Fundamentals of Differential Equations (9th Edition)
9th Edition
ISBN: 9780321977069
Author: R. Kent Nagle, Edward B. Saff, Arthur David Snider
Publisher: PEARSON
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From its past behavior, John knows that the value of a stock has a cyclical component that
increases for the first three months of each year, falls for the next six, and rises again for
the last three. In addition, inflation adds a linear component to the stock's price. John is
seeking a model of the form
f(t) = mt + b + A sin (-)
with t in months since Jan 1. He has the following data in the table:
Jan 1
Apr 1
Oct 1
Jan 1
Date
Jul 1
Price $20.00 $37.50 $35.00 $32.50 $50.00
a) What is the length of the period in his model?
b) Determine two months when the Price is on the slanted midline.
c) Find value ofm, b, and A so that f fits the data.
For f(x) and g(x) given in Problems 35–38, find
(a) (f + g)(x)
(b) (f – g)(x)
(c) (f'g)(x)
(d) (f/g)(x)
35. f(x) = 3x g(x) = x'
36. f(x) = Vx g(x) = 1/x
37. f(x) = V2x g(x) = x²
38. f(x) = (x – 1)? g(x) = 1 – 2x
Click to
%3D
For f(x) and g(x) given in Problems 39–42, find
(a) (fº g)(x)
(b) (g •f)(x)
(c) ƒ(f(x))
(d) f(x) = (f·f)(x)
39. f(x) = (x – 1)³ g(x) = 1 – 2x
40. f(x) = 3x g(x) = x' – 1
41. f(x) = 2Vx g(x) = x* + 5
%3D
%3D
1
42. f(x) = g(x) = 4x + 1
gives the first derivative of a continuousfunction y = ƒ(x). Find y″ and then use Steps 2–4 of the graphingprocedure to sketch the general shape of the graph of ƒ.65. y′ = (8x - 5x2)(4 - x)^2 66. y′ = (x2 - 2x)(x - 5)^2
Chapter 7 Solutions
Fundamentals of Differential Equations (9th Edition)
Ch. 7.2 - Prob. 1ECh. 7.2 - Prob. 2ECh. 7.2 - Prob. 20ECh. 7.2 - Prob. 28ECh. 7.2 - Prob. 29ECh. 7.4 - In Problems 110, determine the inverse Laplace...Ch. 7.4 - Prob. 3ECh. 7.4 - Prob. 5ECh. 7.4 - Prob. 7ECh. 7.4 - Prob. 9E
Ch. 7.4 - Prob. 11ECh. 7.4 - Prob. 13ECh. 7.4 - In Problems 1120, determine the partial fraction...Ch. 7.4 - Prob. 17ECh. 7.4 - Prob. 19ECh. 7.4 - Prob. 21ECh. 7.4 - Prob. 23ECh. 7.4 - Prob. 25ECh. 7.5 - Prob. 1ECh. 7.5 - Prob. 3ECh. 7.5 - Prob. 5ECh. 7.5 - Prob. 7ECh. 7.5 - Prob. 9ECh. 7.5 - Prob. 11ECh. 7.5 - Prob. 13ECh. 7.5 - Prob. 25ECh. 7.5 - Prob. 27ECh. 7.5 - Prob. 33ECh. 7.5 - Prob. 35ECh. 7.6 - In Problems 14, sketch the graph of the given...Ch. 7.6 - Prob. 3ECh. 7.6 - In Problems 510, express the given function using...Ch. 7.6 - Prob. 7ECh. 7.6 - Prob. 11ECh. 7.6 - Prob. 17ECh. 7.6 - Prob. 19ECh. 7.8 - Prob. 1ECh. 7.8 - Prob. 3ECh. 7.8 - Prob. 5ECh. 7.8 - Prob. 23ECh. 7.10 - In Problems 119, use the method of Laplace...Ch. 7.10 - Prob. 3ECh. 7.10 - Prob. 5ECh. 7.10 - Prob. 7ECh. 7.10 - Prob. 9ECh. 7.10 - Prob. 11ECh. 7.10 - Prob. 13ECh. 7.10 - Prob. 15ECh. 7.10 - Prob. 17ECh. 7.10 - Prob. 19E
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- 3. A typical adult human heart contains about 120 milliliters of blood when full (just before contraction) and pumps out about 70 milliliters when it contracts, then fills again to 120 milliliters before contracting again. If a person has a resting heart rate of about 70 beats per minute, find a sinusoidal function V (t) which models the volume of blood in the heart as a function of time (using t = 0 to represent the moment the heart is full of blood, just before contraction), using the following process: a. Explain why we can use a sinusoidal function to model this process; b. Determine whether a sine or cosine function is more appropriate to model this process, and why; c. Identify A, B, C, D and explain how we get each from the context of the problem; d. Write the equation of the function V (t). 