Taylor Series Math Example 4

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Example 4

medium
Use three terms of the Maclaurin series for cos⁡x\cos x to approximate cos⁡(0.1)\cos(0.1).

Solution

  1. 1
    cos⁡x=1−x2/2+x4/24−⋯\cos x = 1 - x^2/2 + x^4/24 - \cdots
  2. 2
    At x=0.1x=0.1: 1−0.005+0.0000042≈0.99500421 - 0.005 + 0.0000042 \approx 0.9950042.

Answer

cos⁡(0.1)≈0.99500\cos(0.1) \approx 0.99500
Taylor series give polynomial approximations. For small xx, few terms suffice. This is how calculators compute cos⁡\cos.

About Taylor Series

A representation of a function as an infinite sum of terms calculated from the function's derivatives at a single point: f(x)=∑n=0∞f(n)(a)n!(x−a)nf(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n
When a=0a = 0, it's called a Maclaurin series.

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More Taylor Series Examples

Example 1 easy

Find the Maclaurin series for [formula] up to the [formula] term.

Example 2 hard

Find the Maclaurin series for [formula] and state the interval of convergence.

Example 3 easy

Write the first four non-zero Maclaurin terms for [formula].

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