Question

Finding Taylor and Maclaurin SeriesIn Exercises 25–34, find the Taylor series generated by f at x = a.f(x) = cos(2x + π/2), a = π/4

Accepted Answer

Recall that the Taylor series of a function $$ f(x) $$ centered at $$ x = a  is $$given by the formula:

$$ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n $$

where $$ f^{(n)}(a)  is $$the $$ n -th $$derivative of $$ f $$ evaluated at $$ x = a . $$Identify the function and the center: here, $$ f(x) = \cos(2x + \frac{\pi}{2}) $$ and $$ a = \frac{\pi}{4} . We $$will need to find the derivatives of $$ f $$ evaluated at $$ x = \frac{\pi}{4} . $$Compute the first few derivatives of $$ f(x) $$:

- $$ f(x) = \cos(2x + \frac{\pi}{2}) $$
- $$ f'(x) = -2 \sin(2x + \frac{\pi}{2}) $$
- $$ f''(x) = -4 \cos(2x + \frac{\pi}{2}) $$
- $$ f^{(3)}(x) = 8 \sin(2x + \frac{\pi}{2}) $$
- $$ f^{(4)}(x) = 16 \cos(2x + \frac{\pi}{2}) $$

Notice the pattern in derivatives and coefficients. Evaluate each derivative at $$ x = \frac{\pi}{4} $$:

Calculate $$ f(a), f'(a), f''(a), f^{(3)}(a), f^{(4)}(a)  by $$substituting $$ x = \frac{\pi}{4} $$ into each derivative expression. Write the Taylor series expansion using the formula:

$$ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n $$

Substitute the values of $$ f^{(n)}(a) $$ found in the previous step to express the series explicitly.

Finding Taylor and Maclaurin Series
In Exercises 25–34, find the Taylor series generated by f at x = a.
f(x) = cos(2x + π/2),a = π/4

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Key Concepts

Taylor Series

Maclaurin Series

Derivatives of Trigonometric Functions

Finding Taylor and Maclaurin SeriesIn Exercises 25–34, find the Taylor series generated by f at x = a.f(x) = cos(2x + π/2),a = π/4