Gradients in three dimensions Consider the following functions f, points P, and unit
- a. Compute the gradient of f and evaluate it at P
- b. Find the unit vector in the direction of maximum increase of f at P.
- c. Find the rate of change of the function in the direction of maximum increase at P.
- d. Find the directional derivative at P in the direction of the given vector.
61.
Want to see the full answer?
Check out a sample textbook solutionChapter 15 Solutions
Calculus: Early Transcendentals (3rd Edition)
Additional Math Textbook Solutions
Precalculus (10th Edition)
Calculus & Its Applications (14th Edition)
Calculus, Single Variable: Early Transcendentals (3rd Edition)
Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (4th Edition)
Precalculus
- aThe average rate of change of a function f between x=a and x=b is the slope of the ___________ line between (a,f(a)) and (b,f(b)).arrow_forwardA. Find the directional derivative of the function at the given point in the direction of the vector v? f(x, y, z) = x2y+y2z P(1, 1, 1), v = (2, -1, 2) B. In which direction does the function f change the most rapidly at point P(1, 1, 1)? C. What is the rate of change of f at point P(1, 1, 1)?arrow_forwardFind the gradient vector of the given function at the given point p. Then find the equation of the tangent plane at p. F(x,y)=x³y+3xy², p=(2,-2)arrow_forward
- Given the problem: Find the gradient of f at P0 Find the unit vector u in the direction of A The derivative of f(x,y) at the point P0 in the direction of Aarrow_forwardThe temperature T in a metal ball is inversely proportional to the distance from the center of the ball, which we take as the origin. The temperature at the point (1, 2, 2) is 120ºC. (a) Determine the rate of change of T at (1, 2, 2) towards the point (2, 1, 3). (b) Show that at any point on the ball the direction of greatest growth in temperature is given by a vector that points to the origin.arrow_forward(1 point) If the gradient of ff is ∇f=2z(i) +x (j) +yx (k) and the point P=(−8,−8,−2) lies on the level surface f(x,y,z)=0 find an equation for the tangent plane to the surface at the point Parrow_forward
- Suppose f(x,y)=x/y, P=(1,−1) and v=−4i−4j. A. Find the gradient of f. ∇f= ____i+____jNote: Your answers should be expressions of x and y; e.g. "3x - 4y" B. Find the gradient of f at the point P. (∇f)(P)= ____i+____j Note: Your answers should be numbers C. Find the directional derivative of f at P in the direction of v. Duf=?Note: Your answer should be a number D. Find the maximum rate of change of f at P. maximum rate of change of f at P=?Note: Your answer should be a number E. Find the (unit) direction vector in which the maximum rate of change occurs at P. ?= ____i+____jNote: Your answers should be numbersarrow_forwardBoth parts of this problem refer to the function f(x, y, z) = x2 + y3 + 5z. (a) Find the directional derivative of f(x, y, z) at the point (1, 1, −2) in the direction of the vector <(1/√(3)), −(1/√(3)), (1/√(3))>(b) Find an equation of the tangent plane to the level surface of f for the function value −8 at the point (1, 1, −2).arrow_forwardConsider the following. f(x,y,z) = xe^5yz, P(1,0,1), u = [2/3,-2/3,1/3]. a)Find the gradient of f. b)Evaluate the gradient at point P c)Find the rate of change of f at P in the direction of the vector u.arrow_forward
- Suppose f(x,y)=x/y, P=(0,−1) and v=3i+3j. A. Find the gradient of f.∇f= ____i+____jNote: Your answers should be expressions of x and y; e.g. "3x - 4y" B. Find the gradient of f at the point P.(∇f)(P)= ____i+____j Note: Your answers should be numbers C. Find the directional derivative of f at P in the direction of v.Duf=?Note: Your answer should be a number D. Find the maximum rate of change of f at P.maximum rate of change of f at P=? Note: Your answer should be a number E. Find the (unit) direction vector in which the maximum rate of change occurs at P.u= ____i+____jNote: Your answers should be numbersarrow_forwardExplain why the function is differentiable at the given point. Then find the linearization L(x,y) of the function at that point. f(x,y)=y+sin(x/y),(0,3)arrow_forwardConsider the function g defined by g(x,y)=cos(πxy)+1/log3(x−y). Do as indicated. 1. Determine ∂2g/∂y∂x 2. Calculate the instantaneous rate of change of g at the point (4,1,2) in the direction of the vector ⟨1,2⟩. 3.In what direction does g attain its maximum directional derivative at the point (4,1)? What is the maximum directional derivative?arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageCollege AlgebraAlgebraISBN:9781305115545Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage LearningFunctions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning
- Algebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning