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

Verified step by step guidance

1. 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 \( x = a \).
2. 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 \( x = \frac{\pi}{4} \).
3. 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.
4. 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.
5. 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.

Key Concepts

Taylor Series

A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point a. It approximates the function near that point using polynomial terms, where each term involves the nth derivative evaluated at a, multiplied by (x - a)^n and divided by n!.

Maclaurin Series

The Maclaurin series is a special case of the Taylor series centered at a = 0. It expresses a function as an infinite sum of derivatives evaluated at zero, useful for approximating functions near zero. Understanding this helps in comparing and deriving series expansions at different points.

Derivatives of Trigonometric Functions

Finding the Taylor series for f(x) = cos(2x + π/2) requires computing successive derivatives of the cosine function, applying the chain rule for the inner function 2x + π/2. Recognizing the pattern of derivatives for cosine and sine functions is essential to formulating the series terms correctly.