Calculus EARLY TRANSCENDENTALS RIGGs COCHRAN
Chapter 3 Integration PEARSON Copyright @2011 Pearson Education, Inc. Publishing as Pearson Addis on-Wesley
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3.1 Introducing the Derivative PEARSON Copyright @2011 Pearson Education, Inc. Publishing as Pearson Addis on-Wesley
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O Slope of tangent Slope of tangent line and line and Instantaneous Instantaneous rate of change rate of change are negative are positive FIGURE 3. 1 Copyright@2011 Pearson Education, Inc. Publishing as Pears on Addison-Wesley Slide 3-4
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Trajectory of moving object Tangents give direction of motion FIGURE 3.2 Copyright@2011 Pearson Education, Inc. Publishing as Pears on Addison-Wesley Slide 3- 5
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slope tan slope m f(r) y=f(r) If()-f(a) a a O lim (x)-f(a) nx→a r-d FIGURE 3.3 Copyright@2011 Pearson Education, Inc. Publishing as Pears on Addison-Wesley Slide 3-6
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DEFINITION Rates of Change and the Tangent line The average rate of change in f on the interval [a, x] is the slope of the correspond ing secant line f(r)-f(a) r- a The instantaneous rate of change in f at x= a is f(x-f(a) tan lim x→ax-al which is also the slope of the tangent line at x=a, provided this limit exists. The tangent line at x a is the unique line through(a, f(a)with slope m an. Its equation is y-f(a)= man(x-a) Copyright@2011 Pearson Education, Inc. Publishing as Pears on Addison-Wesley Slide 3-7
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y=64x+16 150 16x2+96r 80 (1,80) Slope of tangent 50 line at(1, 80) IS m 64. In FIGURE 3.4 Copyright@2011 Pearson Education, Inc. Publishing as Pears on Addison-Wesley Slide 3-8
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slope = mtan sIo sCc f(a+ h f(r) f(a+ h-f( f(a) +h lim f(a+ h)-f(a) tan /→0 h FIGURE 3.5 Copyright@2011 Pearson Education, Inc. Publishing as Pears on Addison-Wesley Slide 3-9
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ALTERNATIVE DEFINITION Rates of change and the Tangent Line The average rate of change in f on the interval [a, a+ h] is the slope of the corre sponding secant line f(a+ h-f The instantaneous rate of change inf at x a is f(a+ h-f( man= lim (2) →0 h which is also the slope of the tangent line at x a, provided this limit exists Copyright@2011 Pearson Education, Inc. Publishing as Pears on Addison-Wesley Slide 3-10
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