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2. Approximate f,(3,5) using the contour diagram of S(x, y) in Figure 14.9.

2. Approximate f,(3,5) using the contour diagram of S(x, y) in Figure 14.9.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Question 22:** Approximate \( f_y(3,5) \) using the contour diagram of \( f(x,y) \) in Figure 14.9. *Explanation for Educational Website:* In this exercise, we are asked to find an approximation of the partial derivative \( f_y(3,5) \) from the contour diagram provided in Figure 14.9. Contour diagrams represent a function \( f(x, y) \) by plotting curves that connect points where the function has the same value. These curves help to visualize changes in function values over the plane. To approximate \( f_y(3,5) \), which represents the rate of change of \( f \) with respect to the variable \( y \) at the point (3,5), locate this point on the contour diagram. Observe how closely or widely spaced the contour lines are near this point. Closely spaced lines indicate a steep change, while widely spaced lines suggest a gradual change. Use the spacing and orientation of these lines to estimate \( f_y \), understanding that steeper gradients correspond to higher magnitudes of partial derivatives.
Transcribed Image Text:**Question 22:** Approximate \( f_y(3,5) \) using the contour diagram of \( f(x,y) \) in Figure 14.9. *Explanation for Educational Website:* In this exercise, we are asked to find an approximation of the partial derivative \( f_y(3,5) \) from the contour diagram provided in Figure 14.9. Contour diagrams represent a function \( f(x, y) \) by plotting curves that connect points where the function has the same value. These curves help to visualize changes in function values over the plane. To approximate \( f_y(3,5) \), which represents the rate of change of \( f \) with respect to the variable \( y \) at the point (3,5), locate this point on the contour diagram. Observe how closely or widely spaced the contour lines are near this point. Closely spaced lines indicate a steep change, while widely spaced lines suggest a gradual change. Use the spacing and orientation of these lines to estimate \( f_y \), understanding that steeper gradients correspond to higher magnitudes of partial derivatives.
The image depicts a contour plot on a Cartesian coordinate system, illustrating the function values over a defined region. The x-axis ranges from 0 to 10, and the y-axis also ranges from 0 to 10. The grid is marked with intervals of 2 units. The contour lines represent different constant values of the function, labeled from 2 to 16 in increments of 2. As the contour lines approach the top-left corner, the values increase, indicating rising function values in that direction. Each curve corresponds to a particular function value: - The innermost contour has a value of 2. - Successive contours represent values of 4, 6, 8, 10, 12, 14, and 16. The plot visually demonstrates the changes in function values across the region, with denser contours suggesting steeper gradients. Figure 14.9 illustrates these details effectively, showcasing how the function changes within the specified domain.
Transcribed Image Text:The image depicts a contour plot on a Cartesian coordinate system, illustrating the function values over a defined region. The x-axis ranges from 0 to 10, and the y-axis also ranges from 0 to 10. The grid is marked with intervals of 2 units. The contour lines represent different constant values of the function, labeled from 2 to 16 in increments of 2. As the contour lines approach the top-left corner, the values increase, indicating rising function values in that direction. Each curve corresponds to a particular function value: - The innermost contour has a value of 2. - Successive contours represent values of 4, 6, 8, 10, 12, 14, and 16. The plot visually demonstrates the changes in function values across the region, with denser contours suggesting steeper gradients. Figure 14.9 illustrates these details effectively, showcasing how the function changes within the specified domain.
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