MATH 53

Midterm 1 | Fall '14 | Agol

Find the area of the region enclosed by one loop of the curve $r^2 = \text{sin}(2\theta)$.

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Decide if the triangle with vertices $P(0, -3, -4), Q(1, -5, -1), R(5, -6, -3)$ is right-angled
(a) using angles between vectors.
(b) using distances and the Pythagorean theorem.

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3. Find an equation for the plane that passes through the point $(-2, 4, -3)$ and is perpendicular to the planes $-x + 3y - 5z = 42$ and $y - 2z = -5$.

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Let $\mathbf{r}(t) = <\text{sin} \; t, 2\; \text{cos} \; t>$.
(a) Sketch the plane curve with the given vector equation.

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(b) Find $\mathbf{r}'(t)$.

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(c) Sketch the position vector $\mathbf{r}(t)$ and the tangent vector $\mathbf{r}'(t)$ for the value $t = \pi / 4$.

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5(a) Find the limit, if it exists, or show that the limit does not exist. $\lim_{(x, y) \to (1, 0)} \frac{xy - y}{(x - 1)^2 + y^2}$

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(b) $\lim_{(x, y) \to (1, 0)} \frac{xy - y}{\sqrt{(x - 1)^2 + y^2}}$

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6. Use the Chain Rule to find $dw / dt$. Express your answer solely in terms of the variable $t$. $w = \text{ln} \; \sqrt{x^2 + y^2 + z^2}, x = \text{sin} \; t, y = \text{cos} \; t, z = \text{tan} \; t$

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7. Find the equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point. $x^2 + y^2 + z^2 + 3xyz, \quad (1, 1, 1)$

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8. Find the extreme values of $f$ on the region described by the inequality. $f(x, y) = 2x^2 + 3y^2 - 4x - 5, \quad x^2 + y^2 \le 16$

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9. If $\mathbf{r}(t)$ is a $3$-dimensional vector-valued function having all derivatives existing, and $\mathbf{u}(t) = \mathbf{r}(t) \cdot [\mathbf{r}(t) \times \mathbf{r}''(t)]$ Show that $\mathbf{u}'(t) = \mathbf{r}(t) \cdot [\mathbf{r}'(t) \times \mathbf{r}'''(t)]$

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