Mathematics 2507 Test number 1 Thursday, 6 October 1994

NAME____________________________

Instructions: Use non-erasable ink. Write the answers where indicated and give clear evidence of your reasoning (or points will be taken off). You may attach extra sheets with your work if it is organized enough to be helpful. Graphs should be clearly labeled. Symbolic and graphing calculators are not permitted. On the other hand, an answer like \pi is actually preferable to an answer like 3.141592653589793238462643, so you do not really need a calculator at all.

0 |

1 |

2 |

3 |

4 |

TOTAL

NAME_____________________

0 (0.5 points). The name of my teaching assistant is ___________

1 (5 points). Find and classify all the critical points of the function

f(x,y) = 3x^2 + 3yx^2 + y^3 -15y.

The critical point(s) __________________ is(are) ______________________.

The critical point(s) __________________ is(are) ______________________.

The critical point(s) __________________ is(are) ______________________.

The critical point(s) __________________ is(are) ______________________.

The critical point(s) __________________ is(are) ______________________.

KEY FORMULA OR METHOD (optional for partial credit)____________________

_________________________________________________

2 (5 points). A cylindrical can of volume 20 cm^3 is to be constructed. Material for the top and bottom costs $0.01/cm^2, and for the side$0.005/cm^2. What dimensions R=radius and H=height should the can have to minimize the cost?

The objective function, if any, is _________________________________

The constraint(s), if any, is(are) __________________________________

______________________________________

______________________________________

______________________________________

The best dimensions are: R_{best} = _______________________________

H_{best} = _______________________________

The cost of the can is: ______________________________________

KEY FORMULA OR METHOD (optional for partial credit)____________________

_________________________________________________

NAME_____________________

3. (5 points - Dang if this doesn't look a lot like homework problem 21!) Find the minimum value of x^3 + y^3 + z^3 for (x,y,z) on the intersection of the planes x + y + z = 2 and x - y + z = 3.

The place(s) where the minimum occurs is (are): _____________________

______________________________________

The minimum value is: ______________________________________

KEY FORMULA OR METHOD (optional for partial credit)____________________

_________________________________________________

4. (5 points) NOTE: The point of this problem is to see if you understand Newton's method. The exact solution of the system is (x,y) = (3,2) or (2,3). Now that we all know the exact answer, you will get 0 points for writing it down!

Suppose

f(x,y) := x + y - 5 = 0 and

g(x,y) := (x - y)^2 - 1 = 0.

Take as your initial guess (x_0,y_0) = (1,-1) and use Newton's method to produce a better guess (x_1,y_1)

The explicit* formula for the improved guess (x_{n+1},y_{n+1}) given (x_n,y_n) is:

x_{n+1} = _________________________________________________

y_{n+1} = _________________________________________________

With (x_0,y_0) = (1,1),

x_1 = _________________________________________________

y_1 = _________________________________________________

KEY FORMULA OR METHOD (optional for partial credit)____________________

_________________________________________________

* "Explicit" means a specific expression using x_n' and y_n'. Do not use the letters f' and g' .