Differential Calculus Introduction: Simple Polynomial Equations


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Polynomial Calculations: Image by blumik

The Main Question in Differential Calculus

“Differential calculus” is a big phrase but a very useful part of mathematics. Several previous articles have built a foundation, and now the first floor will be erected.

The question that differential calculus asks is: What is the slope of a function at a given point?

What Do “Slope” and “Function” Mean?

To Define a Function:

For this article, a function relates one variable to another; it is often written as “y = f(x)”. Calculus uses real numbers rather than integers or rational numbers. (Rational numbers include every fraction, like “7/15,” but exclude irrational numbers such as PI and the square root of two). Let’s ignore imaginary numbers for now, too.

Again, for this article, the best way to think of a function is that it prescribes a line or curve graphed on a Cartesian plane.

Three other necessary features of a function for calculus are “smooth” and “continuous” and “well defined”. A smooth function does not have a sudden change. A continuous function does not have missing pieces. A “well-defined” function has only a single ‘y’ value for any ‘x’ value. Here are some examples.

  • A function: “y = x unless ‘x’ is negative; in that case y = -x” is not smooth. It makes a sudden change at (0, 0). There is no way to calculate the slope at (0, 0), because there are two competing values: one from the negative ‘x’ side, and one from positive ‘x’.
  • A function: “y = x if x > 0; y = -x if x < 0″ is not continuous. It misses the single point (0, 0). Therefore there cannot be a slope at (0, 0); there is no function there at all.
  • A function: “y = 1/x” is not defined where x is zero. Neither is a function that would graph a vertical line; that slope would be infinite because any change is the ‘y’ direction is divided by the zero change in the ‘x’ direction.

To Define Two Types of Slopes

The “average slope” between two points is the vertical change divided by the horizontal change.

For a straight line from point (x[1], y[1]) to (x[2], y[2]), the change in ‘x’ is (x[2] – x[1]), and the change in ‘y’ is (y[2] – y[1]).

The slope then is (y[2] – y[1])/(x[2] – x[1]).

The astute reader has noticed that the calculus question wanted the slope at one “given” point, but the definition of “average slope” uses two points. That’s a good point…er, question…er, observation. Thank you, astute reader.

The value of the “slope” at one point is the limit of the value of the average slope between two points that are arbitrarily close to the one desired point. Last week, “How to Find Limits of Mathematical Functions” showed how to find limits.

What are Polynomial Functions?

Let’s use the notation “x^p” for “‘x’ to the power ‘p’”. So x^2 = x*x; x^3 = x*x*x; and so on.

A polynomial function is an equation of the form y = f(x) = a*x^p + b*x^q…and so on, but only to a finite or limited number of terms. Each term, such as “a*x^p”, has a real numbers ‘a’ as a “coefficient” and ‘x’ as the independant variable which is also the “base” for a non-negative integer ‘p’ as an “exponent”. If ‘p’ is zero, then a*x^p = a.

The Average Slope of a Specific Straight Line Function

"Simple Graph of a Straight Line" by Mike DeHaan

“Simple Graph of a Straight Line” Image by Mike DeHaan

This image is a graph of the function y = 3*x + 1. All along this line, the average slope is “(y[2] – y[1])/(x[2] – x[1])”. For example, the slope from points (1, 4) to (3, 10) is (10-4)/(3-1) = (6/2) = 3.

In general, of course, every ‘y’ value is “3*x + 1″, so the general average slope is…

(y[2] – y[1])/(x[2] – x[1]) = ((3*x[2] + 1) – (3*x[1] + 1))/(x[2] – x[1]).

Let’s just work with the numerator. ((3*x[2] + 1) – (3*x[1] + 1)) = (3*x[2] – 3*x[1] + 1 – 1) = 3*(x[2] – x[1]).

The slope is 3*(x[2] – x[1])/(x[2] – x[1]).

Cancelling “(x[2] – x[1])” in numerator and denominator leaves the number ‘3’ as the slope.

The Average Slope of Any Constant Function “y = a”

In this case, the result ‘y’ has the same value for every ‘x’. Since the change in ‘y’ is zero, the slope of a constant function is zero.

The Average Slope of Any Straight Line Function “y = a*x + b”

The average slope of “y = a*x + b” is still “(y[2] – y[1])/(x[2] – x[1])”.

Let’s simplify the numerator again, substituting the function of ‘x’ for each ‘y’.

(y[2] – y[1]) = ((a*x[2] + b) – (a*x[1] + b)) = a*x[2]- a*x[1] + b – b = a*(x[2]- x[1]).

The slope is a*(x[2]- x[1])/(x[2]- x[1]) = ‘a’ after cancelling “(x[2] – x[1])” in numerator and denominator.

An important note is that the average slope is a constant value, ‘a’, regardless of the value of ‘x’. To repeat: the average slope of a straight line is a constant value everywhere on that line. If every average slope is the same value, then it seems likely that the slope at every point is exactly that average value; but we have not demonstrated this yet.

The Average Slope of the Parabola “y = a*x*x + b*x + c”

"Simple Graph of a Parabola" by Mike DeHaan

“Simple Graph of a Parabola” Image by Mike DeHaan

This image is a graph of the function y = 3*x*x + 2*x + 1. It is a visual aid for this section, but we will only deal with a general parabola rather than this specific one.

The average slope is still “(y[2] – y[1])/(x[2] – x[1])”, but “y = a*x*x + b*x + c”. Again, let’s simplify the numerator.

(y[2] – y[1]) = (a*x[2]*x[2] + b*x[2] + c) – (a*x[1]*x[1] + b*x[1] + c) =

= a*(x[2] – x[1])*(x[2] – x[1]) + b*(x[2] – x[1]) = (x[2] – x[1])*(a*(x[2] – x[1]) + b).

The slope is (x[2] – x[1])*(a*(x[2] – x[1]) + b)/(x[2] – x[1]).

Cancel “(x[2] – x[1])” from numerator and denominator to find the slope is “a*(x[2] – x[1]) + b”.

So for a parabola, the average slope between two points depends directly on the ‘x’ values of those points. The slope varies along the curve!

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