MTH 243

University of Rhode Island

Practice problems for Exam #2
                                                        Kingston 3/20/99

1.   Find the global maximum of the function  f(x,y) = y+  x 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 x - 6xy.

3.   Verify that the function  f(x,y) = y+  x 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 ht - 2 h + 200 t   + 260h.

What are the optimal atmospheric conditions for controlling the missile ?

6.   Let f(x,y) =  3xy -  x - y3 .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 + y + z = 2  and below by the paraboloid    z = x + y .
(b)  bounded above by the  paraboloid    z = 2 - ( x + y ) and the cylinder x + y = 1 and below by the plane  z =0.

11.   Convert the integral

to polar coordinates and hence evaluate it exactly. Sketch  the region over which the integration is being performed.

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

where is the region described in Problem 12.

14.  The path of an object moving in the xyz-space is given by  (x(t), y(t), z(t)) = (cos(t), t , 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 ?