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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 = 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 ) ( 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.

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