MATH 53

Midterm 1 | Spring '15 | Hutchings

1. (10 points) Find the point on the curve $\mathbf{r}(t) = < 1, t, t^2 >$ where the tangent line is parallel to the plane $4x + 5y + 6z = 0$.

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2. (Short questions, 5 points each)
(a) Write the equation of a quadric surface containing the parametrized curve $(x(t), y(t), z(t)) = (t^3, t^4, t^5)$ (Your answer should be a quadratic equation in $x, y$, and $z$.) What kind of quadric surface is it? (e.g. ellipsoid, hyperboloid of one sheet, etc.)

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(b) Suppose that $f$ is a function of $x$ and $y$ such that $f_x = x + 2y$ and $f_y = ax + 3y$ where $a$ is constant. What does $a$ have to be, and why?

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3. (10 points) Let $P$ be the tangent plane to the surface $x^2 + y^2 + xyz = 11$ at the point $(1, 2, 3)$. The plane $P$ intersects that $x$-axis at a point $(a, 0, 0)$. What is $a$?

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(10 points) Does the limit exist, and if so what is the limit? $\lim_{(x, y) \to (0, 0)} \frac{x^3 + y^3}{x^2 + y^2} = ?$ Justify your answer.

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5. (10 points) Find the area of the region that lies inside both of the polar curves $r = 1 + \cos \theta$ and $r = 1 - \cos \theta$. (To evaluate the integral, the identity $\cos^2 \theta = (1 + \cos 2\theta)/2$ may be helpful.)

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6. (10 points) Let $f(x, y) = e^{x^2 + 2y^2}$ Find the unit vector $\mathbf{u} = < a, b >$ which minimizes the directional derivative $D_u f$ at the point $(x, y) = (2, 3)$.

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