42 CHAPTER 2 LIMITS AND DERIVATIVES

(3) (1) 10 7 1 4 93
7. (a) (i) On the interval [1 3], ave = = = = 4 65 m s.
3 1 2 2
(3) (2) 10 7 5 1
(ii) On the interval [2 3], ave = = = 5 6 m s.
3 2 1
(5) (3) 25 8 10 7 15 1
(iii) On the interval [3 5], ave = = = = 7 55 m s.
5 3 2 2
(4) (3) 17 7 10 7
(iv) On the interval [3 4], ave = = = 7 m s.
4 3 1

(b) Using the points (2 4) and (5 23) from the approximate tangent
23 4
line, the instantaneous velocity at = 3 is about 6 3 m s.
5 2




9. (a) For the curve = sin(10 ) and the point (1 0):



2 (2 0) 0 05 (0 5 0) 0
15 (1 5 0 8660) 1 7321 06 (0 6 0 8660) 2 1651
14 (1 4 0 4339) 1 0847 07 (0 7 0 7818) 2 6061
13 (1 3 0 8230) 2 7433 08 (0 8 1) 5
12 (1 2 0 8660) 4 3301 09 (0 9 0 3420) 3 4202
11 (1 1 0 2817) 2 8173

As approaches 1, the slopes do not appear to be approaching any particular value.

(b) We see that problems with estimation are caused by the frequent
oscillations of the graph. The tangent is so steep at that we need to
take -values much closer to 1 in order to get accurate estimates of
its slope.



(c) If we choose = 1 001, then the point is (1 001 0 0314) and 31 3794. If = 0 999, then is

(0 999 0 0314) and = 31 4422. The average of these slopes is 31 4108. So we estimate that the slope of the

tangent line at is about 31 4.