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Prove this therom by using an equation as an example.
15 THEOREM Suppose f is a...
please circle the answer! (1 point) Suppose f (x, y) = , P = (1, 3)...
please circle the answer!
(1 point) Suppose f (x, y) = , P = (1, 3) and v 3i - 2j A. Find the gradient of f Vf = i+ j Note: Your answers should be expressions of x and y; e.g. "3x - 4y" B. Find the gradient of f at the point P. (Vf) (P) it j Note: Your answers should be numbers C. Find the directional derivative of f at P in the direction of v. Duf...
Let U be an open subset of R". Let f:UCR"-R be differentiable at a E U. In this exercise you will prove that if ▽f(a) 0, then at the point a, the function f increases fastest in the direction...
Let U be an open subset of R". Let f:UCR"-R be differentiable at a E U. In this exercise you will prove that if ▽f(a) 0, then at the point a, the function f increases fastest in the direction of V f(a), and the maximum rate of increase is Vf(a)l (a) Prove that for each unit vector u e R" (b) Prove that if ▽/(a)メ0, and u = ▽f(a)/IV/(a) 11, then
Let U be an open subset of R". Let...
(1 point) Suppose f (r,y)= P = (1, -2) and v=3i - 3j. A. Find the...
(1 point) Suppose f (r,y)= P = (1, -2) and v=3i - 3j. A. Find the gradient of 1 Uf = 1 it -x/y^2 Note: Your answers should be expressions of x and y, eg "3x - 4y" j j B. Find the gradient off at the point P. (VA)(P) = 1 i+ -2 Note: Your answers should be numbers C. Find the directional derivative off at P in the direction of v. Duf = 9 Note: Your answer should...
1 1 Consider the function f(x.y,z) 2x y 2 the point P(3,0,1), and the unit vector...
1 1 Consider the function f(x.y,z) 2x y 2 the point P(3,0,1), and the unit vector u 0 Compute the gradient of f and evaluate it at P b. Find the unit vector in the direction of maximum increase of f at P c. Find the rate of change of the function in the direction of maximum increase at P d. Find the directional derivative at P in the direction of the given vector. a.
1 1 Consider the function...
true or false is zero. F 9. The plane tangent to the surface za the point (0,0, 3) is given by the equation 2x - 12y -z+3-0. 10. If f is a differentiable function and zf(x -y), then z +. T 11. If...
true or false
is zero. F 9. The plane tangent to the surface za the point (0,0, 3) is given by the equation 2x - 12y -z+3-0. 10. If f is a differentiable function and zf(x -y), then z +. T 11. If a unit vector u makes the angle of π/4 with the gradient ▽f(P), the directional derivative Duf(P) is equal to |Vf(P)I/2. F 12. There is a point on the hyperboloid 2 -y is parallel to the plane...
Find an equation of the plane tangent to the following surface at the given point. yz...
Find an equation of the plane tangent to the following surface at the given point. yz e XZ - 21 = 0; (0,7,3) An equation of the tangent plane at (0,7,3) is = 0. Find the critical points of the following function. Use the Second Derivative Test to determine if possible whether each critical point corresponds to a local maximum local minimum, or saddle point. If the Second Derivative Test is inconclusive, determine the behavior of the function at the...
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) = L. Prove that f is differentiable at 1-0 with f,(0) = L. (Hint:...
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) = L. Prove that f is differentiable at 1-0 with f,(0) = L. (Hint: Use the definition of derivative and then use mean value theorem)
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) =...
Question 1. (15 pts) Given f(x, y) = 3x 2 + y 3 . (a) Find...
Question 1. (15 pts) Given f(x,
y) = 3x 2 + y 3 . (a) Find the gradient of f. (b) Find the
directional derivative of f at P0 = (3, 2) in the direction of u =
(5/13)i + (12/13)j.
Question 1. (15 pts) Given f(L,y) = 3x2 +y?. (a) Find the gradient of f. (b) Find the directional derivative off at P =(3,2) in the direction of u=(5/13)i + (12/13)j.
7. (20 pts) Consider the surface given by z wy - 4xy + 3x - 2y....
7. (20 pts) Consider the surface given by z wy - 4xy + 3x - 2y. (a) Find the equation of the tangent plane to this surface at the point where (x,y) - (1,3). (b) Find the gradient f at the point where (x,y) = (1,3). (c) Find the directional derivative DaS(1,3) where it is the unit vector in the direction of (1, -2). such that the directional derivative Daf(1.3) is a maximum (d) Find a unit vector in the...
