2.2 Basic Differentiation Rules and Rates of Change.

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Presentation transcript:

2.2 Basic Differentiation Rules and Rates of Change

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

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

Examples This one illustrates the Constant Multiple Rule HW Pg odds, odds, odds, 111, 113, 114

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

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

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:

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

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 , 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.

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.

The velocity at time t is given by the 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 odds, 37, 38, 51, odds, 70, 89, 93, 95