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2. Consider the function f : R2 → R2 given by. (x,y) (a) Compute the Df(x, y) (b) List every vect...
Consider the surface given as a graph of the function g(x, y) = x∗y 2 ∗cos(y). The gradient of g ...
Consider the surface given as a graph of the function g(x, y) = x∗y 2 ∗cos(y). The gradient of g represents the direction in which g increases the fastest. Notice that this is the direction in the xy plane corresponding to the steepest slope up the surface, with magnitude equal to the slope in that direction. 1. At the point (2, π), find the gradient, and explain what it means. 2. Use it to construct a vector in the tangent...
. Consider the function f(x, y) = 3x 2 + 7x 2 y 3 . Compute...
. Consider the function f(x, y) = 3x 2 + 7x 2 y 3 . Compute the gradient, compute the Hessian, and write down the second order approximation to this function at the point (1, 1).
Question 8 (15 marks) Consider the function f: R2 R2 given by 1 (, y)(0,0) f(r,y) (a) Consider the surface z f(x, y). (...
Question 8 (15 marks) Consider the function f: R2 R2 given by 1 (, y)(0,0) f(r,y) (a) Consider the surface z f(x, y). (i Determine the level curves for the surface when z on the same diagram in the r-y plane. 1 and 2, Sketch the level curves (i) Determine the cross-sectional curves of the surface in the r-z plane and in the y- plane. Sketch the two cross-sectional curves (iii) Sketch the surface. (b) For the point (r, y)...
5. Consider the function f: R -> R given by f (x, y) := e°+v* _...
5. Consider the function f: R -> R given by f (x, y) := e°+v* _ 4. (a) Sketch the level curves of f. (5 marks) (b) Find Vf, the gradient of f, and determine at which points Vf is zero. Remark: These points are called the critical points of f (5 marks) (c) Determine whether the critical points of f are local minima, local maxima, or saddle points by considering the level curves of f. (5 marks) (d) Calculate...
Problem 2 (Eigenvalues and Eigenvectors). (a) If R2-R2 be defined by f(x,y) = (y,z), then find al...
please answer both a and b
Problem 2 (Eigenvalues and Eigenvectors). (a) If R2-R2 be defined by f(x,y) = (y,z), then find all the eigenvalues and eigenvectors of f Hint: Use the matrix representation. (b) Let U be a vector subspace (U o, V) of a finite dimensional vector space V. Show that there exists a linear transformation V V such that U is not an invariant subspace of f. Hence, or otherwise, show that: a vector subspace U-o or...
Problem 2 (Eigenvalues and Eigenvectors). (a) If R2 4 R2 be defined by f(x,y) (y,x), then find al...
Problem 2 (Eigenvalues and Eigenvectors). (a) If R2 4 R2 be defined by f(x,y) (y,x), then find all the eigenvalues and eigenvectors of f Hint: Use the matrix representation. (b) Let U be a vector subspace (U o, V) of a finite dimensional vector space V. Show that there exists a linear transformation V V such that U is not an invariant subspace of f Hence, or otherwise, show that: a vector subspace U-0 or U = V, if and...
Consider the following function 6 f(x, y,z)=z - x? cos(my) + xy? (i) Find the gradient...
Consider the following function 6 f(x, y,z)=z - x? cos(my) + xy? (i) Find the gradient of the function f(x, y, z) at the point P,(2,-1,-7). (ii) Find the directional derivative of f(x, y, z) at P,(2,-1,-7) along the direction of the vector ū = 2î+j+2k. (iii) Find the equation of the tangent plane to the surface given below at the point P,(2,-1, -7). 6 :- xcos(ty) + = 0 xy
Tangent and Normal Given the function f(x) = 3x2-1, in the x-y plane Define a unit...
Tangent and Normal Given the function f(x) = 3x2-1, in the x-y plane Define a unit vector tangent to the function at x = 4: Find a unit vector normal to the function at x 4 Find the radius of curvature (p) at x = 4 where the radius of curvature is defined as d-y dx
Let f(x,y) -2(xy 1) be a scalar function in R2. a) Find the vector field F(x, y) for which f(x, y...
Please describe the contour map and list important aspects of
it, thanks!
Let f(x,y) -2(xy 1) be a scalar function in R2. a) Find the vector field F(x, y) for which f(x, y) is a potential function, b) c) sketch a contour map of f (x, y) and, on the same figure, sketch F(x,y) (on R2). Comment on any important aspects of your sketch.
Let f(x,y) -2(xy 1) be a scalar function in R2. a) Find the vector field F(x,...
Question 8 (15 marks) Consider the function f: R2 R2 given by 1 (, y)(0,0) f(r,y) (a) Consider the surface z f(x, y). (i Determine the level curves for the surface when z on the same diagram in the r-y plane. 1 and 2, Sketch the level curves (i) Determine the cross-sectional curves of the surface in the r-z plane and in the y- plane. Sketch the two cross-sectional curves (iii) Sketch the surface. (b) For the point (r, y)...
5. Consider the function f: R -> R given by f (x, y) := e°+v* _ 4. (a) Sketch the level curves of f. (5 marks) (b) Find Vf, the gradient of f, and determine at which points Vf is zero. Remark: These points are called the critical points of f (5 marks) (c) Determine whether the critical points of f are local minima, local maxima, or saddle points by considering the level curves of f. (5 marks) (d) Calculate...
please answer both a and b
Problem 2 (Eigenvalues and Eigenvectors). (a) If R2-R2 be defined by f(x,y) = (y,z), then find all the eigenvalues and eigenvectors of f Hint: Use the matrix representation. (b) Let U be a vector subspace (U o, V) of a finite dimensional vector space V. Show that there exists a linear transformation V V such that U is not an invariant subspace of f. Hence, or otherwise, show that: a vector subspace U-o or...
Problem 2 (Eigenvalues and Eigenvectors). (a) If R2 4 R2 be defined by f(x,y) (y,x), then find all the eigenvalues and eigenvectors of f Hint: Use the matrix representation. (b) Let U be a vector subspace (U o, V) of a finite dimensional vector space V. Show that there exists a linear transformation V V such that U is not an invariant subspace of f Hence, or otherwise, show that: a vector subspace U-0 or U = V, if and...
Consider the following function 6 f(x, y,z)=z - x? cos(my) + xy? (i) Find the gradient of the function f(x, y, z) at the point P,(2,-1,-7). (ii) Find the directional derivative of f(x, y, z) at P,(2,-1,-7) along the direction of the vector ū = 2î+j+2k. (iii) Find the equation of the tangent plane to the surface given below at the point P,(2,-1, -7). 6 :- xcos(ty) + = 0 xy
Tangent and Normal Given the function f(x) = 3x2-1, in the x-y plane Define a unit vector tangent to the function at x = 4: Find a unit vector normal to the function at x 4 Find the radius of curvature (p) at x = 4 where the radius of curvature is defined as d-y dx
Please describe the contour map and list important aspects of
it, thanks!
Let f(x,y) -2(xy 1) be a scalar function in R2. a) Find the vector field F(x, y) for which f(x, y) is a potential function, b) c) sketch a contour map of f (x, y) and, on the same figure, sketch F(x,y) (on R2). Comment on any important aspects of your sketch.
Let f(x,y) -2(xy 1) be a scalar function in R2. a) Find the vector field F(x,...
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