CHAPTER Continuity Derivatives and the Shapes of Curves

The Mean Value Theorem If f is a differentiable function on the interval [a, b], then there exists a number c between a and b such that f’(c) = (f (b) - f (a)) / (b – a) or equivalently, f(b) – f(a) = f’(c) (b – a).

Increasing / Decreasing Test a)If f’ (x) > 0 on an interval, then f is increasing on that interval. b)If f’ (x) < 0 on an interval, then f is decreasing on that interval.

CHAPTER Continuity Example Find the intervals on which f is increasing or decreasing for f (x) = 2 sin x + sin 2 x.

The First Derivative Test Suppose that c is a critical number of a continuous function f. animation animation a)If f’ changes from positive to negative at c, then f has a local maximum at c. b)If f’ changes from negative to positive at c, then f has a local minimum at c. c)If f’ doesn’t change sign at c, then f has no local maximum or minimum at c.

A function (or its graph) is called concave upward on an interval I if f’ is an increasing function on I and concave downward on I if f‘ is decreasing on I. Concavity Test a) If f”(x) > 0 for all x in I, then the graph of f is concave upward on I. b) If f”(x) < 0 for all x in I, then the graph of f is concave downward on I.

The Second Derivative Test Suppose f” is continuous near c. a)If f’(c) = 0 and f”(c) > 0, then f has a local minimum at c. b) If f’(c) = 0 and f”(c) < 0, then f has a local maximum at c.

Example Find for f(x) = ln (tan 2 x). a) vertical and horizontal asymptotes b) intervals of increase and decrease

Example Find for f(x) = x / (x – 1) 2. a) local maximum and minimum b) intervals of concavity and the inflection points