Home  18.013A  Chapter 10 


We address the following questions:
What are these higher, nonlinear 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 $f\text{'}({x}_{0})$ 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 $2b$ , we deduce $a=f\text{\'}({x}_{0}),b=\frac{f"({x}_{0})}{2}$ so that the quadratic approximation to $f$ at ${x}_{0}$ becomes
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 $\mathrm{exp}(x)$ and $\mathrm{exp}({x}_{0})$ .
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.
