Math 113 – Calculus III Exam 4 Practice Problems Spring 2004
1. The energy stored in an idealized stretched spring is proportional to the square of the
length of the string.
Suppose three springs lie in the xy plane. One end of the first spring is fixed at (1,0),
one end of the second spring is fixed at (0,0) and one end of the third spring is fixed at
(0,1). The other ends of the three springs are connected together. This arrangement
of springs will have an equilibrium at the point where the total energy is a minimum.
The total energy is just the sum of the energy of each spring: E=E1+E2+E3=
k1s2
1+k2s2
2+k3s2
3, where siis the length of spring i, and the proportionality constants
kiare all positive.
Let (x, y) be the coordinates of the point where the springs are connected together.
Find the equilibrium point of the springs.
2. Let
f(x, y) = x2−4x+y2−4y+ 16.
(a) Find and classify the critical points of f.
(b) Find the maximum and minimum values of fsubject to the constraint
x2+y2= 18
(c) Find the maximum and minimum values of fsubject to the constraint
x2+y2≤18
(d) Approximate the maximum value of fsubject to the constraint
x2+y2= 18.3
(Explain your answer in terms of Lagrange multipliers.)
3. For each of the following, sketch the region of integration, and rewrite the integral
with the order of integration reversed. Clearly label the boundaries of the region with
an equation that defines the boundary curve, and use shading (or cross-hatches) to
indicate the region.
(a) Z3
0Z9−y2
0
f(x, y)dx dy
(b) Z1
−1Z1
x3
f(x, y)dy dx
1
