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
)
Part (e) of Exercise 14 requires the use of a
graphing calculator or computer.
14. An open-top box is to be made so that its width
is 4 ft and its volume is 40 ft°. The base of the
box costs $4/ft and the sides cost $2/ft.
a. Express the cost of the box as a function of its
length I and height h.
b. Find a relationship between I and h.
c. Express the cost as a function of h only.
d. Give the domain of the cost function.
e. Use a graphing calculator or computer to ap-
proximate the dimensions of the box having
least cost.
Find the linear function whose contour map (with contour interval m = 6) is as shown. What is the linear function if m = 3 (and the curve labeled c = 6 is relabeled c = 3)?
Consider +
4 4
=1
9
Sketch and describe the graph
b. Find the equation of the tangent at the point (2,2,3)
C. Find the linear approximation to the function z = z(x,y) at this point
a.
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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