964 CHAPTER 14 A Preview of Calculus: The Limit, Derivative, and Integral of a Function The instantaneous rate of change of f at c is the derivative of f at c . Finding the Instantaneous Rate of Change The volume V of a right circular cone of height = h 6 feet and radius r feet is π π ( ) = = = V V r r h r 1 3 2 . 2 2 If r is changing, find the instantaneous rate of change of the volume V with respect to the radius r at = r 3. Solution EXAMPLE 5 The instantaneous rate of change of V at = r 3 is the derivative ( ) ′V 3 . π π π π π π ( ) ( ) ( ) ( ) ( )( ) ( ) [ ] ′ = − − = − − = − − = − + − = + = → → → → → V V r V r r r r r r r r r 3 lim 3 3 lim 2 18 3 lim 2 9 3 lim 2 3 3 3 lim 2 3 12 r r r r r 3 3 2 3 2 3 3 At the instant = r 3 feet, the volume of the cone is changing with respect to r at a rate of π ≈ 12 37.699 cubic feet per 1-foot change in the radius. DEFINITION Instantaneous Rate of Change The instantaneous rate of change of f at c is ( ) ( ) ( ) ′ = − − → f c f x f c x c lim x c (4) provided the limit exists. Now Work PROBLEM 43 4 Find the Instantaneous Velocity of an Object If ( ) = s f t is the position of an object at time t , then the average velocity of the object from t0 to t1 is ( ) ( ) = Δ Δ = − − ≠ s t f t f t t t t t Change in position Change in time 1 0 1 0 0 1 (5) DEFINITION Instantaneous Velocity If ( ) = s f t is the position function of an object at time t , the instantaneous velocity v of the object at time t0 is the limit of the average velocity Δ Δ s t as → t t0 so that Δ = − t t t0 approaches 0. That is, ( ) ( ) ( ) ( ) = ′ = − − → v t f t f t f t t t lim t t 0 0 0 0 0 (6)
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