1. Chapter 6, Slide 1 Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Finney Weir Giordano Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.

2. Chapter 6, Slide 2 Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.1: The graph of y = ln/x and its relation to the function y = 1/x, x > 0. The graph of the logarithm rises above the x-axis as x moves from 1 to the right, and it falls below the axis as x moves from 1 to the left.

3. Chapter 6, Slide 3 Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.4: The graphs of inverse functions have reciprocal slopes at corresponding points.

4. Chapter 6, Slide 4 Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.6: The derivative of ƒ(x) = x 3 – 2 at x = 2 tells us the derivative of ƒ –1 at x = 6.

5. Chapter 6, Slide 5 Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.7: The graphs of y = ln x and y = ln –1 x. The number e is ln –1.

6. Chapter 6, Slide 6 Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.9: Exponential functions decrease if 0 1. As x , we have a x  0 if 0 1. As x  – , we have a x  if 0 1.

7. Chapter 6, Slide 7 Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.10: The graph of y = sin –1 x has vertical tangents at x = –1 and x = 1.

8. Chapter 6, Slide 8 Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.12: Slope fields (top row) and selected solution curves (bottom row). In computer renditions, slope segments are sometimes portrayed with vectors, as they are here. This is not to be taken as an indication that slopes have directions, however, for they do not.

9. Chapter 6, Slide 9 Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.16: The growth of the current in the RL circuit in Example 9. I is the current’s steady-state value. The number t = LIR is the time constant of the circuit. The current gets to within 5% of its steady- state value in 3 time constants. (Exercise 33)

10. Chapter 6, Slide 10 Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.19: Three steps in the Euler approximation to the solution of the initial value problem y´ = ƒ(x, y), y (x 0 ) = y 0. As we take more steps, the errors involved usually accumulate, but not in the exaggerated way shown here.

11. Chapter 6, Slide 11 Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.20: The graph of y = 2 e x – 1 superimposed on a scatter plot of the Euler approximation shown in Table 6.4. (Example 3)

12. Chapter 6, Slide 12 Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.21: Notice that the value of the solution P = 4454e 0.017t is 6152.16 when t = 19. (Example 5)

13. Chapter 6, Slide 13 Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.22: Solution curves to the logistic population model dP/dt = r (M – P)P.

14. Chapter 6, Slide 14 Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.23: A slope field for the logistic differential equation = 0.0001(100 – P)P. (Example 6) dP dt

15. Chapter 6, Slide 15 Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.24: Euler approximations of the solution to dP/dt = 0.001(100 – P)P, P(0) = 10, step size dt = 1.

16. Chapter 6, Slide 16 Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.26: The graphs of the six hyperbolic functions.

17. Chapter 6, Slide 17 Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Continued.

18. Chapter 6, Slide 18 Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Continued.

19. Chapter 6, Slide 19 Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.27: The graphs of the inverse hyperbolic sine, cosine, and secant of x. Notice the symmetries about the line y = x.

20. Chapter 6, Slide 20 Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Continued.

21. Chapter 6, Slide 21 Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.28: The graphs of the inverse hyperbolic tangent, cotangent, and cosecant of x.

22. Chapter 6, Slide 22 Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Continued.

23. Chapter 6, Slide 23 Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.30: One of the analogies between hyperbolic and circular functions is revealed by these two diagrams. (Exercise 86)

24. Chapter 6, Slide 24 Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.31: In a coordinate system chosen to match H and w in the manner shown, a hanging cable lies along the hyperbolic cosine y = (H/w) cosh (wx/H).

25. Chapter 6, Slide 25 Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.32: As discussed in Exercise 87, T = wy in this coordinate system.