MTH 132

University of Rhode Island

Exam #3
                                                         Kingston 4/22/98
1.   Which of the tables below represent a linear function  f(x,y) ? For those that represent a linear function find an equation.

(a)
 
 x \ y -1 0 1 2
0 1.5 1 0.5 0
10 3.5 3 2.5 2
20 5.5 5 4.5 4
30 7.5 7 6.5 6
(b)
 
 x \ y -3 -2 -1 0
-3 18 13 10 9
-2 13 8 5 4
-1 10 5 2 1
0 9 4 1 0
2.   Find an equation of the linear function represented by a contour map below. Make sure that your equation is as accurate as possible by choosing points where values of the function are shown on the diagram exactly.


 

3.   Let  f(x,y) = 2y - 3y x + 2xy - x + 1.

Find  fx fy .

4.   Suppose that your weight w, in pounds, is a function  f(c,n) of the number of calories you consume daily and the number n  of minutes you exercise daily. Using units, interpret in everyday terms the statements:

 (a) w(2100,20) = 120               (b)  wc (2100,20) = 0.03                     (c)  wn (2100,20) = -0.025 .

Estimate w(2300,20)  and  w(2100,25).

5.   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.

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

7.   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.

8.  A contour diagram of a function f(x,y)  is given below. The function has critical points at  (0,0) and  (1,1). Classify each of them as a local minimum, maximum or a saddle point. Explain your answer.

9.   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 ?

10.  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.