MATH 2630

Final | Fall '14

1. Suppose you start at the point $(2, 0, 0)$ and move $3$ units along the curve $x = 2 \cos t, y = 2 \sqrt{3} t, z = 2 \sin t$ in the positive direction. Where are you now? (10 points)

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2. Let $r(t) = < 1, t^2, t^3 >$.
(a) Find the unit tangent vector $T(t)$. (5 points)

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(b) Find the unit normal vector $N(t)$. (5 points)

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(c) Find the binormal vector $B(t)$. (5 points)

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3. Use the chain rule to find $\frac{\partial w}{\partial r}$ and $\frac{\partial w}{\partial \theta}$, given $w = xy + yz + zx, x = r\cos\theta, y = r\sin\theta$ and $z = r\theta$. (5 points)

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4. Find the maximum change rate of $f(x, y) = \sin(xy)$ at point $(1, 0)$ and the direction in which it occurs. (5 points)

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5. Find the absolute maximum and minimum values of $f(x, y) = x^2 + y^2 - 2x$ on set $D$, where $D$ is the closed triangular region with vertices $(2, 0), (0, 2)$ and $(0, -2)$. (10 points)

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6. Set up the double integral for the volume of the solid bounded by the coordinate planes and the plane $2x + y + 3z = 6$. You don't need to calculate the integral. (5 points)

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7. Set up $\int \int \int_E x^2 + y^2 \; dV$ using spherical coordinates, where $E$ lies between the sphere $x^2 + y^2 + z^2 = 9$ and $x^2 + y^2 + z^2 = 25$. You don't need to calculate the integral. (5 points)

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8. Find the volume of a sphere of radius $a$. (10 points)

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9. Evaluate the integral using the given transformation: $\int \int_R (x^2) dA$ where $R$ is the region bounded by ellipse $9x^2 + 4y^2 = 36$, and the transformation is $x = 2u, y = 3v$. (10 points)

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10. Use Green's theorem to evaluate $\int_C (xy^2)dx + (2x^y)dy$, where $C$ is the triangular region with vertices $(0, 0), (2, 2), (2, 4)$. (5 points)

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11. Is $F(x, y) = xy^2i + x^yj$ and $F(x, y, z) = yze^{xz}i + e^{xz}j + xye^{xz}k$ conservative? Explain your reasons. (5 points)

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12. Find an equation of the tangent plane to the parametric surface $x = u + v, y = 3u^2, z = u - v$ at the point $(2, 3, 0)$. (5 points)

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13. Find the area of the surface: the part of the plane $x + 2y + 3z = 1$ that lies inside the cylinder $x^2 + y^2 = 3$. (10 points)

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14. (Bonus) Describe Stokes' Theorem and Divergence Theorem.

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