Math 117 Practice Final

Please work out each of the problems below.  Credit will be based on the steps towards the final answer.  Show your work.

Printable Key

Problem 1

Sketch the following.

A.  The point (3,4,1).


B.  The surface z  =  x2 + y2 


C.  Five level curves to the surface z  =  x - y2 


Problem 2

Find fxz for

        f(x,y,z)  =  x2z + y2 - y cos(xz)


Problem 3

A swimwear store sells both men's and women's swimsuits and has determined that the profit, P,  it can make in stocking x men's suits and y women's suits is 

        P(x,y)  =  2x2 + y2 + xy - 900x - 1100y  + 400,000

Determine the number of men's and the number of women's suits that should be stocked in order to maximize profit.  Then determine the maximum profit.


Problem 4

A medical researcher is studying the effect that exercise has on a women's percent body fat.  The table below shows the results of the corresponding study done.  Determine the least squares regression line for this data and use it to estimate the mean percent body fat for a women that exercises 3 hours per week.

Exercise Hours 0 1 2 4 4 5
% Body Fat 31 28 26 18 20 15


Problem 5

Switch the order of integration to evaluate the following



Problem 6

Consider the following game.  Pick a card.  If you select a face card (Jack, Queen, or King) you win $3.  Otherwise, you lose $1.  Find the expected value and comment on how this number shows whether playing this game many times is a good idea.


Problem 7

Consider the following probability density function

        f(x)  =  x + 0.5        0  <  x  <  1

A.  Find P(0.5 < x < 1).


B.  Find the standard deviation.


C.  Find the median.


Problem 8

A pair of baby rabbits is introduced to an island.  Assume that rabbits take two months to reproduce when they have 2 pairs of new baby rabbits. Each month thereafter, the adult rabbits produce two additional pairs of rabbits. Assume that the rabbits never die.

A.  Write down the number of pairs of rabbits on the island during the first 6 months.


B.  Find constants u and v such that

        an  =  u an-1 + v an-2 

Where an represents the number of rabbits after the nth month.


Problem 9

A very long river that runs along a vast area of farming communities has 3000 trout in its first mile.  Because of pollutants, each mile contains only 80% of the fish that the previous mile had.  so that the second mile has 2400 trout and the third has 1820 trout.  Considering the river as running an infinite distance, how many trout live in the river?


Problem 10

Determine whether the following series converge or diverge.  Be sure to cite the test that you are using.





Problem 11

Find a power series representation for 


using the fact that


then find the radius of convergence for your solution


Problem 12

Find the third degree Taylor polynomial centered at x  =  0 for

        f(x)  =  x ex 


Problem 13

Use Newton's method  with initial guess of 1 to estimate 


within three decimal places of accuracy by finding the solution to 

        x2  -  2  =  0


Problem 14

Solve the following differential equation

        y ' -             y  =  2x2 + 1      


Problem 15

The rate of growth of a bacteria is equal to the quotient of the number of bacteria present and one more than the number of hour that the bacteria has been growing.  If initially there were 5 grams of bacteria, how many grams of bacteria will there be in 8 hours?