#
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^{4 } - 3y^{3 }
x^{2 } + 2xy - x^{3 } + 1.

Find f_{x}**
, ** f_{y}**
.**

**4. **Suppose that your weight
w,
in pounds, is a function f(c,n)
of the number
c 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)** w_{c}**
**(2100,20)
= 0.03
**(c)** w_{n}**
**(2100,20)
= -0.025 .

Estimate w(2300,20) and w(2100,25).

**5. **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.

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

**7. **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.

**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 } - 2 ht - 2 h^{2 }
+ 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.