]> 10.2 Higher Approximations and Taylor Series

10.2 Higher Approximations and Taylor Series

What are these higher, non-linear approximations to $f$ in terms of its derivatives?

Why do we do these things?

How accurate are these approximations?

What happens when $f$ is a function of several variables?

The linear approximation to $f$ at $x 0$ is the linear function with value $f ( x 0 )$ and first derivative $f ' ( x 0 )$ there.

The quadratic approximation is the quadratic function whose value and first two derivatives agree with those of $f$ at argument $x 0$ . Being quadratic it can be written as $f ( x 0 ) + a ( x − x 0 ) + b ( x − x 0 ) 2$ .

We determine $a$ and $b$ by applying the condition that its derivatives are those of $f$ at argument $x 0$ . Since its first derivative at $x 0$ is $a$ , and second derivative is $2 b$ , we deduce $a = f ' ( x 0 ) , b = f " ( x 0 ) 2$ so that the quadratic approximation to $f$ at $x 0$ becomes

$f ( x 0 ) + f ' ( x 0 ) ( x - x 0 ) + f " ( x 0 ) ( x - x 0 ) 2 2$

We can extend this argument to create the cubic approximation, etc, when $f$ is suitably differentiable by applying the same steps with still higher derivatives. If we do this on forever, we get the "Taylor series expansion of $f$ at argument $x 0$ ."

Exercises:

10.1 Write down the Taylor series expansion about $x 0$ for a general infinitely differential function $f$ .

10.2 Write down the approximation formula of degree 5 for a general function that is 5 times differentiable, and apply it explicitly for the sine function at $x 0 = 0$ . Give the cubic approximation to the sine, formed at $x 0 = 1$ .

10.3 The exponential function, being its own derivative, can be factored out of its Taylor series expansion. Apply that expansion around $x 0$ , to deduce the relation between $exp ⁡ ( x )$ and $exp ⁡ ( x 0 )$ .

The following applet allows you to enter a standard function and look at what the first three of these approximations look like, as defined over a domain of your choosing.