1. Find the maximum and minimum volumes of a rectangular box whose surface area is \( 1500 \mathrm{~cm}^{2} \) and whose total edge length is 200 cm .
2. The plane \( x+y+2 z=2 \) insects the paraboloid \( z=x^{2}+y^{2} \) in an ellipse. Find the points on this ellipse that are nearest to and farthest from the origin.
3. Find the slope of the tangent line to the polar curve \( r=2 \cos \theta \) at the point \( \theta=\pi / 3 \).
4. Use a graph to estimate the y -coordinate of the highest points on the curve \( r=\sin (2 \theta) \). Then use calculus to find the exact value.
5. Find the area of the region that is bounded by \( r=\sin (\theta)+\cos (\theta) \), for \( 0 \leq \theta \leq \pi \).
6. Find the exact length of the polar curve \( r=2(1+\cos \theta) \).
7. Identify the type of conic sections:
\[
\begin{array}{r}
4 x^{2}=y^{2}+4 \\
4 x^{2}=y+4 \\
x^{2}=4 y-2 y^{2} \\
y^{2}-2=x^{2}-2 x \\
3 x^{2}-6 x-2 y=1
\end{array}
\]
8. Use Desmos to graph the conics \( r=e /(1-e \cos \theta) \) with \( e=0.4,0.6,0.8 \) and 1.0 together. How does the value of \( e \) affect the shape of the curve?
9. Calculate the integrals
\[
\begin{array}{r}
\iint_{D} \frac{x y^{2}}{x^{2}+1} d A, \quad D=\{(x, y) \mid 0 \leq x \leq 1,-3 \leq y \leq 3\} \\
\iint_{D} x \sin (x+y) d A, \quad D=[0, \pi / 6] \times[0, \pi / 3] \\
\iint_{D} \frac{y}{x^{2}+1} d A, \quad D=\{(x, y) \mid 0 \leq x \leq 4,0 \leq y \leq \sqrt{x}\} \\
\iint_{D} x y d A, \quad D \text { is enclosed by the curves } y=x^{2}, y=3 x .
\end{array}
\]
10. (a) Find the volume of the solid that lies under the hyperbolic paraboloid \( z=3 y^{2}-x^{2}+2 \) and above the rectangle \( D=[-1,1] \times[1,2] \).
(b) Find the volume of the solid under the surface \( z=1+x^{2} y^{2} \) and above the region enclosed by \( x=y^{2} \) and \( x=4 \).