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.
Estimate the percentage of students scoring between 2 and 5
Estimate the percentage of students scoring less than 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.
Estimate the percentage of light bulbs which lifetime is between 1 and 4 months
Estimate the percentage of light bulbs which lifetime is less than 2 months.
Estimate the mean and the median value of the light bulbs.
3. Given the following probability density function:
Find its cumulative distribution function.
Find its mean value and the median.
4. Match the contour diagrams to the equations. Explain.
(a) z = 4 - x^{2 } - y^{2} (b) z = 2y^{2}^{ }- x^{2 } (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^{2 } - x y^{2}
(a) Estimate f_{x} (1, 2) and f_{y} (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 f_{x} (1, 2) and f_{y} (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) w_{c}(2000, 71)= 0.07
(c) w_{h}(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^{2 } + 2y - x y^{2}
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 cm^{3} of milk. The sides of the box cost 2 cent/cm^{3} and the top and bottom cost 4 cent/cm^{3} 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 ?