Computing directional derivatives with the gradient Compute the directional derivative of the following functions at the given point P in the direction of the given
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- Computing directional derivatives with the gradient Compute the directional derivative of the following functions at the given point P in the direction of the given vector. Be sure to use a unit vector for the direction vector.arrow_forwardInterpreting directional derivatives Consider the functionƒ(x, y) = 3x2 - 2y2.a. Compute ∇ƒ(x, y) and ∇ƒ(2, 3).b. Let u = ⟨cos θ, sin θ⟩ be a unit vector. At (2, 3), for what values of θ (measured relative to the positive x-axis), with 0 ≤ θ < 2π, does the directional derivative have its maximum and minimum values? What are those values?arrow_forwardDerivatives of vector-valued functions Differentiate the following function. r(t) = ⟨(t + 1)-1, tan-1 t, ln (t + 1)⟩arrow_forward
- f(x,y)=e^xcosy, (a,b)=(0,π/4), and v⃗ =(2,3). Calculate the directional derivative of f at the point (a,b) in the direction defined by v⃗ . Find the direction at (a,b) in which the rate of change of f is greatest. Find the maximum rate of change. Fill in the blank: f decreases the most at (a,b) in the direction ofarrow_forwardSuppose f(x,y)=x/y, P=(−2,−3) and v=4i−3j A. Find the gradient of f. B. Find the gradient of f at the point P. C. Find the directional derivative of f at P in the direction of v. D. Find the maximum rate of change of f at P. E. Find the (unit) direction vector w in which the maximum rate of change occurs at P.arrow_forwardGradients in three dimensions Consider the following functions ƒ, points P, and unit vectors u.a. Compute the gradient of ƒ and evaluate it at P.b. Find the unit vector in the direction of maximum increase of ƒ at P.c. Find the rate of change of the function in the direction of maximumincrease at P.d. Find the directional derivative at P in the direction of the given vector.arrow_forward
- Derivatives of vector-valued functions Differentiate the following function. r(t) = ⟨4, 3 cos 2t, 2 sin 3t⟩arrow_forwardDerivative rules Let u(t) = 2t3 i + (t2 - 1) j - 8 k and v(t) = et i + 2e-t j - e2t k. Compute the derivative of the following function. u(t) ⋅ v(t)arrow_forwardCalculate 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_forward
- a) find the rate of change in f(x,y) when moving in the direction of the vector (1,3) b) find the directional derivative of f(x,y) when moving in the direction of maximum increase c) find the direction of maximum decrease in f(x,y)arrow_forward[Directional Derivatives] Calculate the following directional derivatives. w(x, y, z) = xy^2 / z^2 at the point (6, 1, 2) in the direction toward (5, 2, 5).arrow_forwardDerivatives of vector-valued functions Differentiate the following function. r(t) = ⟨te-t, t ln t, t cos t⟩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