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chapter 2 64

# chapter 2 64 - 128 CHAPTER 2 LIMITS AND DERIVATIVES 14 The...

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Unformatted text preview: 128 CHAPTER 2 LIMITS AND DERIVATIVES 14. The slopes of the tangent lines on the graph of y : P(t) are always positive. so the y—values of y : P’(t) are always positive. These values start out relatively small and keep increasing. reaching a maximum at about t = 6. Then the y—values ofy : P’(t) decrease and get close to zero. The graph of P' tells us that the yeast culture grows most rapidly after 6 hours and then the growth rate declines. 15. It appears that there are horizontal tangents on the graph of M for t = 1963 and t : 1971. Thus. there are zeros for those values oft on the graph of 111'. The derivative is negative for the years 1963 to 1971. 1950 1960 1970 1980 1990 16. See Figure 1 in Section 3.4. 11. _ 18. As ac increases toward 1. f’(a:) decreases from The 510139 at 0 appears to be 1 and the slope at 1 very large numbers to 1. As :1: becomes large, appears to be 2.7. As ac decreases. the slope gets f’(:E) gets closer to 0- As a guess. f’(ac) 2 1/352 closer to 0. Since the graphs are so Similar. we or f'(a:) : l/m make sense. might guess that f’(m) : ex. 19. (a) By zooming in. we estimate that f/(O) : 0. f’(%) : 1. f'(1) : 2. 2‘5 and f/(Z) : 4. (b) By symmetry. f’(—a:) : —f’(m). So f'(-%) : —1. f’(71): -2. and f'(~2) : #4. (c) It appears that f I(:11) is twice the value of cc. so we guess that f’(;z:) : 2:10. 2.5 ...
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