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Preview Activity 10.6.1 Let's consider the function f defined by 1 ƒ(x, y) = 30 — x² – ¹⁄y², and suppose that ƒ measures the temperature, in degrees Celsius, at a given point in the plane, where x and y are measured in feet. Assume that the positive x-axis points due east, while the positive y-axis points due north. A contour plot of f is shown in Figure 10.6.1 6 5- 4 3- 2. 1 Y 1 2 3 4 5 6 Figure 10.6.1 A contour plot of f(x, y) = 30 – x² – ¹⁄y². a. Suppose that a person is walking due east, and thus parallel to the x-axis. At what instantaneous rate is the temperature changing with respect to x at the moment the walker passes the point (2, 1)? What are the units on this rate of change? b. Next, determine the instantaneous rate of change of temperature with respect to distance at the point (2, 1) if the person is instead walking due north. Again, include units on your result.

Preview Activity 10.6.1 Let's consider the function f defined by 1 ƒ(x, y) = 30 — x² – ¹⁄y², and suppose that ƒ measures the temperature, in degrees Celsius, at a given point in the plane, where x and y are measured in feet. Assume that the positive x-axis points due east, while the positive y-axis points due north. A contour plot of f is shown in Figure 10.6.1 6 5- 4 3- 2. 1 Y 1 2 3 4 5 6 Figure 10.6.1 A contour plot of f(x, y) = 30 – x² – ¹⁄y². a. Suppose that a person is walking due east, and thus parallel to the x-axis. At what instantaneous rate is the temperature changing with respect to x at the moment the walker passes the point (2, 1)? What are the units on this rate of change? b. Next, determine the instantaneous rate of change of temperature with respect to distance at the point (2, 1) if the person is instead walking due north. Again, include units on your result.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 94E
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A is attached for reference. PLEASE HELP WITH B

10.6 Directional Derivatives and the Gradient Preview Activity 10.6.1 Let's consider the function f defined by 1 f(x, y) 30 - x² and suppose that ƒ measures the temperature, in degrees Celsius, at a given point in the plane, where x and y are measured in feet. Assume that the positive x-axis points due east, while the positive y-axis points due north. A contour plot of f is shown in Figure 10.6.1 6 5 4 3 2 1 - y X 1 2 3 4 5 Figure 10.6.1 A contour plot of f(x, y) = 30 — x² – ¹⁄y². 6 a. Suppose that a person is walking due east, and thus parallel to the x-axis. At what instantaneous rate is the temperature changing with respect to x at the moment the walker passes the point (2, 1)? What are the units on this rate of change? b. Next, determine the instantaneous rate of change of temperature with respect to distance at the point (2, 1) if the person is instead walking due north. Again, include units on your result.
Transcribed Image Text:10.6 Directional Derivatives and the Gradient Preview Activity 10.6.1 Let's consider the function f defined by 1 f(x, y) 30 - x² and suppose that ƒ measures the temperature, in degrees Celsius, at a given point in the plane, where x and y are measured in feet. Assume that the positive x-axis points due east, while the positive y-axis points due north. A contour plot of f is shown in Figure 10.6.1 6 5 4 3 2 1 - y X 1 2 3 4 5 Figure 10.6.1 A contour plot of f(x, y) = 30 — x² – ¹⁄y². 6 a. Suppose that a person is walking due east, and thus parallel to the x-axis. At what instantaneous rate is the temperature changing with respect to x at the moment the walker passes the point (2, 1)? What are the units on this rate of change? b. Next, determine the instantaneous rate of change of temperature with respect to distance at the point (2, 1) if the person is instead walking due north. Again, include units on your result.
CHAPTER 10. DERIVATIVES OF MULTIVARIABLE FUNCTIONS 81 c. Now, rather than walking due east or due north, let's suppose that the person is walking with velocity given by the vector v = (3, 4), where time is measured in seconds. Note that the person's speed is thus |v| = 5 feet per second. Find parametric equations for the person's path; that is, parametrize the line through (2, 1) using the direction vector v = (3,4). Let x(t) denote the x-coordinate of the line, and y(t) its y-coordinate. Make sure your parametrization places the walker at the point (2, 1) when t = 0. d. With the parametrization in (c), we can now view the temperature f as not only a function of x and y, but also of time, t. Hence, use the chain rule to determine the value of to. What are the units on your answer? What is the practical meaning of this result? dt t=0
Transcribed Image Text:CHAPTER 10. DERIVATIVES OF MULTIVARIABLE FUNCTIONS 81 c. Now, rather than walking due east or due north, let's suppose that the person is walking with velocity given by the vector v = (3, 4), where time is measured in seconds. Note that the person's speed is thus |v| = 5 feet per second. Find parametric equations for the person's path; that is, parametrize the line through (2, 1) using the direction vector v = (3,4). Let x(t) denote the x-coordinate of the line, and y(t) its y-coordinate. Make sure your parametrization places the walker at the point (2, 1) when t = 0. d. With the parametrization in (c), we can now view the temperature f as not only a function of x and y, but also of time, t. Hence, use the chain rule to determine the value of to. What are the units on your answer? What is the practical meaning of this result? dt t=0
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The given function is fx,y=30-x2-12y2 which gives the temperature in degree Celsius at a point x,y, where x feet is due east and y feet is due north.

A man is at the point 2,1 and the person is walking in the north direction.

 

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