Transcript chap.06ET

Chapter
6 ETGiordano,
. FinneyThomas’
Weir Giordano,
Thomas’
Tenth Edition
© 2001.All
Addison
Wesley
Chapter 6 ET.
Finney Weir
Calculus, Tenth
EditionCalculus,
© 2001. Addison
Wesley Longman
rights reserved.
Longman
All rights
reserved.
Chapter
6ET,
Slide
1
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 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Chapter 6ET, Slide 2
Figure 6.2: The graphs of y = ln x and y = ln–1x. The number e
is ln –1 1.
Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Chapter 6ET, Slide 3
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 steadystate value in 3 time constants. (Exercise 33)
Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Chapter 6ET, Slide 4
Figure 6.9: 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.
Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Chapter 6ET, Slide 5
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)
Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Chapter 6ET, Slide 6
Figure 6.11: Notice that the value of the solution
P = 4454e0.017t is 6152.16 when t = 19. (Example 5)
Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Chapter 6ET, Slide 7
Figure 6.12: Solution curves to the logistic population model
dP/dt = r (M – P)P.
Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Chapter 6ET, Slide 8
Figure 6.13: A slope field for the logistic differential equation
dP
= 0.0001(100 – P)P. (Example 6)
dt
Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Chapter 6ET, Slide 9
Figure 6.14: Euler approximations of the solution to
dP/dt = 0.001(100 – P)P, P(0) = 10, step size dt = 1.
Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Chapter 6ET, Slide 10
Figure 6.16: The graphs of the six hyperbolic functions.
Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Chapter 6ET, Slide 11
Continued.
Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Chapter 6ET, Slide 12
Continued.
Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Chapter 6ET, Slide 13
Figure 6.17: The graphs of the inverse hyperbolic sine,
cosine, and secant of x. Notice the symmetries about the
line y = x.
Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Chapter 6ET, Slide 14
Continued.
Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Chapter 6ET, Slide 15
Figure 6.18: The graphs of the inverse hyperbolic tangent,
cotangent, and cosecant of x.
Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Chapter 6ET, Slide 16
Continued.
Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Chapter 6ET, Slide 17
Figure 6.20: One of the analogies between hyperbolic and
circular functions is revealed by these two diagrams.
(Exercise 86)
Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Chapter 6ET, Slide 18
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).
Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Chapter 6ET, Slide 19
Figure 6.22: As discussed in Exercise 87, T = wy in this
coordinate system.
Chapter 6 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Chapter 6ET, Slide 20