2.14 Linear Approximations

Summary

The tangent line to the graph of a function at a point (a,f(a)) is used to give approximate values of the function at nearby points. The function whose graph is the tangent line is called the linearization L(x) of f about the point x = a. After defining the notion of best, it is shown that L is the best linear approximation of f about x = a.

By the end of your studying, you should know:

How to define the linearization of f about x = a.
The geometric meaning of the linearization of a function about a point.
How to define and use an appropriate linearization to approximate values of the function.
What it means for the (tangent line) linearization L to be the best linear approximation.
What a differential is and how to compute it.

On-screen applet instructions: The initial configuration is an implementation of the figure at the beginning of Section 2.14 of the text, except that the slope of the line is not fixed. The applet is to be used to investigate the nature of the error function error(x) = f(x) − L(x) as x → a. First, for a fixed x from the pull-down menu use the slider to change the slope m of the line: in fact, let the slope of the line approach f′(a). For each m, the normalized error function error(t) is shown in the inset: watch its shape. Can you explain the change in the error function as the slope of the line approaches f′(a)? What can you conclude about the error as x → a? Click here for more details and an explanation.

Examples

When solving problems in geometric optics, engineers and physicists often use the simplifying assumption that, for small angles θ, sin(θ) is approximately equal to θ. Find a linear approximation for sin(x) that shows why this is a reasonable assumption.

A pizza restaurant sells an average of 80 pizzas per day at its usual price of $12.95. It experiments with sales and coupons for dollars off the usual price, and finds that the number of pizzas sold when the price decreases by 2 dollars is 135. It estimates that the number of pizzas sold when the price does down by x dollars is modeled by the function 50 ln(x + 1) + 80. Use linear approximation to find the change in the number of pizzas sold when the price drops from $10.95 to $9.95.

Find the linear approximation of the function

about a = 0. Use it to approximate the square root of 0.9.