Chapter 1: Q38P (page 32)

$e^{\cos x - 1}$ . Hint: $e^{c} o s x = e \cdot e^{c o s x - 1}$

Short Answer

Expert verified

$e^{\cos (x)} = e \cdot e^{c o s (x) - 1}$

e^{\cos (x)} = e \cdot [1 - \frac{x^{2}}{3} + \frac{x^{4}}{6} + \dots]

Step by step solution

Given information.

The Maclaurin expansion of the function $e^{\cos x}$ is to be evaluated.

Definition of Maclaurin series.

The definition of the Maclaurin series is defined.

$c o s (x) = 1 - \frac{x^{2}}{2} + \frac{x^{4}}{24} - \frac{x^{6}}{720} + \dots c o s (x) - 1 = - \frac{x^{2}}{2} + \frac{x^{4}}{24} - \frac{x^{6}}{720} + \dots e^{x} = 1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6} + \dots$

Begin by stating the Maclaurin series.

State the Maclaurin series

$\cos (x) = 1 - \frac{x^{2}}{2} + \frac{x^{4}}{24} - \frac{x^{6}}{720} + \dots \cos (x) - 1 = - \frac{x^{2}}{2} + \frac{x^{4}}{24} - \frac{x^{6}}{720} + \dots e^{x} = 1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6} + \dots$

Make appropriate substitution in the Maclaurin series and evaluate.

Replace x with the expression $[- \frac{x^{2}}{2} + \frac{x^{4}}{24} - \frac{x^{6}}{720} + \dots]$ in the third defined Maclaurin series.

role="math" localid="1657382741919" $e^{\cos (x) - 1} = 1 + [- \frac{x^{2}}{2} + \frac{x^{4}}{24} + \dots] + \frac{{[- \frac{x^{2}}{2} + \dots]}^{2}}{2} + \dots e^{\cos (ω) - 1} = 1 + \frac{(- 3 x^{2} + x^{4})}{6} + \dots e^{\cos (x)} = e \cdot e^{c o s (x) - 1} e^{\cos (x)} = e \cdot [1 - \frac{x^{2}}{3} + \frac{x^{4}}{6} + \dots]$

Thus, the final answer is $e^{\cos (x)} = e \cdot [1 - \frac{x^{2}}{3} + \frac{x^{4}}{6} + \dots]$

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$e^{\cos x - 1}$ . Hint: $e^{c} o s x = e \cdot e^{c o s x - 1}$

Short Answer

Step by step solution

Given information.

Definition of Maclaurin series.

Begin by stating the Maclaurin series.

Make appropriate substitution in the Maclaurin series and evaluate.

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ecosx-1. Hint:ecosx=e⋅ecosx-1

Short Answer

Step by step solution

Given information.

Definition of Maclaurin series.

Begin by stating the Maclaurin series.

Make appropriate substitution in the Maclaurin series and evaluate.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Rotational Dynamics

Atoms and Radioactivity

Magnetism

Oscillations

Turning Points in Physics

Electromagnetism

Study anywhere. Anytime. Across all devices.

$e^{\cos x - 1}$ . Hint: $e^{c} o s x = e \cdot e^{c o s x - 1}$