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Section 9.1 Understanding Functions of Two Variables Section 9.1: Understanding Functions of Two Variables Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Example: Wind Chill If the actual temperature is -10°C and the wind speed is 30 km/h, then what is the wind chill? How would you write this using notation? Is wind chill an increasing or decreasing function of actual temperature? Of wind speed? Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Example: Ticket Sales Find a formula for the revenue as a function of the number of discount and full price tickets. Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Example: Auto Loan The total amount, T, of dollars a buyer pays for an automobile is a function of the down payment, P, and the interest rate, r, of the loan on the balance. If the interest rate is held constant, is T an increasing or decreasing function of P? If the down payment is held constant, is T an increasing or decreasing function of r? Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Example: Visualizing f x, y x 2 y 1. 2. 3. 4. Increasing or decreasing? Table of values Visualizing Contour Plots Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. The number of songbirds, B, on an island depends on the number of hawks, H, and the number of insects, I. Thinking about the possible relationship between these three populations, decide which of the following depicts it. (I) B increases as H increases and I remains constant. (II) B decreases as H increases and I remains constant. (III) B increases as I increases and H remains constant. (IV) B decreases as I increases and H remains constant. (a) I and III (b) I and IV (c) II and III (d) II and IV (e) None of these ConcepTest • Section 9.1 • Question 4 ANSWER (c). Under the assumption that hawks eat songbirds, (II) is true. The answer (c) assumes that the songbirds eat the insects. Some songbirds eat only seeds, so in that case, if the insects eat the seeds, the correct answer is (d). COMMENT: You might ask for a reason as well. ConcepTest • Section 9.1 • Answer 4 For a certain function z = f(x, y), we know that f(0, 0) = 50 and that z goes up by 3 units for every unit increase in x and z goes down by 2 units for every unit increase in y. What is f(2, 5)? (a) 51 (b) 46 (c) 1 (d) 55 (e) −4 (f) 16 ConcepTest • Section 9.1 • Question 5 ANSWER (b). At the point (0, 0), the function value is 50. When the x-value goes up by 2, the function value goes up by 3·2 = 6. When the y-value goes up by 5, the function value goes down by 5·2 = 10. We have f(2, 5) = 50 + 6 − 10 = 46. ConcepTest • Section 9.1 • Answer 5 Which of the graphs (a)–(f) shows a crosssection of f(x, y) = 50 − x2 + 5y with y held fixed? ConcepTest • Section 9.1 • Question 6 ANSWER (b) and (f). When y is fixed, we have f(x, y) = 50 − x2 + 5·C = −x2 + (50 + 5·C) = −x2 + C where C is any arbitrary constant. The graph of y = −x2 + C is an upside-down parabola with its vertex on the y-axis, so the only cross-sections with y fixed are graphs (b) and (f). ConcepTest • Section 9.1 • Answer 6