Pat Corn
Math 53
5/5/00
Practice Final
Problem 1: (a) Suppose that we have a function f(r,θ ), where rand θare the usual
polar coordinates. Show that
5f(r, θ) = (x
rfr−sin θ
rfθ,y
rfr+cos θ
rfθ).
(b) Verify that this formula holds for f(r, θ) = rn.
Problem 2: (a) Write the equation for the tangent plane to the surface f(x, y, z) = 0 at
a point (x0, y0, z0) on the surface.
(b) Let ube a vector parallel to this plane. What can you say about Duf(x0, y0, z0)?
Problem 3: Find the points on the sphere of radius 2 centered at the origin which are
closest to and farthest from the point (3,1,−1).
Problem 4: Compute R2
0R√2x−x2
0px2+y2dy dx.
Problem 5: The center of mass of a solid with constant density is called the centroid of
the solid. Find the volume and centroid of the solid bounded by the cone z=px2+y2
and the sphere x2+y2+z2= 1.
Problem 6: Find the volume of the ellipsoid
x2
a2+y2
b2+z2
c2= 1.
Suggestion: change variables so that the ellipsoid is a sphere. Check that your answer
works when a=b=c.
Problem 7: Let Cbe the curve {(cos t, t, sin t):0≤t≤π}. Evaluate RCF·dr, where
F(x, y, z) = (3x2sin y+z ez)i+x3cos yj+ (z+ 1)xezk.
Problem 8: (a) Define 52h=5 · (5h), for any twice differentiable function h. Show
that if fand gare two twice differentiable functions, then
5 · (f5g) = f52g+5f· 5g.
(b) Show that in addition, if Eis a region bounded by a surface Swhich is oriented
outward, then
ZZS
(f5g−g5f)·dS=ZZZE
(f52g−g52f)dV.
Problem 9: Let Cbe the curve obtained by intersecting the plane x+2y+z= 4 with the
cylinder x2+y2= 4. Orient Ccounterclockwise as viewed from above. Calculate RCF·dr,
where F(x, y, z) = zi−xj+yk.
Problem 10: Given a parallelogram in the plane, let aand bbe the lengths of the sides,
and let cand dbe the lengths of the diagonals. Show that c2+d2= 2a2+ 2b2.
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