# MATH 132-URI

Fall 2005

Practice Exam 2

Instructor: M. Kulenović

Kingston 10/10/2005

1. An exam given to a large number of students is scored from 0 to 5 The density function of the scores is shown in the figure below.

1. Estimate the percentage of students scoring between 2 and 5

2. Estimate the percentage of students scoring less than 3

3. Estimate the median and the mean.

2. The density function for the lifetime for certain brand of light bulbs is shown in the figure below. The time t is measured in months and 0 < t < 5.

1. Estimate the percentage of light bulbs which lifetime is between 1 and 4 months

2. Estimate the percentage of light bulbs which lifetime is less than 2 months.

3. Estimate the mean and the median value of the light bulbs.

3. Given the following probability density function:

1. Find its cumulative distribution function.

2. Find its mean value and the median.

4. Match the contour diagrams to the equations. Explain.

(a)  z = 4 - x - y2                                       (b)  z =  2y2 - x2                                                        (c)  z = x + 2y - 1

(i) (ii) (iii)

5. Fill in the blanks in a partial table of values for a linear function:

 x/y 1 2 3 4 2 11 9 5 11 7 8 11

Find an equation for this linear function.

6. Consider the function   f (x, y) = 4 x - x - x y2

(a)   Estimate fx (1, 2)  and fy (1, 2) from a table of values for f with x = 0.9, 1, 1.1 and y = 1.9, 2, 2.1.

(b)    Compare the estimates in part (a) with the exact values of fx (1, 2) and fy (1, 2).

7. Suppose that your weight w in pounds, is a function f (c, h) of the number c of calories you consume daily and the height h in inches. Using units, interpret, in everyday terms the statements:

(a)    w(2000, 71)= 190                                                 (b)   wc(2000, 71)= 0.07

(c) wh(2000, 71)= - 0.05                                                (d)  Use the linear approximation formula to estimate w(2100, 72).

8. Find all critical points of the function:

f (x, y) = 4 x - x + 2y - x y2

Determine if the critical points are local maxima, minima,  or none of these.

9. Design a rectangular milk carton box of width w length l and height h which holds 1024 cm3 of milk. The sides of the box cost 2 cent/cm3 and the top and bottom cost 4 cent/cm3 Find the dimensions of a box that minimize the total cost of materials used.

10. Consider data points (1, 2), (2, 5), (4,7). Suppose you want to find a least square line y = m x + b for these data points without using a calculator program. What function  f (m, b) should be minimized ?