344
CHAPTER 39
39.31
If /W = tan~
1 x, find/
(99> (0).
But
Since
Thus,
39.32
Find/
(100) (0)if f(x) = e"\
Hence,
Hence,
39.33
A certain function f(x) satisfies /(0) = 2, /'(0) = 1, /"(0) = 4, /'"(O) = 12, and /
3.
Find a formula for/(AC).
The Maclaurin series for f(x) is the polynomial
Thus, f(x) = 2 + x + 2x
2 + 2x
3 .
39.34
Exhibit the nth nonzero term of the Maclaurin series for In (1 + x
2 ).
ln(l + x) = x-x
2 /2 + x
3 /3
+ (-l)"
+ Wn + --for |*|<1.
Hence, In (1 -f x
2 ') = x
2 - x*/2 +
x
6 /3 +••• + (-I)
n + 1 x
2 "/n + •••. Thus, the nth nonzero term is (-l)"
+1
i:
2
"/n.
39.35
Exhibit the nth nonzero term of the Maclaurin series for e * .
Then,
The nth nonzero term is (-1)" V"
2 / (n - 1)!.
39.36
Find the Maclaurin series for x sin 3x.
Hence,
and
x sin 3x =
39.37
Find the Taylor series for f(x) = 2*
2 + 4x - 3 about 1.
Method 1. f'(x) = 4x + 4, f"(x) = 4, and f
M (x) = 0 for n>2. Thus, /(I) = 3, /'(I) = 8,
/"(I) = 4, and f(x) = 3 + 8(x - 1) + 2(x - 1)".
Medwd 2. f(x) = 2[(* - 1) + I]2 + 4[(x - 1) + 1] - 3 = 2(x - I)2 + 4(x - 1) + 2 + 4(* - 1) + 4 - 3 = 3 + 8(jc -
1) + 2(x - I)
2 . This is the Taylor series, by Problem 39.11.
39.38 Find the Taylor series for x
4 about -3.
39.39
Find the Maclaurin series for
By Problem 38.104,
Then,
and
The nth term may be rewritten as