1. Homework Homework Assignment #14 Read Section 3.6 Page 165, Exercises: 1 – 49 (EOO) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

2. Homework, Page 165 Calculate the second and third derivatives. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

3. Homework, Page 165 Calculate the second and third derivatives. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

4. Homework, Page 165 Calculate the second and third derivatives. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

5. Homework, Page 165 Calculate the second and third derivatives. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

6. Homework, Page 165 Calculate the derivative indicated. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

7. Homework, Page 165 Calculate the derivative indicated. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

8. Homework, Page 165 Calculate the derivative indicated. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

9. Homework, Page 165 Calculate the derivative indicated. 25. Cont’d Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

10. Homework, Page 165 29. Calculate y (k) (0) where y = x 4 + ax 3 + bx 2 + cx + d, with a, b, c, and d constant for 0≤ k ≤ 5. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

11. Homework, Page 165 Find a general formula for f (n) (x). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

12. Homework, Page 165 37. (a) Find the acceleration at t = 5 min of a helicopter whose height (in ft) is h (t) = – 3t 3 + 400t. (b) Plot the acceleration h″ (t) for 0 ≤ t ≤ 6. How does the graph show that the helicopter is slowing during this interval? The negative values of the function indicate decreasing velocity and, therefore, decreasing speed. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

13. Homework, Page 165 41. Figure 7 shows the graph of the position of an object as a function of time. Determine the intervals on which the acceleration is positive. Acceleration is positive where the slope of the position curve is increasing. This appears to be on (10, 20) and (30,40). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

14. Homework, Page 165 45. Which of the following descriptions could not apply to Figure 8? Explain (a) Graph of acceleration when velocity is constant. (b) Graph of velocity when acceleration is constant. (c) Graph of position when acceleration is zero. The slope of the curve would be units of distance divided by units of time which yields units of velocity, so neither (a) nor (b) apply, but (c) does, as the slope of the curve is not changing. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

15. Homework, Page 165 49. According to one model that attempts to account for air resistance, the distance s(t) (in feet) travel by a falling raindrop satisfies: where D is the raindrop diameter and g = 32 ft/s2. Terminal velocity v term is defined as the velocity at which the drop has zero acceleration. (a) Show that Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

16. Homework, Page 165 49. Continued (b) Find v term for drops of diameter 0.003 and 0.0003ft. (c) In this model do rain drops accelerate more rapidly at higher or lower velocities? Rain drops accelerate more rapidly at lower velocities. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

17. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Chapter 3: Differentiation Section 3.6: Trigonometric Functions Jon Rogawski Calculus, ET First Edition

18. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Recalling that all angles will be measured in radians, unless otherwise stated, the derivatives of the sine and cosine functions are given below.

19. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

20. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Compare the slope of the graph of y = cos (x) to the values of its derivative y = – sin (x) for any x.

21. Example, Page 170 Find an equation of the tangent line at the point indicated. 2. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

22. Example, Page 170 Find the derivative of each function. 6. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

23. Example, Page 170 Find the derivative of each function. 10. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

24. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The derivatives of the other four trig functions may be found by using the definitions of the functions in terms of sine and cosine and differentiating using the quotient rule. The results are given in Theorem 2.

25. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

26. Example, Page 170 Calculate the second derivative. 28. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

27. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

28. Example, Page 170 Find an equation of the tangent line at the point specified. 32. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

29. Example, Page 170 44. Find y (157), where y = sin x. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

30. Homework Homework Assignment #15 Read Section 3.7 Page 170, Exercises: 1 – 49 (EOO), 43 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company