Find the equation of the linear function (y = ax + b) that best fits the data points (1,2), (2,3), (4,6) based on a least squares regression analysis criteria.

Section 13.9 #6

DPGraph Picture

Pictured above is the region over which we are maximizing P(x,y).

Pictured below is the graph of P(x,y) over the region pictured above.

Maple Computation Worksheet

Find the point on the graph of z = x² + y² closest to the point (5,5,0).

In this DPGraph Picture of z = x² + y² and the point (5,5,0) each unit on the z-axis corresponds to 10.

d = [(x-5)² + (y-5)² + (z - 0)²]^1/2

d(x,y) = [(x-5)² + (y-5)² + (x²+y²)²]^1/2

d² = f(x,y) = (x-5)² + (y-5)² + (x²+y²)²

     Solve the system below.

f_x = 2(x-5) + 2(x²+y²)(2x) = 0

f_y = 2(y-5) + 2(x²+y²)(2y) = 0

     y(x-5) + y(x²+y²)(2x) = 0

     x(y-5) + x(x²+y²)(2y) = 0

Subtraction produces -5y + 5x = 0 which yields

     x = y

Back substitution produces

     x - 5 + 2x(2x²) = 0

     4x³ + x - 5 = 0

The solutions are 

x = 1, x = -1/2 - i and x = -1/2 + i

For x=1, y=1, and z=2.

The point on the graph of z = x² + y² closest to the point (5,5,0) is (1,1,2).  

d(1,1) = 6 

The bottom graph at the right is the graph of d(x,y).   DPGraphPicture of d(x,y)

Here is a DPGraph Picture of z = x² + y² and the points (5,5,0) and (1,1,2).

Click on the picture above to see a related Maple worksheet.  The graph below is of the distance function d(x,y).

In constructing an open topped box it costs $3 per square foot for the base and $2 per square foot for the sides.  Find the dimensions of the box of maximum volume that can be constructed for $100.

Let x = width of base, y = length of base, z = height

3xy + 4xz + 4yz = 100

z = (100 - 3xy)/(4x + 4y)     (1)

V = xyz

V(x,y) = xy(100 - 3xy)/(4x + 4y)

V_x = (400y² - 12x²y² - 24xy³)/(4x + 4y)² = 0

V_y = (400x² - 12x²y² - 24yx³)/(4x + 4y)² = 0

This leads to the system

100 - 3x² - 6xy = 0

100 - 3y² - 6xy = 0

and subtraction leads to x = y.

Back substitution produces

100 - 9x² = 0 so x = 10/3 and y = 10/3

Substituting into (1) above produces z = 5/2.

Thus the volume is a maximum if the dimensions are

10/3 ft by 10/3 ft by 5/2 ft 

and the maximum volume is 250/9 cubic ft.

The graphs at the right are graphs of V(x,y).  Click on the top graph to see it rotate.  DPgraphPicture of V(x,y)

You might enjoy staring at the pictures below of the box of maximum volume for 15 minutes or so if you have no life.

DPGraphPicture of the box of maximum volume

DPGraphPicture of the box with shading

Maple Computation Worksheet