# MATH137 Lecture Notes - Lecture 36: Differentiable Function, Inverse Trigonometric Functions

Higher Order Error

Recall for

, we defined the error to be |f(x) -

|. Similarly for the Taylor Polynomial ,

we define the error as |f(x) - |.

The expression f(x) - is often called the nth degree Taylor Remainder and denoted by

That is = f(x) - so that the error = ||

Again for the linear approximation

, we had the result |f(x) -

| ≤

provided that

|f ’’| ≤ M

For the Taylor polynomial approximation, we have:

Taylor’s Theorem

For an n+1 times differentiable function f on an interval I, if , then there is a c between x and a such

that

Note this is basically a higher other version of the Mean Value Theorem

Recall:

i..e. when n = 0, = f(a) and we get by Taylor’s theorem:

(i.e. MVT)

Furthermore when n =1

And Tayler’s theorem says

Which is our previous result assuming |f ‘’| ≤ M

More generally:

Taylor’s Inequality

Assuming the condition for Taylor’s theorem, if | )| ≤ M, then

Error =

Ex. Approximate using of f(x) = . Find an upper bound on the error.

Soln:

We previously calculated