Final Exam

1. Find the quadratic Taylor polynomials about $(0,0)$ for $e^{x} \cos y$ .
2. Find the critical points of $f(x,y)=\frac{1}{2} x^{2} + 3 y^{3} + 9 y^{2} - 3 x y + 9 y - 9 x$ and classify them as local maxima, local minima, saddle points, or none of these.

3. A boat is heading due south at $25$ km/hr (relative to the water). The current is moving toward the northwest at $10$ km/hr.
   1. How fast is the boat going, relative to the ground?
   2. By what angle does the current push the boat off its due south course?

4. Evaluate $${\int_{0}^{1} \int_{e^{y}}^{e} \frac{x}{\ln x} \, dx dy}$$ . It might be useful to reverse the order of integration.

5. Evaluate the integral of the function $f(\rho, \theta, \phi) = \sin \phi$ over the region $0 \le \theta \le 2 \pi$ , $0 \le \phi \le \pi/4$ , $1 \le \rho \le 2$ .

6. The path of a particle is described by
   $$x=(t-1)^{2} , \quad y = 2 , \quad z = 2 t^{3} - 3 t^{2}$$
   Find the velocity $\mathbf{v}(t)$ and the speed $\vert\vert\mathbf{v}(t)\vert\vert$ . Also find any times at which the particle stops.

7. Calculate the line integral $\int_{C} y \, dx + z \, dy + x \, dz$ , where $C$ consists of the line segments from $(0,0,0)$ to $(1,1,2)$ and from $(1,1,2)$ to $(3,1,4)$ .

8. Use Green's Theorem to calculate the circulation of $\mathbf{F}=(2x^{2}+3y) \, \hat{\mathbf{i}} + (2x + 3y^{2}) \, \hat{\mathbf{j}}$ around the triangle with vertices $(2,0)$ , $(0,3)$ , $(-2,0)$ , oriented counterclockwise.

9. Show that $\mathbf{F} = (4 x^{3} y^{2} - 2 x y^{3}) \, \hat{\mathbf{i}} + (2 x^{4} y - 3 x^{2} y^{2} + 4 y^{3}) \, \hat{\mathbf{j}}$ is conservative and use this fact to evaluate $\int_{C} \mathbf{F} \cdot d \mathbf{r}$ along the curve $C$ given by $\mathbf{r}(t) = (t + \sin \pi t) \, \hat{\mathbf{i}} + (2 t + \cos \pi t) \, \hat{\mathbf{j}}$ , $0 \le t \le 1$ .

10. Calculate the surface integral $\iint_{S} \mathbf{F} \cdot d \mathbf{S}$ with $\mathbf{F} = x^{2} z^{3} \, \hat{\mathbf{i}} + 2 x y z^{3} \, \hat{\mathbf{j}} + x z^{4} \, \hat{\mathbf{k}}$ and $S$ is the surface of the box with vertices $(\pm 1, \pm 2, \pm 3)$ . The Divergence Theorem might be useful.

11. Use Stokes' Theorem to evaluate $\int_{C} \mathbf{F} \cdot d \mathbf{r}$ , where $\mathbf{F} = (z-2y) \, \hat{\mathbf{i}} + (3x-4y) \, \hat{\mathbf{j}} + (z+3y) \, \hat{\mathbf{k}}$ , and $C$ is the circle $x^{2} + y^{2} = 4$ , $z=1$ , oriented counterclockwise as viewed from above.