MATH 21 D (Section A) NAME:
Quarter: Spring SECTION:
Date: June 08, 2002
1 2 3 4 5 total 6
Final
Problem 1. (30 pts; estimated time: 20 mn)
Find the dimensions of the box of largest volume whose surface area is to be 6
square inches (see figure 1).
Problem 2. (40 pts; estimated time: 25 mn)
Denote by Sthe surface of equation: z=−1
2x2+y2+ 2x+ 1.
(a) Verify that the point (0,1,2) belongs to the surface S.
(b) Verify that the vector i+j+ 4kis tangent to the surface Sat the point
(0,1,2).
(c) Find the parametric equations of the vertical plane passing through the
point (0,1,2) and containing the vector i+j+ 4k.
(d) Find the parametric equations of a curve drawn on the surface Swhose
tangent at the point (0,1,2) is the vector i+j+ 4k.
Problem 3. (40 pts; estimated time: 25 mn)
(a) Show that the curve parameterized by
G(t) = et+e−t
2i+et−e−t
2j= cosh ti+ sinh tj,
for 0 ≤t≤a, with a > 0, lies on the parabola x2−y2= 1, and joins the
point A= (1,0) to the point B= (cosh a, sinh a).
(b) Let Abe the region bounded by the line segment OA, the line segment OB,
and the curve in (a) joining Ato B(figure 2.) Show that the area of A
is a/2.
Problem 4. (60 pts; estimated time: 30 mn)
Let
F(x, y) = −yi
x2+y2+xj
x2+y2.
(a) Show that the divergence of Fis zero.
(b) Show that the curl of Fis zero.
(c) Let f(x, y) = tan−1(y/x). Does Fequal ∇fwhere both are defined? Justify your answer.
(d) Evaluate HCF·dr, where Cis the circle x2+y2= 1 in the xy plane, taken
counterclockwise.
(e) Is F conservative? Justify your answer.
(f) Does (e) contradict (c)? Explain.
