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
20
Want to see the full answer?
Check out a sample textbook solutionChapter 12 Solutions
Calculus: Early Transcendentals (2nd Edition)
Additional Math Textbook Solutions
University Calculus: Early Transcendentals (3rd Edition)
Calculus, Single Variable: Early Transcendentals (3rd Edition)
Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (4th Edition)
Glencoe Math Accelerated, Student Edition
- 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_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
- Derivatives of vector-valued functions Differentiate the following function. r(t) = ⟨(t + 1)-1, tan-1 t, ln (t + 1)⟩arrow_forwardDerivatives of vector-valued functions Differentiate the following function. r(t) = ⟨4, 3 cos 2t, 2 sin 3t⟩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) = ⟨te-t, t ln t, t cos t⟩arrow_forwardDerivatives of vector-valued functions Differentiate the following function. r(t) = ⟨cos t, t2, sin t⟩arrow_forwardf(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_forward
- Derivative rules Let u(t) = ⟨1, t, t2⟩ , v(t) = ⟨t2, -2t, 1⟩ , and g(t) = 2√t. Compute the derivatives of the following function. v(et)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_forwardf(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_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