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Chapter 6, Slide 1Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Finney Weir GiordanoFinney Weir Giordano
Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Chapter 6, Slide 2Chapter 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.
Chapter 6, Slide 3Chapter 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.
Chapter 6, Slide 4Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 6.6: The derivative of ƒ(x) = x3 – 2 at x = 2 tells us the derivative of ƒ –1 at x = 6.
Chapter 6, Slide 5Chapter 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.
Chapter 6, Slide 6Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
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.
Chapter 6, Slide 7Chapter 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.
Chapter 6, Slide 8Chapter 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.
Chapter 6, Slide 9Chapter 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)
Chapter 6, Slide 10Chapter 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.
Chapter 6, Slide 11Chapter 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)
Chapter 6, Slide 12Chapter 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 = 4454e0.017t is 6152.16 when t = 19. (Example 5)
Chapter 6, Slide 13Chapter 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.
Chapter 6, Slide 14Chapter 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)dPdt
Chapter 6, Slide 15Chapter 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.
Chapter 6, Slide 16Chapter 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.
Chapter 6, Slide 17Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Continued.
Chapter 6, Slide 18Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Continued.
Chapter 6, Slide 19Chapter 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.
Chapter 6, Slide 20Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Continued.
Chapter 6, Slide 21Chapter 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.
Chapter 6, Slide 22Chapter 6. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Continued.
Chapter 6, Slide 23Chapter 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)
Chapter 6, Slide 24Chapter 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).
Chapter 6, Slide 25Chapter 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.
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