Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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**(part 2 of 4)** Determine \( f_{xx} + f_{yy} \) when \[ f(x, y) = (x - 5)(y + 3)(x + y + 1) \] 1. \( f_{xx} + f_{yy} = 2(x + y - 2) \) 2. \( f_{xx} + f_{yy} = x + y - 2 \) 3. \( f_{xx} + f_{yy} = x + y + 8 \) 4. \( f_{xx} + f_{yy} = 2(x + y + 8) \) 5. \( f_{xx} + f_{yy} = 2(x + y - 8) \) --- **(part 3 of 4)** Find \( f_{xy} \) when \[ f(x, y) = (x + 2y) e^{xy} \] 1. \( f_{xy} = (2 - xy)(x + 2y) e^{xy} \) 2. \( f_{xy} = (2 + xy)(x + 2y) e^{xy} \) 3. \( f_{xy} = (1 + xy) e^{xy} \) 4. \( f_{xy} = (1 + xy)(x + 2y) e^{xy} \) 5. \( f_{xy} = (1 - xy) e^{xy} \) 6. \( f_{xy} = (2 + xy) e^{xy} \)
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Transcribed Image Text:**(part 2 of 4)** Determine \( f_{xx} + f_{yy} \) when \[ f(x, y) = (x - 5)(y + 3)(x + y + 1) \] 1. \( f_{xx} + f_{yy} = 2(x + y - 2) \) 2. \( f_{xx} + f_{yy} = x + y - 2 \) 3. \( f_{xx} + f_{yy} = x + y + 8 \) 4. \( f_{xx} + f_{yy} = 2(x + y + 8) \) 5. \( f_{xx} + f_{yy} = 2(x + y - 8) \) --- **(part 3 of 4)** Find \( f_{xy} \) when \[ f(x, y) = (x + 2y) e^{xy} \] 1. \( f_{xy} = (2 - xy)(x + 2y) e^{xy} \) 2. \( f_{xy} = (2 + xy)(x + 2y) e^{xy} \) 3. \( f_{xy} = (1 + xy) e^{xy} \) 4. \( f_{xy} = (1 + xy)(x + 2y) e^{xy} \) 5. \( f_{xy} = (1 - xy) e^{xy} \) 6. \( f_{xy} = (2 + xy) e^{xy} \)
### Task: From the contour map of \( f \) shown below, decide whether \( f_x, f_y \) are positive, negative, or zero at \( P \). ### Diagram Explanation: The contour map features lines of constant values of a function \( f \), shown as blue curves. The x-axis and y-axis are marked in red. The values increase from left to right and from bottom to top. Point \( P \) is marked on the map, and the contour lines appear closer together as they move away from \( P \) towards the top right, indicating an increase in function value. ### Potential Answers: 1. \( f_x > 0, \, f_y < 0 \) 2. \( f_x < 0, \, f_y < 0 \) 3. \( f_x < 0, \, f_y = 0 \) 4. \( f_x < 0, \, f_y > 0 \) 5. \( f_x > 0, \, f_y > 0 \) 6. \( f_x > 0, \, f_y = 0 \) ### Analysis: - **\( f_x \):** Positive if moving right at \( P \) increases \( f \), negative if it decreases \( f \). - **\( f_y \):** Positive if moving up at \( P \) increases \( f \), negative if it decreases \( f \). Evaluate the directions of increasing and decreasing values to determine the correct relationship at point \( P \).
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Transcribed Image Text:### Task: From the contour map of \( f \) shown below, decide whether \( f_x, f_y \) are positive, negative, or zero at \( P \). ### Diagram Explanation: The contour map features lines of constant values of a function \( f \), shown as blue curves. The x-axis and y-axis are marked in red. The values increase from left to right and from bottom to top. Point \( P \) is marked on the map, and the contour lines appear closer together as they move away from \( P \) towards the top right, indicating an increase in function value. ### Potential Answers: 1. \( f_x > 0, \, f_y < 0 \) 2. \( f_x < 0, \, f_y < 0 \) 3. \( f_x < 0, \, f_y = 0 \) 4. \( f_x < 0, \, f_y > 0 \) 5. \( f_x > 0, \, f_y > 0 \) 6. \( f_x > 0, \, f_y = 0 \) ### Analysis: - **\( f_x \):** Positive if moving right at \( P \) increases \( f \), negative if it decreases \( f \). - **\( f_y \):** Positive if moving up at \( P \) increases \( f \), negative if it decreases \( f \). Evaluate the directions of increasing and decreasing values to determine the correct relationship at point \( P \).
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