http://www.ck12.org Chapter 3. Applications of Derivatives
3.8 Approximation Errors
Learning Objectives
A student will be able to:
- Extend the Mean Value Theorem to make linear approximations.
- Analyze errors in linear approximations.
- Extend the Mean Value Theorem to make quadratic approximations.
- Analyze errors in quadratic approximations.
Introduction
In this lesson we will use the Mean Value Theorem to make approximations of functions. We will apply the Theorem
directly to make linear approximations and then extend the Theorem to make quadratic approximations of functions.
Let’s consider the tangent line to the graph of a functionfat the point(a,f(a)).The equation of this line isy=
f(a)+f′(a)(x−a).We observe from the graph that as we considerxneara,the value off(x)is very close tof(a).
In other words, forxvalues close toa,the tangent line to the graph of a functionfat the point(a,f(a))provides an
approximation off(x)orf(x)≈f(a)+f′(a)(x−a).We call this thelinearortangent line approximationoffat
aand indicate it by the formulaL(x) =f(a)+f′(a)(x−a).