 1 + x + (x2/2) + (x3/6) + (x4/24) is used to
estimate the value of ex at x = 1/2.
37. For approximately what values of x can you replace sinx by x — (x /6)
with an error of magnitude no greater than 5 X 10"4? Give reasons for your
answer.
3
to establish the following test.
Let / have continuous first and second derivatives and suppose that
f ' ( a ) = O.Then
a. / has a local maximum at a if /" ^ 0 throughout an interval whose
interior contains a;
b. / has a local minimum at a if/" — 0 throughout an interval whose
interior contains a.