Taylor Polynomials

$Taylor Polynomials Example 2$

$Taylor Polynomials Example 1$

$Taylor Polynomials Example 3$

In this section we’ll discuss infinite series that can frequently be used to represent functions.

The Taylor Series of an infinitely differentiable function in a neighborhood of $a$ is defined to be $\sum\limits_{n=0}^\infty \frac{f^n(a)}{n!} (x-a)^n = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f^3(a)}{3!}(x-a)^3 + \cdot \cdot \cdot$ .

Taylor Polynomial
For any $d > 0$ , the finite sum $\sum\limits_{n=0}^d \frac{f^n(a)}{n!} (x-a)^n = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f^3(a)}{3!}(x-a)^2 + \cdot \cdot \cdot + \frac{f^d(a)}{d!} (x-a)^d$ is called a Taylor Polynomial, and may be used to approximate $f(x)$ near $a$ .
That is:

$f(x) \approx \sum\limits_{n=0}^d \frac{f^n(a)}{n!} (x-a)^n$

for $x$ sufficiently close to $a$ , and $d$ sufficiently large.

Justification:
Note this is trivially true for $x = a$ .

Now consider the first-degree (linear) approximation for $f(x)$ near $a$ :

$f(x) \approx f(a) + f'(a)(x-a)$

This is just the tangent line to $y = f(x)$ at $x = a$ , and so it is a reasonably good approximation of $f(x)$ near $a$ .

In order to discuss higher degree Taylor Polynomials, we’ll look at a specific function.

Example:
Consider the function $f(x) = \sin(x)$ . We will find the first, second, third, and fourth-degree Taylor Polynomial approximations for $f(x)$ near $a = \pi/2$ .

Linear Approximation:

$\sin(x) \approx \sin(\pi/2) + \cos(\pi/2)(x-\pi/2) = 1$ .

(Graph of sin and $y = \sin(x)$ and $y = 1$ )

This is only a good approximation very close to $\pi/2$ .
Note that this is a constant function, rather than a first-degree polynomial, because $\cos(\pi/2) = 0$ .

Second Degree Taylor Polynomial:

$\sin(x) \approx \sin(\pi/2) + \cos(\pi/2)(x-\pi/2) - \sin(\pi/2) (x - \pi/2)^2 = 1 - (x - \pi/2)^2$

(Graph of sin and $y = \sin(x)$ and $y = 1 - (x - \pi/2)^2$ )

Third Degree Taylor Polynomial:

$\sin(x) \approx \sin(\pi/2) + \cos(\pi/2)(x-\pi/2) - \sin(\pi/2) (x - \pi/2)^2 - \cos(\pi/2) (x - \pi/2)^3 = 1 - (x - \pi/2)^2$

(This is the same as the second-degree polynomial, because $\cos(\pi/2) = 0$ )

Fourth Degree Taylor Polynomial:

$\sin(x) \approx \sin(\pi/2) + \cos(\pi/2)(x-\pi/2) - \sin(\pi/2) (x - \pi/2)^2 - \cos(\pi/2) (x - \pi/2)^3 + \sin(\pi/2)(x-\pi/2)^4= 1 - (x - \pi/2)^2 + (x - \pi/2)^4$

(Graph of sin and $y = \sin(x)$ and $y = 1 - (x - \pi/2)^2 + (x - \pi/2)^4$ )

Note that, as the degree $d$ gets larger, the polynomial approximation for the function gets better, even farther away from $a$ .

(Graph of sin and $y = \sin(x)$ and linear, second degree, fourth degree polynomials)

Many functions are equal to their Taylor Series on the entire number line or an open interval, but some are not.

Taylor Polynomials

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