MATH 53

Midterm 1 | Fall '16 | Steel

1. Let $f(x, y, z) = ye^{-xz}$. Find the rate of change of $f$ at the point $P = (2, 1, 0)$, in the direction of the point $Q = (3, 2, 1)$.

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2. Let $f = f(x, y)$ have continuous partial derivatives. Let $h(u, v) = f(u^2 + v^3, u)$. Compute $\frac{\partial^2 h}{\partial u \partial v}$, in terms of $u, v$, and the partials of $f$.

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3. Let $f(x, y) = x^2y - x^2 - 2y^2$. Find the critical points of $f$, and classify them.

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Let $C$ be the curve of intersection of the surfaces $z = x^2 + y^2$ and $4x^2 + y^2 + z^2 = 9$. Find a nonzero vector that is tangent to $C$ at $(-1, 1, 2)$.

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True or False. (There is no penalty for guessing wrong)
a) For any vectors $\vec{a}, \vec{b}$, and $\vec{c}, \vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \times \vec{b}) \times \vec{c}$

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b) If $\vec{a} \cdot \vec{c} = \vec{b} \cdot \vec{c}$ for all vectors $\vec{c}$, then $\vec{a} = \vec{b}$.

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c) If a particle moves with constant speed, then its velocity and acceleration vectors are always orthogonal to each other.

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d) If all partials of $F$ exist and are continuous everywhere, then the equation $F(x, y, z) = F(a, b, c)$ defines a surface near $(a, b, c)$.

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e) If $f$ has continuous partials of all orders, and $\vec{r}(t)$ has continuous derivatives of all orders, then $\frac{d^2}{dt^2}(f(\vec{r}(t))) = \nabla f(\vec{r}(t)) \cdot \frac{d^2 \vec{r}}{dt^2}$.

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f) If $\vec{r}(t)$ describes the motion of a particle whose acceleration vector always points toward the origin, then $\frac{d}{dt}(\vec{r} \times \frac{d\vec{r}}{dt}) = \vec{0}$.

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