Short answer questions. Does there exist a function f with continuous second-ord

Question

Short answer questions. Does there exist a function f with continuous second-order partial derivatives such that f_x(x, y) = x + y^2 and f_y(x, y) = x - y^2? Explain. The following four questions involve the diagram shown below. Determine whether f_x(3, 2) negative. Which is greater, f_y(2, 1) or f_y (2, 2)? Sketch in the vector nabla f(2, 1) Sketch in the direction of greatest decrease at (2, 1). Would you expect D_ f(2, 1) where u = (1/Squareroot 2, 1/Squareroot 2) to be positive or negative? Explain. Evaluate the following limit along the curves y = x and y = 2x. What do your evaluations tell you about the limit? lim_(x, y) rightarrow (0, 0) xy/Squareroot x^2 + y^2 Sketch the domain of the function f(x, y, z) = Squareroot 4 - x^2 - y^2 - z^2 Use a tree diagram to write out the Chain Rule for the function w = f(x, y, z) where x = x(m, v), y = Assume all functions are differentiable. Find the equation of the tangent plane and the normal line to the surface sin(xyz) = x + 2y + 3z at the point (2, -1, 0). Use Lagrange multipliers to find the maximum and minimum values of f(x, y) a with the constraint x^2 + y^2 = 1.

Explanation / Answer

b)

From a we have fx(x,y) = x + y2 , and fy (x,y) =x-y2

We need to check fx(3,2) is positive or negative ?

fx(3,2 ) = 3 + 22 = 3+4 =7 (which is positive )

fy (x,y) =x-y2

=> fy (2,1) =2-12 = 1 ( positive)

fy (2,2) =2-22 = 2- 4 = -2 ( nehative )

therefore fy (2,1) is greater than fy (2,2)

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