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Chapter 1: Q29MP (page 45)

Use Maclaurin to do problem 26 to 29 and check your results by computer.
${\frac{d^{6}}{d x^{6}} (x^{4} e^{x^{2}})}_{x = 0}$

Short Answer

Expert verified

Answer using Maclaurin series is

${\frac{d^{6}}{{dx}^{6}} (x^{4} e^{x^{2}})}_{x = 0} = 720$

Step by step solution

01

Maclaurin series and given information:

Consider . ${\frac{d^{6}}{{dx}^{6}} (x^{4} e^{x^{2}})}_{x = 0}$

Maclaurin series: $f (x) = f (0) + \frac{f^{'} (0)}{1!} (x) + \frac{f^{''} (0)}{2!} (x^{2}) +$

02

Use the Maclaurin series:

Let $f (x) = x^{4} e^{x^{2}}$

Consider the expansion of $e^{x}$ .

$e^{x} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + 鈰� 鈥� 鈥� {e^{x}}^{2}) = 1 + x^{2} + \frac{x^{4}}{2!} + \frac{x^{6}}{3!} + 鈰� 鈥� 鈥� . x^{4} {e^{x}}^{2}) = x^{4} + x^{6} + \frac{x^{8}}{2!} + \frac{x^{10}}{3!} + 鈰� 鈥� . . . .$

Therefore,

$f (x) = x^{4} e^{(} x^{2}) = x^{4} + x^{6} + \frac{x^{8}}{2!} + \frac{x^{10}}{3!} + 鈰� 鈥� 鈥�$

Find the derivatives of f(x) at x=0 .

f^{'} (x) = 4 x^{3} + 6 x^{5} + 鈰� 鈥� 鈥� . 鈥� f^{''} (x) = 12 x^{2} + 30 x^{4} + 鈰� 鈥� 鈥� 鈥� . . . . {\frac{d^{6}}{d x^{6}} f (x)}_{x = 0} = 720

Hence, ${\frac{d^{6}}{d x^{6}} f (x)}_{x = 0} = 720$

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Most popular questions from this chapter

$\frac{1}{x} s i n x$

By computer or tables, find the exact sum of each of the following series.

$(a) {鈭�}_{n = 1}^{鈭�} \frac{n}{{(4 n^{2} - 1)}^{2}} (b) {鈭�}_{n = 1}^{鈭�} \frac{n^{3}}{n!} (c) {鈭�}_{n = 1}^{鈭�} \frac{n (n + 1)}{3^{n}}$

Use Maclaurin series to evaluate each of the following. Although you could do them by computer, you can probably do them in your head faster than you can type them into the computer. So use these to practice quick and skillful use of basic series to make simple calculations.

$\frac{d^{10}}{{dx}^{10}} (x^{8} \tan^{2} x) at x = 0 .$

$x^{2} l n (1 - x)$

$e^{x} s i n x$

See all solutions

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