1. Find the global maximum of the function f(x,y) = y2 + x2 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) = 2y3 + 3 x2 - 6xy.
3. Verify that the function f(x,y) = y4 + x3 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),
miles, as a function of the temperature
in degrees Fahrenheit, and
percent humidity h:
f(t,h) = 12,000 - t2 - 2 ht - 2 h2 + 200 t + 260h.
What are the optimal atmospheric conditions for controlling the missile ?
6. Let f(x,y) = 3xy - x2 - y3 .Find all critical points, local minima, maxima and saddle points of f(x,y).
7. Let a function f(x,y)
a point (a,b)
f x (a,b) = f y(a,b) = 0, f xx (a,b) > 0, f yy(a,b) > 0, f xy (a,b) = 0.
can you conclude about the behavior of f(x,y)
(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
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
above by the sphere x2
+ y2 + z2 = 2 and
below by the paraboloid
z = x2
+ y2 .
(b) bounded above by the paraboloid z = 2 - ( x2 + y2 ) and the cylinder x2 + y2 = 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
path of an object moving in the xyz-space
is given by (x(t), y(t), z(t)) = (cos(t),
t2 , 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 ?