Fall 2004

Practice Exam 2

Instructor: M. Kulenović

Kingston 10/10/2004

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 $4$ and $5.$

  2. Estimate the percentage of students scoring less than $2.$

  3. Estimate the median and the mean.


2. The density function for the lifetime for certain type of switching device is shown in the figure below. The time $t$ is measured in months and $0\leq t\leq 5.$

  1. Estimate the percentage of switching devices which lifetime is between $2$ and $4$ months$.$

  2. Estimate the percentage of switching devices which lifetime is less than $3$ months.

  3. Estimate the mean and the median value of this switching device.


3. Given the following probability density function:


  1. Find its cumulative distribution function.

  2. Find its mean value and the median.

4. Match the graphs of surfaces to the equations. Explain.

(a) $z=4-x^{2}-y^{2}$ (b) $z=2y^{2}-x^{2}$ (c) $z=x+4y-1$

(i) m132f04pracex2__18.gif(ii) m132f04pracex2__19.gif

(iii) m132f04pracex2__20.gif

5. Match the contour diagrams to the equations in the previous problem. Explain.

(i) m132f04pracex2__21.gif(ii) m132f04pracex2__22.gif

(iii) m132f04pracex2__23.gif

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

$x/y$ $1$ $2$ $3$ $4$
$2$ $4$ $5$
$4$ $9$ $11$

Find an equation for this linear function.

7. Consider the function MATH

(a) Estimate $f_{x}(2,1)$ and $f_{y}(2,1)$ from a table of values for $f$ with $x=1.9,2,2.1$ and $y=.9,1,1.1.$

(b) Compare the estimates in part (a) with the exact values of $f_{x}(2,1)$ and $f_{y}(2,1).$

8. 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) $w(2000,70)=180$ (b) MATH

(c) MATH

Estimate $w(2200,70).$

9. Find all critical points of the function:

$f(x,y)=x^{3}$ $-xy-3y.$

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

10. Design a rectangular milk carton box of width $w,$ length $l,$ and height $h$ which holds $512\;cm^{3}$ of milk. The sides of the box cost $1$ cent/$cm^{2}$ and the top and bottom cost $2$ cent/$cm^{2}.$ Find the dimensions of a box that minimize the total cost of materials used.

11. Consider data points MATH. Suppose you want to find a least square line $y=mx+b$ for these data points without using a calculator program. What function $f(m,b)$ should be minimized ?

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