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

Maclaurin series and given information:

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

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

Use the Maclaurin series:

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

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

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

Therefore,

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

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

f^{'} (x) = 4 x^{3} + 6 x^{5} + \dots \dots \dots . \dots f^{''} (x) = 12 x^{2} + 30 x^{4} + \dots \dots \dots \dots . . . . {\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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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

Step by step solution

Maclaurin series and given information:

Use the Maclaurin series:

One App. One Place for Learning.

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Use Maclaurin to do problem 26 to 29 and check your results by computer.d6dx6(x4ex2)x=0

Short Answer

Step by step solution

Maclaurin series and given information:

Use the Maclaurin series:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Physics of Motion

Electricity and Magnetism

Waves Physics

Nuclear Physics

Mechanics and Materials

Atoms and Radioactivity

Study anywhere. Anytime. Across all devices.

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