Interpreting directional derivatives A function f and a point P are given. Let θ correspond to the direction of the directional derivative. a. Find the gradient and evaluate it at P. b. Find the angles θ ( with respect to the positive x-axis ) associated with the directions of maximum increase, maximum decrease, and zero change. c. Write the directional derivative at P as a function of θ; call this function g. d. Find the value of θ that maximizes g ( θ ) and find the maximum value. e. Verify that the value of θ that maximizes g corresponds to the direction of the gradient. Verify that the maximum value of g equals the magnitude of the gradient . 33 . f ( x , y ) = 8 + x 2 + 3 y 2 ; P ( − 3 , − 1 )
Interpreting directional derivatives A function f and a point P are given. Let θ correspond to the direction of the directional derivative. a. Find the gradient and evaluate it at P. b. Find the angles θ ( with respect to the positive x-axis ) associated with the directions of maximum increase, maximum decrease, and zero change. c. Write the directional derivative at P as a function of θ; call this function g. d. Find the value of θ that maximizes g ( θ ) and find the maximum value. e. Verify that the value of θ that maximizes g corresponds to the direction of the gradient. Verify that the maximum value of g equals the magnitude of the gradient . 33 . f ( x , y ) = 8 + x 2 + 3 y 2 ; P ( − 3 , − 1 )
Interpreting directional derivativesA function f and a point P are given. Let θ correspond to the direction of the directional derivative.
a. Find the gradient and evaluate it at P.
b. Find the angles θ (with respect to the positive x-axis) associated with the directions of maximum increase, maximum decrease, and zero change.
c. Write the directional derivative at P as a function of θ; call this function g.
d. Find the value of θ that maximizes g(θ) and find the maximum value.
e. Verify that the value of θ that maximizes g corresponds to the direction of the gradient. Verify that the maximum value of g equals the magnitude of the gradient.
33.
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An airport can be cleared of fog by heating the air. The amount of heat required depends on the air temperature and the wetness of the fog. The figure below shows the heat H(T,w) required (in calories per cubic meter of fog) as a function of the temperature T (in degrees Celsius) and the water content w (in grams per cubic meter of fog). Note that this figure is not a contour diagram, but shows cross-sections of H with w fixed at 0.1, 0.2, 0.3, and 0.4.
(a) Estimate HT (10,0.2): HT (10,0.2) ≈ (Be sure you can interpret this partial derivative in practical terms.)
(b) Make a table of values for H(T,w) from the figure, and use it to estimate HT (T,w) for each of the following: T = 20,w = 0.2 : HT (T,w) ≈ T = 30,w = 0.2 : HT (T,w) ≈ T = 20,w = 0.3 : HT (T,w) ≈ T = 30,w = 0.3 : HT (T,w) ≈
(c) Repeat (b) to find Hw(T,w) for each of the following: T = 20,w = 0.2 : Hw(T,w) ≈ T = 30,w = 0.2 : Hw(T,w) ≈ T = 20,w = 0.3 : Hw(T,w) ≈ T = 30,w = 0.3 : Hw(T,w) ≈ (Be sure you can interpret this…
To determine the weather, a meteorologist uses the function T = x^2y^3z, where T represents the temperature in degrees Celsius in terms of the position in meters. A probe is sent in a straight line, determined by the direction (−2, −1, 2) passing through the point (1, −2,1). What is the rate of the temperature change as the probe passes through the point?
A) −9.333 °C/mB) 16.000 °C/mC) −32.000 °C/mD) 5.333 °C/m
(e) The instructor adds 4 points to the average exam score ofeveryone in the class. Describe the changes in the positionsof the plotted points and the change in the equation of theline.
Chapter 15 Solutions
Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)
Thomas' Calculus: Early Transcendentals (14th Edition)
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Differential Equation | MIT 18.01SC Single Variable Calculus, Fall 2010; Author: MIT OpenCourseWare;https://www.youtube.com/watch?v=HaOHUfymsuk;License: Standard YouTube License, CC-BY