1. 2.2 Basic Differentiation Rules and Rates of Change

2. Now for a little review. What is the derivative of f(x) = 3? This is called the “constant rule” and since the graph is a straight horizontal line, it would have a slope of 0 Now break into groups of 2 or 3 and find the derivatives of the following functions 12x2x 3x23x2 -x -2 4x 3

3. This is called the Power Rule and you will learn to love it.

4. Examples This one illustrates the Constant Multiple Rule HW Pg. 115 3-13 odds, 39-49 odds, 53-59 odds, 111, 113, 114

5. Let’s try these 2 Want proof? We can generalize this by saying that

6. Let’s look at some trig functions now You have to remember, in trig functions, “co-” means opposite in derivatives.

7. Find the slope and equation of the tangent line of the graph of y = 2 cos x at the point Therefore, the equation of the tangent line is:

8. The average rate of change in distance with respect to time is given by… change in distance change in time Also known as average velocity

9. Ex. If a free-falling object is dropped from a height of 100 feet, its height s at time t is given by the position function s = -16t 2 + 100, where s is measured in feet and t is measured in seconds. Find the average rate of change of the height over the following intervals. a. [1, 2] b. [1, 1.5] c. [1, 1.1] a. b. c.

10. At time t = 0, a diver jumps from a diving board that is 32 feet above the water. The position of the diver is given by where s is measured in feet and t in seconds. a.When does the diver hit the water? b.What is the diver’s velocity at impact? To find the time at which the diver hits the water, we let s(t) = 0 and solve for t. t = -1 or 2 -1 doesn’t make sense, so the diver hits at 2 seconds.

11. The velocity at time t is given by the derivative. @ t = 2 seconds, s’(2) = -48 ft/sec. The negative gives the direction, which in this case is down. The General Position Function HW Pg. 115 19-23 odds, 37, 38, 51, 61-69 odds, 70, 89, 93, 95