Worksheet for Section 3.9

Secton 3.9 is about using the derivative of a function to construct a linear approximation to the function, and investigating the accuracy of this approximation. Suppose $f(x)$ is differentiable at $x=a$. Then the tangent line to the graph of $f(x)$ at $x=a$ is given by $y=f(a)+f'(a)(x-a)$. In fact, for values of $x$ near $a$, this tangent line is approximately the same as the graph of $f(x)$:

$$f(x)\approx f(a)+f'(a)(x-a)$$

So the tangent line is sometimes called the local linearization of $f(x)$ near $x=a$. The error in this approximation is given by

$$E(x)=f(x)-f(a)-f'(a)(x-a)$$

With this definition for error, then:

$$\lim_{x\to a}\frac{E(x)}{x-a}=0$$

As an example, consider the function $f(x)=\sin x$ at $x=\frac\pi4$. Find the tangent line approximation to $f(x)$ at this point, and use it to find $E(x)$. Now use the Limits applet to investigate $$\lim_{x\to \frac\pi4}\frac{E(x)}{x-\frac\pi4}$$. Also look at $$\lim_{x\to \frac\pi4}\frac{E(x)}{(x-\frac\pi4)^2}$$. For comparison, compute $f''(\frac\pi4)$. This is an indication of what turns out to be a general pattern for the error in the tangent line approximation:

$$E(x)\approx\frac{f''(a)}{2}(x-a)^2$$

In fact, similar error formulas work out for higher degree polynomial approximations constructed using higher order derivatives. These polynomial approximations are called Taylor polynomials, and are covered in Chapter 10 (in Calculus II). For our purposes, though, this indicates that the tangent line approximation is actually a pretty good approximation to the differentiable function $f(x)$ near $x=a$, so that graphs of differentiable functions ``look linear'' if you look at them on a small enough scale, for example by ``zooming in'' to the graph of $f(x)$ on your calculator. I will demonstrate this with the function $f(x)=\sin x$ and its tangent line approximation from above.