Spring '98 M403L Final Exam

Please show all work and label solutions clearly. Please put boxes around your final answers. Calculators are allowed, but books and notes are not. You are, however, allowed two handwritten ``crib sheets''.

Since calculators are allowed, and some of these problems are numerical, it is EXTREMELY important to show your work. A correct answer without supporting work will be assumed to come from a calculator, and will NOT get you full credit. For example, if the question was ``use a 2nd-order Taylor series to approximate '', the answer should be something like `` ''. Just writing `` '' will get part credit at best, and writing will get you no credit at all.

1. A population of bacteria is growing exponentially. At noon today there were 1,000. At 2 PM there were 1,500 bacteria.

a) How many bacteria will there be t hours after noon?

b) How many bacteria will there be at noon tomorrow?

c) At what time will there be 10,000 bacteria?

2. Consider the function .

a) Which of the following points are critical points: (1,-1), (1,0), (1,1), and (0,0)?

b) Of the above, classify which critical points are local maxima, which are local minima, and which are saddle points.

3. Evaluate the double integral of the function f(x,y)=24xy over the region between the line y=x and the parabola . [Hint: you can slice either vertically or horizontally, but one approach is a bit easier than the other].

4. Consider the function .

a) Find the 2nd order Taylor polynomial for f(x) around x=0.

b) Use this polynomial to approximate f(-0.1).

c) Use this polynomial to approximate .

5. Use Newton's method to find the point (x,y) where the curves and intersect. Your answer should be correct to at least 5 decimal places.

6. a) Evaluate exactly.

b) Approximate this integral numerically using Simpson's rule with . Express your result to at least four decimal places.

7. a) A certain baseball player has a 0.300 chance of getting a hit every time he comes to bat. Assuming he bats 500 times this season, write down the exact probability that he has 160 or more hits this season. You can express your answer as a sum - you need not attempt to evaluate the sum numerically.

b) Estimate this probability numerically using the normal distribution (see attached table).

8. Until 1964, dimes (and quarters and half-dollars) were made of silver. There are a few of these old silver coins still in circulation - about one in a thousand dimes is silver. Assuming you handle 5 dimes every day, what is the probability of your encountering 2 or more silver dimes in the next year? (For the record, I was handed a silver dime at Burger King last month, and my daughter found a silver half-dollar on the ground earlier this year.)

9. A continuous random variable is described by the probability density function

where C is a constant.

a) Find the constant C that makes f(x) into a legal probability density function.

b) Find the probability that 2 < X < 3.

c) Find the mean and standard deviation of X.

10. For the system of linear equations:

a) Write down the augmented matrix.

b) Apply row operations to put this matrix in reduced form.

c) Find all solutions to the system of linear equations.

11. Maximize subject to the constraints:

12. Express the following situation as a linear programming problem. That is, a) set up the inequalities, b) choose slack variables, and c) write down the matrix of the initial system. You do not have to actually solve the resulting system:

``An investor has at most $100,000 to invest in government bonds, mutual funds and money market funds. The average yields for these investments are 8%, 13% and 15%, respectively. To avoid undue risk, the total amount invested in mutual and money market funds cannot exceed the amound invested in government bonds. How should the money be invested to maximize expected return?''