Path of steepest descent Consider each of the following surfaces and the point P on the surface. a. Find the gradient of f. b. Let C’ be the path of steepest descent on the surface beginning at P and let C be the projection of C’ on the xy-plane. Find an equation of C in the xy-plane. 52 f ( x , y ) = y + x ( a plane ) ; P ( 2 , 2 , 4 )
Path of steepest descent Consider each of the following surfaces and the point P on the surface. a. Find the gradient of f. b. Let C’ be the path of steepest descent on the surface beginning at P and let C be the projection of C’ on the xy-plane. Find an equation of C in the xy-plane. 52 f ( x , y ) = y + x ( a plane ) ; P ( 2 , 2 , 4 )
Solution Summary: The author explains how the gradient of f(x,y)=y+x is computed as follows.
Path of steepest descentConsider each of the following surfaces and the point P on the surface.
a.Find the gradient of f.
b.Let C’ be the path of steepest descent on the surface beginning at P and let C be the projection of C’ on the xy-plane. Find an equation of C in the xy-plane.
52
f
(
x
,
y
)
=
y
+
x
(
a plane
)
;
P
(
2
,
2
,
4
)
Let f(x, y) = √3x + 2y. (a) Find the slope of the surface z = f(x,y)in the x direction at the point (4, 2). (b) Find the slope of the surface z = f(x,y)in the y direction at the point (4, 2).
(a) Two surfaces are called orthogonal at a point of intersection if their normal lines are perpendicular at that point. Show that surfaces with equations F(x, y, z) = 0 and G(x, y, z) = 0 are orthogonal at a point P where ∇F ≠ 0 and ∇G ≠ 0 if and only if FxGx + FyGy + FzGz = 0 atP(b) Use part (a) to show that the surfaces z2 = x2 + y2 and x2 + y2 + z2 = r2 are orthogonal at every point of intersection. Can you see why this is true without using calculus?
A bowl has inner surface given by the graph of the function z = f(x, y) = 2x^2 + 3y^2. A drop of oil is placed on this surface at the point (2, 1, 11) and moves along the surface under the influence of gravity toward the point (0, 0, 0), with position function (x(t), y(t), z(t)). The projection into the xy-plane of its position is the pair (x(t),y(t)). Assume that gravity causes the drop to move so that the projection moves in the direction of the negative of the gradient vector of f. Find the curve in the xy-plane above which the drop moves. Give your answer in the form y = some function of x.
this is all the information we were given
Chapter 15 Solutions
Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)
Calculus, Single Variable: Early Transcendentals (3rd Edition)
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