MTH 251 – Differential Calculus Chapter 3 – Differentiation Section 3.11 Linearization and Differentials Copyright © 2010 by Ron Wallace, all rights reserved.

Zooming in on a Function x  [-3, 3] y  [-2, 5] (1,2)

Zooming in on a Function x  [0, 2] y  [-2, 5] (1,2)

Zooming in on a Function x  [0.5, 1.5] y  [-2, 5] (1,2)

Zooming in on a Function x  [0.9, 1.1] y  [-2, 5] (1,2) As you zoom in on a function at a point of continuity, the graph tends approach a line. Therefore, a function can be approximated near a point by the tangent line at that point.

Linearization Using the tangent line to a curve at a point as an approximation of a function near that point.  AKA: Standard Linear Approximation  NOTE: Linearization and finding the tangent line to a function at a point … is the same thing! If the function is differentiable at x = a, then the linearization of the function at a is … Purpose of Linearization?  It provides a quick and easy approximation of a complex function near a known point.

Preview of things to come … Linearization approximates a function with a line through a point of the function (a,f(a)). In Chapter 10, we’ll see how to …  approximate a function with a quadratic  approximate a function with a cubic  etc … notice the pattern?

Preview of things to come … Example: f(x) = sin a =  /4

Preview of things to come … Example: f(x) = sin a =  /4

Preview of things to come … Example: f(x) = sin a =  /4

Preview of things to come … Example: f(x) = sin a =  /4

Preview of things to come … Example: f(x) = sin a =  /4

Linearization Example … Approximate  Fist note that …  Therefore, use

Along the function: Differentials f(x) Tangent Line: Along the tangent line: SAME

Differentials f(x) Tangent Line:

Differentials Examples …  Determine dy for each of the following functions. dy is an estimate of the change in the function at x when there is a given change in x (called dx ).

Differentials - Example The diameter of a tree was 10 inches. During the following year, the circumference increased by 2 inches. About how much did the tree’s diameter increase? About how much did the tree’s cross-sectional area increase? Hints: