Computing directional derivatives Compute the gradient of the following functions, evaluate it at the given point P, and evaluate the directional derivative at that point in the direction of the given
46. f(x, y, z) =
Want to see the full answer?
Check out a sample textbook solutionChapter 15 Solutions
Calculus: Early Transcendentals (3rd Edition)
Additional Math Textbook Solutions
Precalculus Enhanced with Graphing Utilities (7th Edition)
Precalculus (10th Edition)
Calculus, Single Variable: Early Transcendentals (3rd Edition)
Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (4th Edition)
Calculus & Its Applications (14th Edition)
- Calculate the directional derivative in the direction of v at the given point. Remember to normalize the direction vector. g(x, y, z) = z^2 − xy^2, v =〈−1, 2, 2〉, P = (2, 1, 3)arrow_forwardUsing Properties of the Derivative In Exercise 26, use the properties of the derivative to find the following. (a) r′(t) (b) d dt [u(t) − 2r(t)] (c) d dt [(3t)r(t)] (d) d dt [r(t) ∙ u(t)] (e) d dt [r(t) × u(t)] (f) d dt [u(2t)] 26. r(t) = sin ti + cos tj + tk, u(t) = sin ti + cos tj + 1 t k * only d ,e, f *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_forward
- I would need help with a, b, and c as mention below. (a) Find the gradient of f.(b) Evaluate the gradient at the point P.(c) Find the rate of change of f at P in the direction of the vector u.arrow_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_forwardPlease answer all parts completely. Suppose f (x, y) = x2exy (a) Find ∇f (x, y)(b) Find the directional derivative of f at the point (1, 0) in the direction of the vector <1.3>.(c) At the point (1, 0), in what direction will the directional derivative of f be greatest?(d) What will the directional derivative of f at (1, 0) be in the direction you found in part (c)? (e) What will the smallest possible directional derivative of f be at (1, 0)?arrow_forward
- Find the directional derivative of the function at the given point in the direction of the vector v. f(x,y) = x/(x^2+y^2), (1,2), v = <3,5>arrow_forwardEavluate the gradient vector at the given point. a) f(x,y) = cos2(πxy) at (1,-1) b) f(x,y,z) = xeyz at (2,1,0)arrow_forwardVECTOR DIFFERENTIATION: If R = e^(−t) i + ln(t^2+ 1) j - tant k. Find: (a) dR/dt, (b) d^2R/dt^2,(c) |dR/dt| ; (d) |d^2R/dt^2| at t = 0arrow_forward
- The three components of the derivative of the vector-valued function r are positive at t = t0. Describe the behavior of r at t = t0. (b) Consider the vector-valued function r(t) = t 2 i + (t − 3)j + tk. Write a vector-valued function s(t) that is the specified transformation of r. i. a vertical translation 3 units upward ii. a horizontal translation 2 units in the direction of the negative x-axis iii. a horizontal translation 5 units in the direction of the positive y-axisarrow_forwardConsider the function f (x, y) = 3 + 5x2 + 2y2, which is differentiable in an environment D from point P (6, 6). The value of the directional derivative of f at P in the direction of the vector u = (3, 3)corresponds to: Answers in the picturearrow_forwardUsing a Function (a) find the gradient of the function at P, (b) find a unit normal vector to the level curve f (x, y) = c at P, (c) find the tangent line to the level curve f (x, y) = c at P, and (d) sketch the level curve, the unit normal vector, and the tangent line in the xy-plane. f(x, y) = 9x2 + 4y2, c = 40, P(2, −1)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