MATH 1170 Section 3.9 Worksheet

NAME

Linear Approximations and Differentials

Linearizations The idea behind linear approximations is that it might be easy to calculate the value f(a) for some a but difficult to compute the value f(x) for some x near a. But before we figure out how to do this, let’s review a couple of points.

Assume that we have function f that is differentiable at a point a. How would you write an equation for the line tangent to f at the point (a, f(a))?

Sketch a graph f at this linear function that displays the relationship.

Our goal is estimate the value f(x) for some x near a when f(x) is difficult to compute.

Since we cannot compute f(x), we will compute values of the linear function L(x) whose graph is tangent to f at (a, f(a)). In other words, we use the tangent line at (a, f(a)) as an approximation to the curve y = f(x) when x is near a.

The approximation f(x) ≈ f(a) + f 0(a)(x − a) is called the linear approximation. The linear function whose graph is the tangent line, that is

L(x) = f(a) + f 0(a)(x − a)

is called the linearization of f at a.

1 Examples √ 1. Find the linearization if the function f(x) = 1 − x at a = 0.

√ Use this to approximate the number 0.9.

√ Approximate the number 0.99.

Illustrate the relationships by graphing f and the langent line to f at a = 0.

Differentials If y = f(x), where f is a differentiable function, then the differential dx is an independent variable: that is, dx can be given the value of any real number. The differential dy is then defined in terms of dx by the equation

dy = f 0(x)dx.

So dy is a dependent variable; it depends on the value of x and dx. If dx is given a specific value and x is taken to be some specific number in the domain of f, then the numerical value of dy is determined.

2 To get a better understanding of what the differential represents, re-sketch the diagram of a differentiable function, f, and it’s derivative at (a, f(a)). Label the values dx and dy.

This means that dy represents the amount that the tangent line rises or falls. And this serves a a a good approximation for how much f rises or falls. In other words, ∆y = f(x + ∆x) − f(x). Now, if we let dx = x − a, then x = a + dx and we have the following approximation

f(a + dx) ≈ f(a) + dy.

We can use this fact in order to make an approximation

Examples 1. Find the differential of the function y = x6.

Use this to approximate the number (1.01)6 (a = 1 and dx = .01).

Approximate the number (1.001)6.

3 2. Use linear approximation or differentials to estimate the given numbers.

a. (2.001)5

b. e−0.015

c. ln 1.05

4