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InstRatesOfChange

InstRatesOfChange - 1 RATES OF CHANGE AVERAGE AND...

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1 RATES OF CHANGE AVERAGE AND INSTANTANEOUS RATES OF CHANGE We have already considered the average rate of change back in another set of notes, but we review the definition anyway. DEFINITION: The average rate of change of a function f ( x ) with respect to x over the interval from x 0 to x 0 + h is Now here is the definition of the instantaneous rate of change. Note how it contrasts with the average rate of change definition. DEFINITION: The instantaneous rate of change of f with respect to x at x 0 is the derivative provided the limit exists. Sometimes this derivative is called the rate of change. If the word “instantaneous” does not appear before “rate of change” you must assume it is there. The word “average” is always expressed, never implied. The instantaneous rate of chance in f ( x ), namely f ' (x), is the instantaneous rate of change in f ( x ) per unit increase in x . EXAMPLE 1: The volume V = (4/3) π r 3 of a spherical balloon changes with the radius. At what rate does the volume change with respect to the radius when r = 2 ft.? SOLUTION: Here we are concerned with the instantaneous rate of change, so let us find the derivative dV dr . dV dr = 4 3 3 r 2 = 4 r 2 When r = 2 ft., the volume is changing at a rate of 16 π ft 3 /ft. This means that a small change in r ft in the radius would result in a change of (16 π )( r ) cubic feet in the volume of the sphere. MOTION ALONG A LINE - DISPLACEMENT, VELOCITY, SPEED, AND
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InstRatesOfChange - 1 RATES OF CHANGE AVERAGE AND...

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