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 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."

Exercises:

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.