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11. } d d y x } 5 lim h 0 } y ( x 1 h h ) 2 y ( x ) } 5 lim h 0 5 lim h 0 5 lim h 0 5 lim h 0 (4 x 1 2 h 2 13) 5 4 x 2 13 At x 5 3, } d d y x } 5 4(3) 2 13 5 2 1, so the tangent line has slope 2 1 and passes through (3, y (3)) 5 (3, 2 16). y 5 2 1( x 2 3) 2 16 y 5 2 x 2 13 12. Let f ( x ) 5 x 3 . f 9 (1) 5 lim h 0 } f (1 1 h h ) 2 f (1) } 5 lim h 0 } (1 1 h h ) 3 2 1 3 } 5 lim h 0 5 lim h 0 (3 1 3 h 1 h 2 ) 5 3 (a) The tangent line has slope 3 and passes through (1, 1). Its equation is y 5 3( x 2 1) 1 1, or y 5 3 x 2 2. (b) The normal line has slope 2 } 1 3 } and passes through (1, 1). Its equation is y 5 2 } 1 3 } ( x 2 1) 1 1, or y 5 2 } 1 3 } x 1 } 4 3 } . 13. Since the graph of y 5 x ln x 2 x is decreasing for 0 , x , 1 and increasing for x . 1, its derivative is negative for 0 , x , 1 and positive for x . 1. The only one of the given functions with this property is y 5 ln x . Note also that y 5 ln x is undefined for x , 0, which further agrees with the given graph. (ii) 14. Each of the functions y 5 sin x , y 5 x , y 5 ˇ x w has the property that y (0) 5 0 but the graph has nonzero slope (or undefined slope) at x 5 0, so none of these functions can be its own derivative. The function y 5 x 2 is not its own derivative because y (1) 5 1 but y 9 (1) 5 lim h 0 } (1 1 h h ) 2 2 1 2 } 5 lim h 0 } 2 h 1 h h 2 } 5 lim h 0 (2 1 h ) 5 2. This leaves only e x , which can plausibly be its own derivative because both the function value and the slope increase from very small positive values to very large values as we move from left to right along the graph. (iv) 15. (a) The amount of daylight is increasing at the fastest rate when the slope of the graph is largest. This occurs about one-fourth of the way through the year, sometime around April 1. The rate at this time is approximately } 2 4 4 h d o a u y rs s } or } 1 6 } hour per day. (b) Yes, the rate of change is zero when the tangent to the graph is horizontal. This occurs near the beginning of the year and halfway through the year, around January 1 and July 1. (c) Positive: January 1 through July 1 Negative: July 1 through December 31 16. The slope of the given graph is zero at x < 2 2 and at x < 1, so the derivative graph includes ( 2 2, 0) and (1, 0). The slopes at x 5 2 3 and at x 5 2 are about 5 and the slope at x 5 2 0.5 is about 2 2.5, so the derivative graph includes ( 2 3, 5), (2, 5), and ( 2 0.5, 2 2.5). Connecting the points smoothly, we obtain the graph shown. 17. (a) Using Figure 3.10a, the number of rabbits is largest after 40 days and smallest from about 130 to 200 days. Using Figure 3.10b, the derivative is 0 at these times. (b) Using Figure 3.10b, the derivative is largest after 20 days and smallest after about 63 days. Using Figure 3.10a, there were 1700 and about 1300 rabbits, respectively, at these times. 18. (a) The slope from x 5 2 4 to x 5 0 is } 0 2 2 2 ( 2 0 4) } 5 } 1 2 } . The slope from x 5 0 to x 5 1 is } 2 1 2 2 2 0 2 } 5 2 4. The slope from x 5 1 to x 5 4 is } 2 2 4 2 2 ( 2 1 2) } 5 0.

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