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
Copyright © 2011 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 6
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
Copyright © 2011 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 7
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
Copyright © 2011 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 8
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
Copyright © 2011 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 9
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
Copyright © 2011 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3 - 10