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

2. Chapter 6ET, Slide 2 Chapter 6 ET. 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 6ET, Slide 3 Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.2: The graphs of y = ln x and y = ln –1 x. The number e is ln –1 1.

4. Chapter 6ET, Slide 4 Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.6: 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)

5. Chapter 6ET, Slide 5 Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.9: 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.

6. Chapter 6ET, Slide 6 Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.10: The graph of y = 2e x – 1 superimposed on a scatter plot of the Euler approximation shown in Table 6.4. (Example 3)

7. Chapter 6ET, Slide 7 Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.11: Notice that the value of the solution P = 4454e 0.017t is 6152.16 when t = 19. (Example 5)

8. Chapter 6ET, Slide 8 Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.12: Solution curves to the logistic population model dP/dt = r (M – P)P.

9. Chapter 6ET, Slide 9 Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.13: A slope field for the logistic differential equation = 0.0001(100 – P)P. (Example 6) dP dt

10. Chapter 6ET, Slide 10 Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.14: Euler approximations of the solution to dP/dt = 0.001(100 – P)P, P(0) = 10, step size dt = 1.

11. Chapter 6ET, Slide 11 Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.16: The graphs of the six hyperbolic functions.

12. Chapter 6ET, Slide 12 Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Continued.

13. Chapter 6ET, Slide 13 Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Continued.

14. Chapter 6ET, Slide 14 Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.17: The graphs of the inverse hyperbolic sine, cosine, and secant of x. Notice the symmetries about the line y = x.

15. Chapter 6ET, Slide 15 Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Continued.

16. Chapter 6ET, Slide 16 Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.18: The graphs of the inverse hyperbolic tangent, cotangent, and cosecant of x.

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

18. Chapter 6ET, Slide 18 Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.20: One of the analogies between hyperbolic and circular functions is revealed by these two diagrams. (Exercise 86)

19. Chapter 6ET, Slide 19 Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.21: 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).

20. Chapter 6ET, Slide 20 Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Figure 6.22: As discussed in Exercise 87, T = wy in this coordinate system.