Equal volumes Consider the region R bounded by the curves y = ax2 + 1, y = 0, x = 0, and x = 1, for a -1. Let S1 and S2 be solids generated when R is revolved about the x- and y-axes, respectively. a. Find V1 and V2, the volumes of S1 and S2, as functions of a. b. What are the values of a -1 for which V11a2 = V21a2?

Calculus notes for the week of 10/3/16 4.1 Maxima and Minima and 4.2 What Derivatives Tell Us 15 10 5 01 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 -5 -10 -15 f has a local maximum at c if f(c) > f(x) for all x sufficiently close to c. f has a local minimum at c if f(c) < f(x) for all x sufficiently close to c. We see that, if f is differentiable at a local extremum (c), then f’(c) = 0. It is impossible that f is not differentiable at a local extremum. Definition: f has a critical point at x if f ’(x) = 0 or f ’(x) DNE. Coordinates for local extremum will be critical points. We see that, if f ‘(x) is negative on an interval I, then f is decreasing on I. If f ‘(x) is positive on an interval I, then f is