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 x0 is the linear function with value f(x0) and first derivative f '(x0) there.
The quadratic approximation is the quadratic function whose value and first two derivatives agree with those of f at argument x0. Being quadratic it can be written as f(x0) + a(x - x0) + b(x - x0)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 x0."
10.1 Write down the Taylor series expansion about x0 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 x0 = 0. Give the cubic approximation to the sine, formed at x0 = 1.
10.3 The exponential function, being its own derivative, can be factored out of its Taylor series expansion. Apply that expansion around x0, to deduce the relation between exp(x) and exp(x0).
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.