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.
53
Want to see the full answer?
Check out a sample textbook solutionChapter 15 Solutions
Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)
Additional Math Textbook Solutions
Precalculus (10th Edition)
Precalculus
Calculus, Single Variable: Early Transcendentals (3rd Edition)
University Calculus: Early Transcendentals (3rd Edition)
Single Variable Calculus: Early Transcendentals (2nd Edition) - Standalone book
- (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 ∇F≠ 0 if and only if FxGx +FyGy+FzGz=0 at P (b) Use part (a) to show that the surfaces z2 = x2 +y2 and x2 +y2 + z2= 12are orthogonal at every point of intersection. Can you see why this is true without using calculus?arrow_forwardfind the absolute maximum and minimum of the functionf(x,y,z)=x+2yon the intersection curve between the plane x+y+z=1 and the cylinder y2 +z2=4.arrow_forwardThe plane y = 1 intersects the surface z = x^4 + 6xy − y^4 in a certain curve. Find the slope of the tangent line to this curve at the point P = (1, 1, 6).arrow_forward
- 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).arrow_forwardA-) Find the equation of tangent plane to the surface x^2 + 2y^2 + 3z^2 = 21 which is parallel to the plane 2x + 4y + 6z = 3. B-)Find the extremum and saddle points of the function f (x, y) = x^3 − 3xy + y^3 if any.arrow_forwardA 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 givenarrow_forward
- Find the slopes of the surface in the x- and y-directions at the given point.h(x, y) = x2 − y2(-2, 1, 3)arrow_forwardA bug crawls on the surface z = x2 - y2 directly above a path in the xy-plane given by x = ƒ(t) and y = g(t). If ƒ(2) = 4, ƒ′(2) = -1, g(2) = -2, and g′(2) = -3, then at what rate is the bug’s elevation z changing when t = 2?arrow_forwardFind an equation of the tangent plane to the surface at the given point. X+y+z=12, (3,5,4) And find a set of symmetric equation for the normal line to the surface at the given point ○x-3 = y-5 = z-4 ○x-3/3 = y-5/5 = z-4/4 ○3x = 5y = 4z ○x-3/3 = y-5/4 = z-4/5 ○x/3 = y/5 = z/4arrow_forward
- Find an equation of the plane tangent to the surface z = x2 + y2 at (2, 3, 13). [Hint: Consider the function f(x,y,z) = x2 + y2 - z.]arrow_forwardFind the point of intersection between the surface y=9 and the tangent line to the curve of intersect at the point (-3,0,3) of the following two surfaces given z > 0. x2+y2+z2=18 z2=x2+y2arrow_forwardThe base of the closed cubelike surface shown here is the unit square in the xy-plane. The four sides lie in the planes x = 0, x = 1, y = 0, and y = 1. The top is an arbitrary smooth surface whose identity is unknown. Let F = x i - 2y j + (z + 3)k, and suppose the outward flux of F through Side A is 1 and through Side B is -3. Can you conclude anything about the outward flux through the top? Give reasons for your answer.arrow_forward
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning