Calculus III Final Exam. April 2003. Name
Mathematically justify your answers (show work). Simplify and complete all
computations as much as possible. Circle answers.
1. Near the surface of the planet X-12 a falling body undergoes a downward
acceleration of 5m/sec2.A projectile is fired on X-12 from ground level with an
initial speed of 300m/s at a 30◦angle of elevation above the horizontal. Ignoring
friction, find (A) the position −→
r(t) at time t(before it strikes the ground), and
(B) the range of the projectile.
2. The position of a particle at time tis given by −→
r(t) = cos(3t)−→
i+
sin(3t)−→
j+t−→
k . Find a parametric equation for the line tangent to the path
when t=π/4.
3. Let f(x, y) = tan−1(y
x).Find (and simplify)
(A) fx
(B) fy
(C) fxy
4. Find an equation for the plane tangent to the surface x2+ 3y2+ 5z2= 9
at the point (x, y, z) = (−1,1,−1).
5. Express in two different iterated integrals RRDf(x, y)dif Dis the region
bounded by y= 3xand y=x2.
6. Use the method of Lagrange multipliers to find the maximum and mini-
mum values of f(x,y , z) = x3+y3+z3on the sphere x2+y2+z2= 1.
7. Let f(x, y)=4xy −1
2x2−y4.Find (and LIST) all critical points and
use the second derivative test to identify all local minima, maxima, and saddle
points.
8. Use polar coordinates and integration to find the volume of the region
inside the sphere x2+y2+z2= 16 and outside the cylinder x2+y2= 4.
9. Let a vector field be −→
F(x, y) = y−→
i−x−→
j. Find the work done if a unit
mass in the field traverses once in the counterclockwise direction around the
circle x2+y2= 9.
10. Suppose the force exerted on a unit mass at the point (x, y) is −→
F(x, y) =
(3x2−y2−4)−→
i+ (3y2−2xy + 1)−→
j .
(A) Show the field is conservative and find its potential.
(B) Find the work done in moving the mass from (0,0) along straight line
to (3,19) and then along another line to (1,2).
11. Use Green’s Theorem to show that the area of the ellipse x2
a2+y2
b2= 1
is πab. Explain in a coherent manner your reasoning. (Hint: let x=acos t,
y=bsin t).
12. Let w=g(x, y, z ) be continuous and have continuous first and second
derivatives. Show that curl(∇g(x, y, z)) = 0.
13. Let a vector field be −→
F(x, y, z) = (x−y+ 3z)−→
i+ (y−x−3z)−→
j+ (3x−
3y+ 9z)−→
k.
Show that −→
Fis a conservative field and find its potential.
14. Use cylindrical coordinates and integrals to show that the volume of a
right circular cylinder of radius Rand altitude His 1
3πR2H.
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