|
Practice Problems For Exam III
These problems should be looked at in advance at home.
Some of them (the same or very similar) will be on Exam III.
| 1.
For the data points (1,1), (2,3), (3,6), (4,7), and (5,9) find the best fit linear function
(y = ax + b) based on a least squares criteria. Show the
system of equations to be solved in finding a and b by setting Sa and
Sb each equal to zero. S(a,b) is the function giving the sum of
the squares of the errors. Make sure you follow the instructions for this
problem given in class. |
 |
| 2. Find the box of largest
volume that can be inscribed in the ellipsoid whose equation is:
click here to see a larger picture
State both
the dimensions and volume of your "largest" box. You may
not use the method of
Lagrange
Multipliers. |
 |
| 3.
In the problem pictured below, determine the values for x and y
that will minimize total construction costs. The idea is to lay pipe from
point P to Point Q. It costs 3 million dollars per mile to lay the pipe
through the blue area, 2 million dollars per mile to lay the pipe through the
green area, and 1 million dollars per mile to lay the pipe along the boundary
between the green area and the brown area. Consider the colored regions to
be rectangles, x to represent the horizontal distance for the pipe in the blue
region and y to be the horizontal distance for the pipe in the green
region. The blue region and the green region are each 1 mile wide and the
horizontal distance from P to Q is 5 miles. You must thoroughly
investigate the costs on the boundary of the region over which you would be
applying the cost function. You may assume you would not minimize total
cost by laying pipe in the negative x direction, negative y direction, or
by laying pipe past point Q and then coming back. Thus this region
(see the bottom picture on the right) would be described by
Click on the top picture at the right to see an
animation of some of the possible paths.
|
|
4. (From your homework, Section 13.9:
9) The sum of the length and girth (perimeter of a cross section) of
packages carried by parcel post cannot exceed 108 inches. Find the
dimensions of the rectangular package of largest volume that may be sent by
parcel post. 5. Find the absolute
maximum and absolute minimum values of the given function over the region R and
identify the corresponding points on the surface. Demonstrate your
evaluation along each boundary.
 |