6. (a) Find the directional derivative of f(, y,z)at Po(1,10 in the direction of u-i-2j+k. 4 marks] (b) Letz201), where f is a differentiable function, show that 08z 4 marks] (c) Use Lagrange Multipl...
6. (a) Find the directional derivative of f(, y,z)at Po(1,10 in the direction of u-i-2j+k. 4 marks] (b) Letz201), where f is a differentiable function, show that 08z 4 marks] (c) Use Lagrange Multiplier to find the absolute maximum and minimum of 10 (x,y)-x +y subject to 2 12 marks
6. (a) Find the directional derivative of f(, y,z)at Po(1,10 in the direction of u-i-2j+k. 4 marks] (b) Letz201), where f is a differentiable function, show that 08z 4 marks]...
please circle the answer!
(1 point) Suppose f (x, y) = , P = (1, 3) and v 3i - 2j A. Find the gradient of f Vf = i+ j Note: Your answers should be expressions of x and y; e.g. "3x - 4y" B. Find the gradient of f at the point P. (Vf) (P) it j Note: Your answers should be numbers C. Find the directional derivative of f at P in the direction of v. Duf...
Let U be an open subset of R". Let f:UCR"-R be differentiable at a E U. In this exercise you will prove that if ▽f(a) 0, then at the point a, the function f increases fastest in the direction of V f(a), and the maximum rate of increase is Vf(a)l (a) Prove that for each unit vector u e R" (b) Prove that if ▽/(a)メ0, and u = ▽f(a)/IV/(a) 11, then
Let U be an open subset of R". Let...
(1 point) Suppose f (r,y)= P = (1, -2) and v=3i - 3j. A. Find the gradient of 1 Uf = 1 it -x/y^2 Note: Your answers should be expressions of x and y, eg "3x - 4y" j j B. Find the gradient off at the point P. (VA)(P) = 1 i+ -2 Note: Your answers should be numbers C. Find the directional derivative off at P in the direction of v. Duf = 9 Note: Your answer should...
1 1 Consider the function f(x.y,z) 2x y 2 the point P(3,0,1), and the unit vector u 0 Compute the gradient of f and evaluate it at P b. Find the unit vector in the direction of maximum increase of f at P c. Find the rate of change of the function in the direction of maximum increase at P d. Find the directional derivative at P in the direction of the given vector. a.
1 1 Consider the function...
true or false
is zero. F 9. The plane tangent to the surface za the point (0,0, 3) is given by the equation 2x - 12y -z+3-0. 10. If f is a differentiable function and zf(x -y), then z +. T 11. If a unit vector u makes the angle of π/4 with the gradient ▽f(P), the directional derivative Duf(P) is equal to |Vf(P)I/2. F 12. There is a point on the hyperboloid 2 -y is parallel to the plane...
Find an equation of the plane tangent to the following surface at the given point. yz e XZ - 21 = 0; (0,7,3) An equation of the tangent plane at (0,7,3) is = 0. Find the critical points of the following function. Use the Second Derivative Test to determine if possible whether each critical point corresponds to a local maximum local minimum, or saddle point. If the Second Derivative Test is inconclusive, determine the behavior of the function at the...
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) = L. Prove that f is differentiable at 1-0 with f,(0) = L. (Hint: Use the definition of derivative and then use mean value theorem)
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) =...
Question 1. (15 pts) Given f(x,
y) = 3x 2 + y 3 . (a) Find the gradient of f. (b) Find the
directional derivative of f at P0 = (3, 2) in the direction of u =
(5/13)i + (12/13)j.
Question 1. (15 pts) Given f(L,y) = 3x2 +y?. (a) Find the gradient of f. (b) Find the directional derivative off at P =(3,2) in the direction of u=(5/13)i + (12/13)j.
7. (20 pts) Consider the surface given by z wy - 4xy + 3x - 2y. (a) Find the equation of the tangent plane to this surface at the point where (x,y) - (1,3). (b) Find the gradient f at the point where (x,y) = (1,3). (c) Find the directional derivative DaS(1,3) where it is the unit vector in the direction of (1, -2). such that the directional derivative Daf(1.3) is a maximum (d) Find a unit vector in the...
6. (a) Find the directional derivative of f(, y,z)at Po(1,10 in the direction of u-i-2j+k. 4 marks] (b) Letz201), where f is a differentiable function, show that 08z 4 marks] (c) Use Lagrange Multiplier to find the absolute maximum and minimum of 10 (x,y)-x +y subject to 2 12 marks
6. (a) Find the directional derivative of f(, y,z)at Po(1,10 in the direction of u-i-2j+k. 4 marks] (b) Letz201), where f is a differentiable function, show that 08z 4 marks]...
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