PROBLEMS y g(x) y=h(x) y=f(x) In problems 1-3, sketch the graph of the derivative of each function. 1. Use Fig. 20. T Fig. 21 Fig. 22 Fig. 20 2. Use Fig. 21. 3. Use Fig. 22 300-+ height (feet) 300-+ height (feet) 300-+ height (feet) 200-+ 200- 200- In problems 4-6, the graph of 100- 100 100- the height of a helicopter is shown. Sketch the graph of 10 time (minutes) time (minutes) time (minutes) the upward velocity of the Fig. 23 Fig. 24 Fig. 25 helicopter. 4. Use Fig. 23. 5. Use Fig. 24. 6. Use Fig. 25. Functions f Height Velocity 7. In Fig. 26, match the graphs of the functions with those of their derivatives. 8. In Fig. 27, match the graphs showing the heights of rockets with those B showing their velocities. I T. P .. 9. Use the Second Shape Theorem to show that f(x) = In(x) is monotonic increasing on ( 0, 00 ). 10. Use the Second Shape Theorem to show Fig. 27 Fig. 26 that g(x) = e" is increasing on the entire real number line. 11. A student is working with a complicated function f and has shown that the derivative of f is always positive. A minute later the student also claims that f(x) = 2 when x = 1 and when x = It. Without checking the student's work, how can you be certain that it contains an error?12. Fig. 28 shows the graph of the derivative of a continuous function f. (a) List the critical numbers of f. {b} For what values of x does f have a local maximum? (c) For what values of x does f have alocalminimum'? 13. Fig. 29 shows the graph of the derivative of a continuous function g. (a) List the critical numbers of g. (b) For what values of x does g have a local maximum? (c) For what values of x does g have alocal minimum? In problems 1416, the graphs of the upward velocities of several helicopters are shown. Use each graph to determine when each helicopter was at a relatively maximum and minimum height. 14. Use Fig.30. 15. Use Fig.31. 16. Use Fig. 32. \" a: e a '6 5 T: T3 g 5 ' 5 5 F' 30 time time time 'g' Fig. 31 Fig. 32 In problems l? 22 , use information from the derivative of each function to help you graph the function. Find all local maximums and minimums of each function. 1?. f(x): x33x29x5 18. g(x)=2x315x2+6 l9. h(x):x43x2+3 2 2 3 20. s(t):t+sin{t) 21. r{t)= 1 22. f{x):x : t+1 23. f(x) : 2x + cos(x) so f(0) : 1. Without graphing the function, you can be certain that f has how many positive roots? (zero, one, two. more than two) 24. g{x) = 2x cos{x) so g(0) = l. Wimout graphing the function. you can be certain that g has how many positive roots? (zero, one, two. more than two) 25. h{x) : x3 + 9x 10 has a root at x : 1. Without graphing h. show that h has no other roots. 26. Sketch the graphs of monotonic decreasing functions which have exactly (a) zero roots, (b) one root. and (c) two roots. 2?. Each of the following statements is false. Give (or sketch) a counterexample for each statement. (a) If f is increasing on an interval 1. then f r(x) )0 for all x in l. {b} If f is increasing and differentiable on I. then f '{x) > 0 for all x in l. (c) If cars A and B always have the same speed, then they will always be the same distance apart