Math 117 Practice Midterm I

Please work out each of the given problems.  Credit will be based on the steps that you show towards the final answer.  Show your work.

Printable Key

Problem 1

Let

        f(x,y)  =  xy - ln(2x - y)

Find

1. 
   
   Solution
   
2. 
   
   Solution
   
    
3. f_xy(x,y)
   
   Solution

Problem 2

Find the length of the sides of the triangle with vertices (1,-1,2), (3,1,5), and (-1,2,4) and determine whether the triangle is a right triangle, isosceles triangle, or neither of these.

Solution

Problem 3

Name the following surfaces and draw a rough sketch of each.

A.  x² + 4y2  =  36 + 9z²

Solution

B.  3x + y + 2z  =  6

Solution

Problem 4

Use a graphing calculator to draw at least five level curves for the following function.  Copy them on the test and describe in words to what extent the surface rises above or below the paper.  Describe in words how the surface differs from quadrant to quadrant.

                          1
        f(x,y)  =             
                         xy²

Solution

Problem 5

Locate and classify the critical points of

        f(x,y)  =  x² + 2xy + 3y² - 4x - 6y + 5

Solution

Problem 6

Common blood types are determined genetically by three alleles A, B, and O.  A person whose blood type is AA, AB, or AO is homozygous.  A person whose blood type is AB, AO, or BO is heterozygous.  A Hardy Weinberg law states that the proportion P of heterozygous individuals in any population is

        P(p,q,r)  =  2pq + 2pr + 2qr

where p, q, and r represent the proportion of A, B, and C respectively in the population.  Use Lagrange multipliers and the fact that

        p + q + r  =  1

to show that the maximum proportion of heterozygous individuals in any population is 2/3.

Solution

Problem 7

The number of students taking math at LTCC over the past years is shown in the following table.

A.  Use a calculator to find the least squares regression line for this data.

B.  Estimate the number of students that took math in 2003.

Solution

Problem 8

Evaluate the following double integral.  (Hint:  You may need to change the order).

Solution

Problem 9

Use a double integral to find the volume of the solid bounded by

        z  =  x,    z  =  0,    y  =  x,     and    y  =  x²

Solution