10.2 Higher Approximations and Taylor Series

]> 10.2 Higher Approximations and Taylor Series

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10.2 Higher Approximations and Taylor Series

We address the following questions:

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 $a = f' (x_{0}), b = \frac{f " (x_{0})}{2}$ so that the quadratic approximation to $f$ at $x_{0}$ becomes

f (x_{0}) + f' (x_{0}) (x - x_{0}) + f " (x_{0}) \frac{{(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.