asic Differentiation Rules and Rates of Chan
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
HW Pg. 115 3-13 odds, 39-49 odds,53-59 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 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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