A discussion with a student starting the term

1-What is Average?  Average is the regular or the ordinary. In business average is when the product or the profit is not top class but not mediocre either. In other words, what you expect most of the time.  2- What is a straight line in Business? A straight line in business is when the procedure to make  profit is just an one step to follow. you don't have to go through a lot of procedures in order to reach your goal. Straight line is a relationship between two variables and two coefficients  that represents proportionality. WHAT YOU HAVE TO DO IS: to know what is the variable,  & what is the coefficient is the step I would like you to think now, and from the general mathematical presentation, to move into the real life translating the abstract algebraic world into the specific real. This is the first step for modeling.   3- What is the relationship between a line and a plan? The relationship between a line and a plan is a plan consists of a bunch of lines or a plan makes from more than one line. Correct more or less - what is the relationship between a point and a line, a line and a plane? How many dimensions does a point, a line and a plane have? 4- What is the relationship between a line and a curve? The relationship between a line and a curve is a curve is a bent line. i.e. lines or curves have the same dimensions - two - but while lines present proportionality, curves are not proportional, but something more dynamic in the relationship of two variables. The the question is how many types of curves do you know, and  what each type of a relationship each curve represents?

The code in Mathematica for some work and review

A review of your basic algebra skills using Mathematica 8 is available in a related file.

It is important that you can understand the meaning of a line.

The equation relates variable x (horizontal axis) to y (vertical axis) with a line given by the equation:

y = a + b x

a provides the distance that the line intersects to the vertical axis.

If a is negative then the intersection is below the origin point (0,0),

if a is zero then it crosses the origin point (0,0), and

if a is positive then the line intersects to the vertical axis above the origin point.

This magnitude provides the value of y that is independent of any value of x, in other ways when b is zero.

If b is zero, the equation is reduced to y = a.

If b is positive, then the line has an upward slope.

That means that as the a increases, so y, and the value of b, the slope of the line indicates how x affects y. If the slope is 45o then any change of x results equal change in y.

If the slope is greater than 45o, any change of x results greater change in y, while if the slope is less than 45o, the change of x results smaller change in y.

If b is negative, then the line has a downward slope.

That implies substitution, as the value of x increases, the value of y decreases.

Manipulate[Plot[a + b x == c, {x, -10, 10}],

{a, -20, 20}, {b, -.01, .6}, {c, -10, 10}]

When we move from a two dimensions to three dimensions we have:

y = a + b x + c z

Manipulate[Plot3D[a + b x + c z == 0, {x, -10, 10}, {z, -10, 10}],

{a, -20, 20}, {b, -5, 5}, {c, -5, 5}]

Now we do not have a line but a surface plane.

You can play around using the Manipulate command in Mathematica, so you can verify the change on the plane's position when coefficients a, b, & c change.

When the dependent variable is not in its simple (linear) form that indicates that x affects y in a proportional way, the relationship becomes non linear, i.e. non proportional.

This meas that either the variable x interrelates to itself, becoming x*x or x^2.

Then we have a second degree polynomial.

y = a + b x + c x^2

Manipulate[Plot[a + b x + c x^2 == 0, {x, -5, 5}],

{a, -20, 20}, {b, -10, 10}, {c, -50, 50}]

In a three dimensional form we may have a case that one variable affect the other.

Then we have the case

y = a + b x * c z => y = a + (bc) x*z

Manipulate[Plot3D[a + b x c y == 0, {x, -10, 10}, {y, -10, 10}],

{a, 0, 20}, {b, -5, 5}, {c, -5, 5}]

This implies that the surface bends. The opposite type of a bend we have whenever we have a division as:

y = a + ( b x / c z )

Manipulate[Plot3D[a + b x / c y == 0, {x, -10, 10}, {y, -10, 10}],

{a, 0, 20}, {b, -5, 5}, {c, -5, 5}]

The general case in the three dimensional relationship is:

y = a + b x + c x^2 + d x z + e z^2 + f z

Manipulate[Plot3D[a + b x + c x^2 + d x y + e y^2 + f y == 0, {x, -10, 10}, {y, -10, 10}], {a, -200, 200}, {b, -50, 50}, {c, -10, 10}, {d, -100, 100}, {e, -10, 10}, {f, -50 , 50}]

Manipulate[

Plot[a + b x + c x^2 + d x^3 == 0, {x, -1, 1}], {a, 0, 20}, {b, -10,

10}, {c, -5, 5}, {d, -20, 20}]

Manipulate[Plot[a + b x + c x^2 + d x^3 + e x^4 == 0, {x, -1, 10}],

{a, 0, 20}, {b, -10, 10}, {c, -5, 5}, {d, -20, 20}, {e, -3, 3}]

Then you can practice to see the planes defined on page 19 using:

Plot3D[3 x + 4 y == 18, {x, -10, 10}, {y, -20, 20}]

and

Plot3D[9 x + y == 21, {x, -10, 10}, {y, -20, 20}]

Then you can solve the system of these two equations by:

Solve[{3 x + 4 y == 18, 9 x + y == 21}, {x, y}]

and have its solution.

{{x -> 2, y -> 3}}

Finally, you can control the relationships as:

Manipulate[Plot[a x ^b == 0, {x, -1, 1}], {a, -20, 20}, {b, -6, 6}]

and

Manipulate[Plot[a + b x , {x, 0, 10}], {a, 0, 20}, {b, 0, .99}]

Manipulate[Plot[a * x^b , {x, 0, 10}], {a, 0, 20}, {b, 0, .99}]

Manipulate[Plot[a * E^(-b x) , {x, 0, 10}], {a, 0, 20}, {b, 0, .99}]

Manipulate[Plot[a * (1 - E^(-b x)) , {x, 0, 10}], {a, 0, 20}, {b, 0, .99}]

Manipulate[Plot[a x ^b == 0, {x, 0, 10}], {a, 0, 20}, {b, 1, 5}]

Manipulate[Plot[a x ^b == 0, {x, 0, 10}], {a, 0, 20}, {b, -6, .99}]

If you want to indicate an influence diagram 

you may use:

LayeredGraphPlot[{"Price" -> "Revenue", "Revenue" -> "Profit",

"Advertising" -> "Units Sold", "Advertising" -> "Cost",

"Units Sold" -> "Revenue", "Seasonal Factors" -> "Units Sold",

"Unit Cost" -> "Cost of Goods", "Cost of Goods" -> "Cost",

"Sales Expence" -> "Cost", "Overhead" -> "Cost" ,

"Cost" -> "Profit"}, Left, VertexLabeling -> True]