Computing gradients Compute the gradient of the following functions, evaluate it at the given point P, and evaluate the directional derivative at that point in the given direction.
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Calculus: Early Transcendentals (3rd Edition)
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- Find the directional derivative of the function at the given point in the direction of the vector v.. f(x,v) = e3xV – y?, (0,- 1), v= (2,3) -5 а. V5 7 b. V5 -7 C. d. -7 e V3arrow_forwardUse the contour diagram of f to decide if the specified directional derivative is positive, negative, or approximately zero. 2.4 1.6 1. At the point (-1,1) in the direction of 0.8 of (-i +3)/v2, -0.8 v 2. At the point (0, –2) in the direction of -1.6 4.0 12.0 -2.4 (i – 2j)//5, -2.4 -1.6 -0.8 0.8 1.6 2.4 (Click graph to enlarge) ? 3. At the point (-2, 2) in the direction of i, 4. At the point (-1,1) in the direction of | (-i - 5)/v2, 5. At the point (0, 2) in the direction of j, 6. At the point (1,0) in the direction of - j, 12.0 10.0 12.0 10.0 10.0 8.0 10.0 12.0arrow_forwardFind the directional derivative of f at the given point in the direction indicated by the f(x, y) = 2ye-X, (0,4), 0 = 2π/3 Duf(0, 4) = 2 + √3 2 Need Help? Watch It Submit Answerarrow_forward
- Find the gradient of the function at the given point. Function Point f(x, у, 2) : x2 + y2 + z? (7, 8, 4) %3D Vf(7, 8, 4) = Find the maximum value of the directional derivative at the given point.arrow_forwardInterpreting directional derivatives A function ƒ and a point P are given. Let θ correspond to the direction of the directional derivative. a. Find the gradient and evaluate it at P. b. Find the angles θ (with respect to the positive x-axis) associated with the directions of maximum increase, maximum decrease, and zero change. c. Write the directional derivative at P as a function of θ; call this function g. d. Find the value of θ that maximizes g(θ) and find the maximum value. e. Verify that the value of θ that maximizes g corresponds to the direction of the gradient. Verify that the maximum value of g equals the magnitude of the gradient. ƒ(x, y) = e-x2 - 2y2; P(-1, 0)arrow_forwardInterpreting directional derivatives A function ƒ and a point P are given. Let θ correspond to the direction of the directional derivative. a. Find the gradient and evaluate it at P. b. Find the angles θ (with respect to the positive x-axis) associated with the directions of maximum increase, maximum decrease, and zero change. c. Write the directional derivative at P as a function of θ; call this function g. d. Find the value of θ that maximizes g(θ) and find the maximum value. e. Verify that the value of θ that maximizes g corresponds to the direction of the gradient. Verify that the maximum value of g equals the magnitude of the gradient.arrow_forward
- Find the derivative of the function at the given point in the direction of A. f(x,y,z)= 8x + 5y-10z, (-7, -2,-8), A=3i - 6j - 2k O A. O B. - - 34 7 19 7 O c. -2 O D. -7arrow_forwardFind the gradient of the functions below.arrow_forwardCalculate the gradient of the given function; evaluate the gradient of the function at the pointP, and calculate the directional derivative of the function in the direction of u.arrow_forward
- Use the contour diagram of f to decide if the specified directional derivative is positive, negative, or approximately zero. 2.4 Negative 1. At the point (-2,2) in the direction of i, 1.6 Positive 2. At the point (0, 2) in the direction of j, 0.8 Positive 3. At the point (0, –2) in the direction of (i – 2j)/V5, ? 4. At the point (-1, 1) in the direction of (-i + j)//2, 0.8 -1.6 ? v 5. At the point (1, 0) in the direction of –i, 4.0 -2.4 Zero 6. At the point (-1, 1) in the direction of (-i – )/V2, 2.4 -1.6 -0.8 0.8 1.6 2.4 (Click graph to enlarge) 12.0 10.0 12.0 10.0 O'g 10.0 12.0 10.0 12.0arrow_forwardUse the contour diagram of ƒ to decide if the specified directional derivative is positive, negative, or approximately zero. 1. At the point (0, 2) in the direction of 7, 2. At the point (−1, 1) in the direction of ? ? (−i +j)/√2, ? (i - 2j)/√5, ? (-i-j)/√2, ? ? 3. At the point (0, −2) in the direction of 4. At the point (-1, 1) in the direction of 5. At the point (-2, 2) in the direction of i, 6. At the point (1, 0) in the direction of -j, 2.4 1.6 0.8 0 -0.8 -1.6- -2.4 12.0 12.0 10.0 6.0 10.0 -2.4 -1.6 -0.8 0 X 0.8 4.0 1.6 12.0 10.0 8. 10.0 12.0 2.4 (Click graph to enlarge)arrow_forwardDescribe the two main geometric properties of the gradient V f.arrow_forward
- Calculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,