Vector calculus: final

Grading

These are exercises for Vector calculus: course.

1. An artillery gun with a muzzle velocity of 1000 ft/s is located atop a seaside cliff 500 ft high. At what initial inclination angle should it fire a projectile in order to hit a ship at sea 20,000 ft from the foot of the cliff? Assume $g=32ft/s^{2}.$
2. Find the center of curvature of the parabola $y=x^{2}$ at the point $(1,1).$
3. Find the arc-length of the curve $x=2e^{t},y=e^{-t},z=2t,$ from $t=0$ to $t=1.$
4. Find the volume of the region bounded by the surfaces $z=y,y=4,z=0,y=x^{2}.$
5. Maximize the area of a right triangle with a given perimeter.
6. Find the best affine approximation of the vector function $F(x,y,z)=(xz+y,x^{2}+zy^{2})$ at the point where $x=1,y=1$ and $z=0.$
7. Find the volume of a sphere of radius $a.$
8. Sketch the velocity vector field $F(x,y)=(x,-y)$ identifying the most important features. Describe the motion in detail.
9. Evaluate $\int_{C}(y^{2}+2xy)dx+(x^{2}+2xy)dy,$ where $C$ is the part of the graph $y=2x^{2}$ from $(0,0)$ to $(1,2).$
10. Let $F(x,y)=(\dfrac{-y}{x^{2}+y^{2}},\dfrac{x}{x^{2}+y^{2}})$ be a vector field, and let $C$ a simple (i.e., without self-intersections) closed path that encloses the origin. Find the work of $F$ along $C.$ Hint: it is equal to the work along a certain circle.