966 CHAPTER 14 A Preview of Calculus: The Limit, Derivative, and Integral of a Function Figure 20(b) (e) At = t 2 sec, the instantaneous velocity of the ball is ( ) ( ) ( ) ′ = − ⋅ − =− − = s 2 16 2 2 5 16 1 16 ft sec (f) The instantaneous velocity of the ball is zero when ( ) ( ) ′ = − − = = = s t t t 0 16 2 5 0 5 2 2.5 sec (g) The ball passes the rooftop on the way down when = t 5. The instantaneous velocity at = t 5 is ( ) ( ) ′ = − − = − s 5 16 10 5 80 ft sec At = t 5sec, the ball is traveling −80 ft sec. When the instantaneous rate of change is negative, it means that the direction of the object is downward.The ball is traveling 80 ft/sec in the downward direction when = t 5 sec. Desmos can be used to calculate the derivative. See Figure 20(b). (h) The ball strikes the ground when = t 6. The instantaneous velocity when = t 6 is ( ) ( ) ′ = − − = − s 6 16 12 5 112 ft sec The velocity of the ball at = t 6sec is −112 ft sec. Again, the negative value implies that the ball is traveling downward. Exploration Determine the vertex of the quadratic function given in Example 6. What do you conclude about the velocity when ( ) s t is a maximum? SUMMARY The derivative of a function ( ) = y f x at c is defined as ( ) ( ) ( ) ′ = − − → f c f x f c x c lim x c provided the limit exists. • In geometry, ( ) ′ f c equals the slope of the tangent line to the graph of f at the point ( ) ( ) c f c , . • In physics, ( ) ′ f t0 equals the instantaneous velocity of an object at time t0 , where ( ) = s f t is the position of the object at time t . • In applications, if two variables are related by the function ( ) = y f x , then ( ) ′ f c equals the instantaneous rate of change of y at c . ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 14.4 Assess Your Understanding 1. Find an equation of the line with slope 5 containing the point ( ) − 2, 4 . (p. 36) 2. True or False The average rate of change of a function f from a to b is ( ) ( ) + + f b f a b a (pp. 93–94) 3. If ( ) ( ) − − → f x f c x c lim x c exists, it equals the slope of the to the graph of f at the point ( ) ( ) c f c , . 4. If ( ) ( ) − − → f x f c x c lim x c exists, it is called the of f at c . 5. Multiple Choice If ( ) = s f t denotes the position of an object at time t , the derivative ( ) ′ f t0 is the of the object at t0 . (a) slope (b) speed (c) velocity (d) time Concepts and Vocabulary 6. True or False The tangent line to a function is the limiting position of a secant line. 7. True or False The slope of the tangent line to the graph of f at ( ) ( ) c f c , is the derivative of f at c . 8. True or False The velocity of an object whose position at time t is ( ) s t is the derivative ( ) ′s t . 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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