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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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Chapter 12 Solutions
Student Solutions Manual, Single Variable for Calculus: Early Transcendentals
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- Describe the two main geometric properties of the gradient V f.arrow_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 gradient of the functions below.arrow_forward
- Calculate 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_forwardUse the gradient to find the directional derivative of the function at P in the direction of v. g(a, y) = a2 + y² + 1, P(1,2), v (2, 3) %3D V13 O 16 V13 16 V13arrow_forwardI 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_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_forwardFind 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_forwardUse the contour diagram of f to decide if the specified directional derivative is positive, negative, or approximately zero. ? ✓1. At the point (1, 0) in the direction of -3. ✓ 2. At the point (0, -2) in the direction of (2-23)/√5, ✓3. At the point (0, 2) in the direction of 3, 4. At the point (-1, 1) in the direction of (-7+3)/√2, ? ? ? ? ? V 5. At the point (-1, 1) in the direction of (-7-3)/√2, ✓6. At the point (-2, 2) In the direction of 7, V > 2.4 1.6 0.8 0 0.8 -1.6- -2.4 12.0 12,0 10.0 10.0 -2.4 6.0 -1.6 -0.8 0 X 0.8 4.0 (Click graph to enlarge) 1.6 12.0 10.0 8.0 10.0 12.0 2.4arrow_forward
- Interpreting 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_forwardUse 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_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,
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