MATH 131 section 1 – Calculus for Physical and Life Sciences Discussion 2 – October 3, 2007

Background

Recall that last time we introduced the derivative at a of a function f(x):

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

(if the limit exists). If f′(a) exists for all a in an interval, we can also use this to define the values of a new function, also called f′.

Discussion Questions

Below is the graph y = f(x).

A. In terms of the graph, what do each of the following quantities represent?

f(a) − f(b), \frac{f(b) − f(a)}{b − a}, f′(a), f′(b)

B. Based on the information in the graph, put the following quantities in numerical order from smallest to largest: f′(a), f′(b), the slope of the secant line containing P and Q, the slope of the secant line containing Q and R, \frac{f(c)−f(a)}{c−a}. (Don’t forget that for negative numbers we have −10 < −4, for instance!)

C. Redraw the same graph y = f(x) as accurately as you can, and indicate new points S = (s, f(s)), T = (t, f(t)), and U = (u, f(u)) on the graph such that s < t < u and

f′(u) < m_{T U} < f′(t) < f′(s)

where m_{T U} is the slope of the secant line containing your T and U.

D. Draw the graph of the function f′(x), where as above f′(x) is the derivative function – its value at x is the slope of the tangent to y = f(x) at x.