Finney Weir Giordano 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.

Figure 6.4: The graphs of inverse functions have reciprocal slopes at corresponding points.

Figure 6.6: The derivative of ƒ(x) = x3 – 2 at x = 2 tells us the derivative of ƒ –1 at x = 6.

Figure 6. 7: The graphs of y = ln x and y = ln–1 x Figure 6.7: The graphs of y = ln x and y = ln–1 x. The number e is ln –1.

Figure 6.9: Exponential functions decrease if 0 < a < 1 and increase if a > 1. As x , we have ax  0 if 0 < a < 1 and ax  if a > 1. As x  –  , we have ax if 0 < a < 1 and ax  0 if a > 1.

Figure 6.10: The graph of y = sin–1 x has vertical tangents at x = –1 and x = 1.

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.

Figure 6. 16: The growth of the current in the RL circuit in Example 9 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)

Figure 6.19: Three steps in the Euler approximation to the solution of the initial value problem y´ = ƒ(x, y), y (x0) = y0. As we take more steps, the errors involved usually accumulate, but not in the exaggerated way shown here.

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)

Figure 6. 21: Notice that the value of the solution P = 4454e0 Figure 6.21: Notice that the value of the solution P = 4454e0.017t is 6152.16 when t = 19. (Example 5)

Figure 6.22: Solution curves to the logistic population model dP/dt = r (M – P)P.

Figure 6.23: A slope field for the logistic differential equation dP dt = 0.0001(100 – P)P. (Example 6)

Figure 6. 24: Euler approximations of the solution to dP/dt = 0 Figure 6.24: Euler approximations of the solution to dP/dt = 0.001(100 – P)P, P(0) = 10, step size dt = 1.

Figure 6.26: The graphs of the six hyperbolic functions.

Continued.

Continued.

Figure 6.27: The graphs of the inverse hyperbolic sine, cosine, and secant of x. Notice the symmetries about the line y = x.

Continued.

Figure 6.28: The graphs of the inverse hyperbolic tangent, cotangent, and cosecant of x.

Continued.

Figure 6.30: One of the analogies between hyperbolic and circular functions is revealed by these two diagrams. (Exercise 86)

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).

Figure 6.32: As discussed in Exercise 87, T = wy in this coordinate system.