1. Homework Homework Assignment #17 Read Section 3.9 Page 184, Exercises: 1 – 49 (EOO) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

2. Homework, Page 184 1. Show that if you differentiate both sides of x 2 + 2y 3 = 6, the result is 2x + 6y 2 y′ = 0. Then solve for y′ and calculate dy/dx at the point (2, 1). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

3. Homework, Page 184 Differentiate the expression with respect to x. 5. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

4. Homework, Page 184 Calculate the derivative of y with respect to x. 9. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

5. Homework, Page 184 Calculate the derivative of y with respect to x. 13. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

6. Homework, Page 184 Calculate the derivative of y with respect to x. 17. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

7. Homework, Page 184 Calculate the derivative of y with respect to x. 21. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

8. Homework, Page 184 Calculate the derivative of y with respect to x. 25. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

9. Homework, Page 184 Find the equation of the tangent line at the given point. 29. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

10. Homework, Page 184 Find the equation of the tangent line at the given point. 33. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

11. Homework, Page 184 37. Show that no point on the graph of x 2 – 3xy +y 2 = 1 has a horizontal tangent line. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

12. Homework, Page 184 41. Differentiate the equation x 3 + 3xy 2 = 1 with respect to the variable t and express dy/dt in terms of dx/dt. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

13. Homework, Page 184 45. The folium of Descartes is the curve with the equation x 3 + y 3 = 3xy. Find an equation of the tangent line to the curve at the point (2/3, 4/3). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

14. Homework, Page 184 49. The equation xy = x 3 – 5x 2 + 2x – 1 defines a trident curve. Find the points where the tangent to the trident is horizontal. (a) Show that xy′ + y = 3x 2 – 10x + 2. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

15. Homework, Page 184 49. (b) Set y′ = 0 in (a), replace y by x –1 (x 3 – 5x 2 + 2x – 1), and solve for x. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

16. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Chapter 3: Differentiation Section 3.9: Derivatives of Inverse Functions Jon Rogawski Calculus, ET First Edition

17. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Recall that inverse functions are defined for one-to-one functions, and their graphs are reflections about the line y = x of the graph of the basic function, and for every point (a, b) on the graph of the basic function there is a corresponding point (b, a) on the graph of the inverse function. We then have Theorem 1, which states:

18. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company In Figure 1 (A), we see that inverse linear functions have slopes that are the reciprocals of one another. For other functions with inverse functions The slope at a point is given by the two relationships on the previous slide.

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

20. Example, Page 191 2. Let g(x) be the inverse of f (x) = x 3 + 1. find a formula for g(x) and calculate g′(x) in two ways, using Theorem 1 and by direct calculation. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

21. Example, Page 191 Use Theorem 1 to calculate g′(x) where g(x) is the inverse of f (x). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

22. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company To find the slope of the graph of arcsine or arccosine, we need to first differentiate the function, as shown in Theorem 2.

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

24. Example, Page 191 Compute the derivative at the point without using a calculator. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

25. Example, Page 191 Find the derivative. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

26. Example, Page 191 Find the derivative. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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

29. Homework Homework Assignment #18 Read Section 3.10 Page 191, Exercises: 1 – 37 (EOO) Quiz next time Rogawski Calculus Copyright © 2008 W. H. Freeman and Company