Kingston 3/20/99

**1. **Find the global maximum
of the function f(x,y)
= y^{2 }+ x^{2 } in
the region R: 0<=x<=1, 0<=y<=2.
Explain your reasoning.

**2. **Find local minima, maxima
and saddle points for the function f(x,y)
= 2y^{3 } ^{ }+ 3 x^{2 }
- 6xy.

**3. **Verify that the function f(x,y)
= y^{4 }+ x^{3 } has
a critical point at (0,0).
Is it a local maximum, minimum or a saddle point ? Explain your answer.

**4. **A contour diagram of a function
f(x,y)
is given below. The function has critical
points at (0,0)
and
(2,2).
Classify each of them as a local minimum,
maximum or a saddle point. Explain your answer.

**5. **A cruise missile has a remote
guidance device which is sensitive to both temperature and humidity. Army
engineers have worked out a formula to show the range at which the missile
can be controlled, f(t,h),
in
miles, as a function of the temperature
t,
in degrees Fahrenheit, and
percent humidity h:

f(t,h) = 12,000 - t^{2 } - 2 ht - 2 h^{2 }
+ 200 t ^{ } + 260h.

What are the optimal atmospheric conditions for controlling the missile ?

**6. **Let f(x,y)
= 3xy - x^{2 } - y^{3 }.Find
all critical points, local minima, maxima and saddle points of f(x,y).

**7. **Let a function f(x,y)
and
a point (a,b)
be
such that

f _{x}** **(a,b) = f
_{y}(a,b)
= 0, f
_{xx}** **(a,b) > 0, f
_{yy}(a,b)
> 0, f
_{xy}** **(a,b) = 0.

**(a) **What
can you conclude about the behavior of f(x,y)
near
(a,b)
?
**(b) **Sketch
a possible contour diagram for f(x,y)
near
(a,b).

**8. **Consider data points (1,1),
(2,2.5), (3,2.5), (4,5). Suppose you want
to find a least squares line y = mx
+ b for these data points without using
a calculator program. What function f(m,b)
would you have to minimize ? Write the function f(m,b)
but do not perform actual calculations.

**9. **Consider the integral

**(a) **Sketch
and label the region over which the integration is being performed.
**(b) **Rewrite
the integral with the integration performed in the opposite order.

**10.** Set up
*but do not evaluate** *a
triple integral that gives the volume of the solid

**(a) **bounded
above by the sphere x^{2 }
+ y^{2 } + z^{2 } = 2 and
below by the paraboloid
z = x^{2 }
+ y^{2 } .
**(b) **bounded
above by the paraboloid
z = 2 - ( x^{2 }
+ y^{2 } ) and the cylinder x^{2 }
+ y^{2 } = 1 and below by the
plane z =0.

**11. **Convert the integral

**12. **Consider the volume W
between a cone centered along the positive
z -axis, with vertex at the origin and conaining
the point (0,1,1),
and a sphere of radius 2
centered at the origin.
**(a) **Write
a triple integral which represents this volume. Use spherical coordinates.
**(b) **Evaluate
the integral.

**13. **Evaluate
the integral

**14. **The
path of an object moving in the xyz-space
is given by (x(t), y(t), z(t)) = (cos(t),
t^{2 } , t + 1).
**(a) **At
time t = 1, what
is the object's velocity ? What is the speed ?
**(b) **What
is the acceleration at arbitrary moment t
?