4. Graph the function V (t) using the same process you used to graph f (0) in Question 1. Note: The differences in scale, between transformations, may mean that it makes more sense to graph…arrow_forwardFrom its past behavior, John knows that the value of a stock has a cyclical component that increases for the first three months of each year, falls for the next six, and rises again for the last three. In addition, inflation adds a linear component to the stock's price. John is seeking a model of the form f(t) = mt + b + A sin with t in months since Jan 1. He has the following data in the table: Jan 1 Apr 1 Jul 1 Price $20.00 $37.50 $35.00 $32.50 $50.00 Date Oct 1 Jan 1 a) What is the length of the period in his model? b) Determine two months when the Price is on the slanted midline. c) Find value of m, b, and A so that f fits the data.arrow_forwardIn Problems 1 through 6, express the solution of the given ini- tial value problem as a sum of two oscillations as in Eq. (8). Throughout, primes denote derivatives with respect to time t. In Problems 1–4, graph the solution function x(t) in such a way that you can identify and label (as in Fig. 3.6.2) its pe- riod. 3. x" + 100x = 225 cos 5t + 300 sin 5t; x(0) = 375, x'(0) = 0arrow_forward
- In Problems 1 through 6, express the solution of the given ini- tial value problem as a sum of two oscillations as in Eq. (8). Throughout, primes denote derivatives with respect to time t. In Problems 1–4, graph the solution function x(t) in such a way that you can identify and label (as in Fig. 3.6.2) its pe- riod. 4. x" + 25x = 90 cos 41; x (0) = 0, x'(0) = 90arrow_forward1. Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the given function. g(x) = e [² et²-t dtarrow_forwardProblem 4. Find the total derivative of the following function: df af of dy + dr Dy dr F(x,y) = (x²-3y)-(x+y³) if y = 2x² + 3xarrow_forward
- In Problems 1 through 6, express the solution of the given ini- tial value problem as a sum of two oscillations as in Eq. (8). Throughout, primes denote derivatives with respect to time t. In Problems 1–4, graph the solution function x(t) in such a way that you can identify and label (as in Fig. 3.6.2) its pe- riod. 1. x" + 9x = 10 cos 2t; x(0) = x'(0) = 0arrow_forward10. [Motion in R³] A drone has an initial position of (3,1,2) and an initial velocity of (-2,5,1). The drone's acceleration can be given by the function a(t) = (4 - 6t, 9 sin(3t), 10e-2t). (a) Derive a formula for the position of the drone as a function of t. (b) If the drone flies for 4 seconds, where does the drone end up?arrow_forwardDetermine whether the pairs of functions in Problems 20 through 26 are linearly independent or linearly dependent on the real line. 20. f(x) = 1, g(x) = cos² x + sin? x 21. f(x) = x³, g(x) = x²|x| 22. f(x) = 1+ x, g(x) = 1+ |x| 23. f(x) = xe*, g(x) = |x|e* 24. f(x) = sin? x, g(x) = 1 – cos 2.x 25. f(x) = e* sin x, g(x) = e* cos x 26. f(x) = 2 cos x + 3 sin x, g(x) %3D = 3 cos x – 2 sin xarrow_forward
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