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 horizontalline, it would have a slope of 0

Now break into groups of 2 or 3 and find the derivatives of the following functions

1 2x 3x2

-x-24x3

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

Examples

This one illustrates the Constant Multiple Rule

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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 trigfunctions, “co-” means oppositein derivatives.

Find the slope and equation of the tangent lineof 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 withrespect to time is given by…

change in distancechange in time

Also known asaverage velocity

Ex. If a free-falling object is dropped from aheight of 100 feet, its height s at time t is givenby the position function s = -16t2 + 100, wheres is measured in feet and t is measured in seconds.Find the average rate of change of the height overthe 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 derivative.

@ t = 2 seconds, s’(2) = -48 ft/sec.

The negative gives the direction, which in this case is down.

The General Position Function

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