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 . 38 . f ( x , y ) = ln ( 1 + 2 x 2 + 3 y 2 ) ; P ( 3 4 − 3 )
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 . 38 . f ( x , y ) = ln ( 1 + 2 x 2 + 3 y 2 ) ; P ( 3 4 − 3 )
Solution Summary: The author explains how the gradient of f(x,y)=mathrmln left is computed as follows.
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.
38.
f
(
x
,
y
)
=
ln
(
1
+
2
x
2
+
3
y
2
)
;
P
(
3
4
−
3
)
The altitude of a plane is a function of the time since takeoff. Identify the
A. time
B. speed
C. altitude
D. acceleration
Oil imports Crude oil and petroleum products areimported continuously by the United States. Thetable shows the billions of dollars of net expendituresfor the imports of these products for several years.a. Write a linear function that models this data, withx equal to the number of years after 2015 and yequal to the billions of dollars of expenditures.Report the model with two decimal places.b. Is the model an exact fit for the data?
c. What does the model predict the billions of dollars of net expenditures will be for the imports in2023?d. When does the model predict the billions of dollars of net expenditures for the imports will be208.2 $billion?
Hampton is a small town on a straight stretch of coastline running north and south. A lighthouse
is located 3 miles offshore directly east of Hampton. The light house has a revolving searchlight
that makes two revolutions per minute. The angle that the beam makes with the east-west line
through Hampton is called o. Find the distance from Hampton to the point where the beam
strikes the shore, as a function of o. Include a sketch.
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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