MATH 53

Midterm 2 | Fall '15 | Givental

1. Find a value of $k$ for which there exists a function $f(x, y)$ such that $f_x = kx + 6y, f_y = kx - 6y$, and find such a function.

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2. For each critical points of the function $x^2y - y + y^3/3$, determine whether it is a local maximum, a local minimum, or a saddle.

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3. Find possible values at the point $(x, y) = (1, 2)$ of a differentiable function $z(x, y)$ implicitly defined by the equation $x^2 + 2xy + 3yz + 2z^2 = 5$, and compute the gradient vector of each branch of this function at this point.

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4. Find the maximum and minimum values of the function $f(x, y) = x$ in the region $2x^2 + 6xy + 9y^2 \le 9$.

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5. Verify that for arbitrary twice differentiable functions $f$ and $g$ in one variable each, the function $u(x, t) = f(x - ct) + g(x + ct)$ satisfies the so called wave equation $u_{tt} = c^2u_{xx}$ (where $c$ is a given constant).

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6. Give an example of a two-times differentiable function $f(x, y)$ which has a critical point at the origin $(x, y) = (0, 0)$, and satisfies $f_{xx}(0, 0) = 4, f_{xy}(0, 0) = 6, f_{yy}(0, 0) = 9$, but does not have a local minimum at the origin.

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