Mathematics 133 – Intensive Calculus for Science

Discussion 3 – The Derivative Function, Graphically, Numerically, and Symbolically October 4, 2000

Background

We have seen how the derivative of a function f(x) at x = a,

f′(a) = lim_{h→0} \frac{f(a + h) − f(a)}{h}

measures the slope of the graph y = f(x) at x = a (that is, the slope of the tangent line to y = f(x) at x = a). Letting x = a vary, we get the derivative function f′(x):

(1)  f′(x) = lim_{h→0} \frac{f(x + h) − f(x)}{h}.

Today, we want to look at some additional examples of derivative functions graphically, numerically, and symbolically.

Discussion Questions

A) The following graph shows y = f(x) for some function. At which x is f′(x) = 0? On which intervals is f′(x) > 0? On which intervals is f′(x) < 0? Using this information, give a rough sketch of the graph y = f′(x).

B) Suppose we wanted to understand f′(x) for the function f(x) = 6^{x} (exponential growth). Using (1) above, explain why

f′(x) = 6^{x} · lim_{h→0} \frac{6^{h} − 1}{h}

To go farther at this point, we need to resort to some numerical work using a calculator. Compute \frac{6^{h}−1}{h} for h = ±1, ±.1, ±.01, ±.001. Using this information, guess an approximate value for lim_{h→0} \frac{6^{h}−1}{h} to 2 decimal places. Then this constant times 6^{x} gives an approximation to the derivative function of f(x) = 6^{x}.

C) What is f′(x) for f(x) = \frac{1}{x}? Use the limit definition:

\begin{aligned}f′(x) & = lim_{h→0} \frac{f(x + h) − f(x)}{h} \\ & = lim_{h→0} \frac{\frac{1}{x+h} − \frac{1}{x}}{h} \end{aligned}

then try to simplify and take the limit as h → 0.

Assignment

Group writeups due on Wednesday, October 11 (this is the first day back from Columbus Day break).