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
24
Want to see the full answer?
Check out a sample textbook solutionChapter 12 Solutions
Calculus: Early Transcendentals (2nd Edition)
Additional Math Textbook Solutions
Precalculus Enhanced with Graphing Utilities (7th Edition)
University Calculus: Early Transcendentals (3rd Edition)
Calculus, Single Variable: Early Transcendentals (3rd Edition)
Precalculus (10th Edition)
Calculus & Its Applications (14th Edition)
- Derivatives of vector-valued functions Differentiate the following function. r(t) = ⟨(t + 1)-1, tan-1 t, ln (t + 1)⟩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_forwardMotion around a circle of radius a is described by the 2D vector-valued function r(t) = ⟨a cos(t), a sin(t)⟩. Find the derivative r′ (t) and the unit tangent vector T(t), and verify that the tangent vector to r(t) is always perpendicular to r(t).arrow_forward
- 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_forwardDerivatives of vector-valued functions Differentiate the following function. r(t) = ⟨4, 3 cos 2t, 2 sin 3t⟩arrow_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_forward
- f(x,y)=3 e^x cos y, (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_forwardDerivatives of vector-valued functions Differentiate the following function. r(t) = ⟨te-t, t ln t, t cos t⟩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
- 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_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_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_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