2.  Exploration:  Sketch a rectangular coordinate plane on a piece of paper.  Label the points (1, 3) and (5, 3).  Draw the graph of a differentiable function that starts at (1, 3) and ends at (5, 3).  WHAT DO YOU NOTICE ABOUT YOUR GRAPH?

3.  If  f is continuous on [a, b],  f is differentiable on (a, b) AND  f(a) = f(b)  Then there is at least one number c in (a, b) such that f’(c) = 0. In other words, if the stipulations above hold true, there is at least one point where the derivative equals 0 (there exists at least one critical point in (a, b) ALSO f has at least an absolute max or min on (a, b)).

4.  Example: Show that f(x) = x 4 – 2x 2 satisfies the stipulations to Rolle’s Theorem on [-2, 2] and find where Rolle’s Theorem holds true. f is continuous on [-2, 2] f is differentiable on (-2, 2) f(-2) = f(2) = 8

5.  Consider the graph below: c a b Slope of tangent at c? f’(c) Slope of secant from a to b?

6.  If  f is continuous at every point on a closed interval [a, b]  f is differentiable at every point on its interior (a, b)  Then there is at least one point c in (a, b) at which In other words, if the stipulations above hold true, there is at least one point where the instantaneous slope equals the average slope.

7.  Example: a.) Show that the function f(x) = x 2 + 2x – 1 on [0, 1] satisfies the Mean Value Theorem. b.) Find each value of c that satisfies the MVT. a.) f is continuous and differentiable on [0, 1] b.) So,

8.  Example: Given f(x) = 5 – (4/x), find all values of c in the open interval (1, 4) that satisfies the Mean Value Theorem. So,

9. Let f be a function defined on an interval I and let x 1 and x 2 be any two points in I.  f increases on I if x 1 < x 2 results in f(x 1 ) < f(x 2 )  f decreases on I if x 1 > x 2 results in f(x 1 ) > f(x 2 ) More importantly relating to derivatives: Let f be continuous on [a, b] and differentiable on (a, b): o If f’ > 0 at each point of (a, b), then f increases on [a, b] o If f’ < 0 at each point of (a, b), then f decreases on [a, b] o If f’ = 0 at each point of (a, b), then f is constant on [a, b] (called monotonic)

10.  Example: Find the open intervals on which is increasing or decreasing. 1.) Find critical points Interval(- ∞, 0) (0, 1)(1, ∞) Test Valuex = -1x = ½x = 2 f’(x)6 > 0-3/4 < 06 > 0 ConclusionIncreasingDecreasingIncreasing

11.  Find a function F whose derivative is f(x) = 3x 2  F(x) = x 3  A function F is an antiderivative of f on an interval I if F’(x) = f(x) for all x in I. To find an antiderivative of a power function, increase the exponent by 1 and divide the original coefficient by the new exponent. What is the antiderivative of 4x 5 ? (2/3)x 6 Look at example 8 on pg. 201!!!