340 Chapter 6 • Differentiable Functions
- Find the sixth Taylor polynomial T 6 (x) for the function f(x) = ln x about
- Also, write the formula for the Lagrange form of R5(x).
- When a Taylor polynomial Tn(x) is used as an approximation to f(x),
IR(x)I is called the "error." Use Taylor's theorem to find an upper bound
on the error when T 6 ( x) about 0 is used to approximat e ex, for -2 :::; x :::;
2. [See the inequality in Example 6.5. 1 3.] Repeat for -1:::; x:::; 1. - As in Exercise 9, use Taylor's theorem to find an upper bound on the
"error" IR(x)I when T5(x) about 0 is used to approximate sinx, for - i :::;
x :::; l Repeat for -i :::; x :::; i. - Find a Taylor polynomial about 0 that approximates ex to within 3 dec-
imal place accuracy for all x in [-2, 2]. Repeat for [-1, l]. - Find a Taylor polynomial about 0 that approximates sin x to within 3
decimal place accuracy for all x in [ -i , i]. Repeat for [ -i , i]. - Use Taylor's theorem and the methods of Example 6.5.13 to prove that
. oo (- 1 rx2n+i oo (-l)nx2n
Vx E JR, smx = L ( 2 )' , and cosx = L ( 2 )'
n=O n + l · n=O n ·
14. Use Taylor's theorem to prove that for all x in the interval [1, 2), ln(x) =
oo (-l)n-1
L (x - l t. [See Exercise 8 and Example 6.5.13.]
n=l n- Prove Theorem 6.5.14. [Hint: See how Example 6 .5. 13 was done.]
- Use the nth derivative test to lo cate all maxima and minima of the given
function. Justify your answer.
(a) f(x) = ex
3
(b) f(x) =ex•
17. Use the nth derivative test to determine whether sin^3 (x^2 ) has a local
maximum, local minimum, or neither at x = 0.18. Proof of the nth Derivative Test: Prove Theorem 6.5.15, as follows:(a) Write the (n - 2)nd Taylor polynomial for f(x).
(b) Show that if n is even and f (n)(a) > 0, then for all x and c within
a small enough neighborhood of a, Rn-i(x) ~ 0. Show that this
implies t hat f has a local minimum at a.
(c) Show that if n is even and f(n)(a) < 0, then for all x and c within
a small enough neighborhood of a, Rn-i(x) :::; 0. Show that this
implies that f has a local maximum at a.