Worksheet for Section 2.2

Section 2.2 looks more closely at the limit computations from Section 2.1, introducing some new terminology and notation for the concept. The derivative of $f$ at $a$, written $f'(a)$, is the instantaneous rate of change of $f$ at $a$ as computed in Section 2.1. So:

$$f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$$

Look first at the example $f(x)=x^2$ -- using the above limit, what is the slope of the tangent line to $f(x)$ at $x=2$? What is the equation for this tangent line? (Hint: plug into point-slope form.) Look also at the derivative of $\sin x$ at $x=0$ and at $x=\pi$, by looking at the graph of $\sin x$. Set up the limits for these derivatives, too - don't expect to be able to complete the evaluation for these limits, though (you will see how to finish these in Section 3.5). See if you can estimate the derivative from your graph, and use the Limits applet to test your estimate numerically. You can use similar numerical and graphical techniques to estimate the derivative of the exponential function $f(x)=e^x$ when $x=0$.

For an example that can be completed using the techniques from Section 1.8 to compute the limit, consider the function $f(x)=1/x$. Use the above definition of the derivative of $f(x)$ at $a$ to compute $f'(2)$. Use this to find an equation for the line tangent to the graph of $f(x)$ at $x=2$, and graph both $f(x)$ and this tangent line together.