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 '(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 . Since its first derivative at is a, and second derivative is 2b, we deduce so that the quadratic approximation to f at 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 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.
