Chapter 8 - Taylor Series

  1. f(x) = cos(x^2)

To find the Taylor series for \(f(x) = \cos(x^2)\), we need to find the derivatives of \(f(x)\) and evaluate them at \(x = 0\) to get the coefficients of the series.

The first few derivatives of \(f(x)\) are:

  1. \(f'(x) = -2x \sin(x^2)\)
  2. \(f''(x) = -2 \sin(x^2) - 4x^2 \cos(x^2)\)
  3. \(f'''(x) = -4x \cos(x^2) - 12x \sin(x^2)\)
  4. \(f''''(x) = -4 \cos(x^2) + 48x^3 \sin(x^2)\)

Evaluating these derivatives at \(x = 0\) gives:

  1. \(f'(0) = 0\)
  2. \(f''(0) = -2\)
  3. \(f'''(0) = 0\)
  4. \(f''''(0) = -4\)

The Taylor series for \(f(x) = \cos(x^2)\) centered at \(x = 0\) is:

\[ \cos(x^2) \approx 1 - \frac{2x^2}{2!} - \frac{4x^4}{4!} = 1 - x^2 - \frac{x^4}{3} \]