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" 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" 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!
Use Limits to Define the Slope at a Point
Define the Terms “Function”, “Absolute Value”, and “Delta-Epsilon Limits”
Writing the change in a variable as “(x[2] – x[1])” is somewhat lengthy.
The notation “y = f(x)” is familiar. It means that “‘y’ is a function, ‘f’, of ‘x’”.
The absolute value of a variable, “A = |a|”, means that “A = -a if ‘a’ is negative, and A = a otherwise”.
The “limit” article introduced the “delta-epsilon” ( ‘δ’-'ε’>>) terminology for the value of a limit. The limit value ‘F’ of “f(x)” at a point ‘X’ means that “|(f(x+δ) – F| < ε whenever zero < |x – X| < δ”. In other words, the computed values for “f(x)” get very close to ‘F’ as ‘x’ gets very close to the target ‘X’.
Define the Slope Function
For a smooth, continuous, real function “y = f(x)”, let’s define “f’(x)” as the function that gives the slope of “f(x)” at every point ‘x’. This definition does not tell us how to determine “f’(x)”, however.
Use a Tiny Slice of the Average Slope
Earlier we had described two points on “y = f(x)” as (x[1], y[1]) to (x[2], y[2]). Now, let’s say that these points are (x, y) and (x + δ, y + ε).
The average slope from (x + δ, y + ε) to (x, y) is (y + ε – y)/(x + δ – x) = ε/δ.
Next, of course, we need to determine ‘ε’ in terms of “f(x)”. The average slope is (f(x + δ) – f(x))/(x + δ – x) = (f(x + δ) – f(x))/δ.
We then define f’(x) as the limit, as δ approaches zero, of the average slope “(f(x + δ) – f(x))/δ”.
Derive the Slope of “f(x) = a*x”
Let’s substitute “a*x” for the “f(x)” function and see what “f’(x)” becomes.
f’(x) = the limit, as δ approaches zero, of (f(x + δ) – f(x))/δ = limit of (a*(x + δ) – a*(x))/δ = a*(x + δ – x)/δ = a*δ/δ = a. Amazingly, “f’(a*x) = a”. This is, of course, the same as the average slope found for the same function.
Derive the Slope of “f(x) = a*x*x”
Let’s substitute “a*x*x” for the “f(x)” function and see what “f’(x)” becomes.
f’(x) = limit of f(x + δ) – f(x))/δ = (a*(x + δ)*(x + δ) – a*x*x)/δ =…
= a*((x + δ)*(x + δ) – x*x)/δ = a*((x*x + 2*(x*δ) + δ*δ – x*x)/δ = a(2*x*δ + δ*δ)/δ = a*δ*(2*x + δ)/δ.
Cancel the ‘δ’ term from numerator and denominator to find f’(x) = the limit as δ approaches zero of (2*a*x + δ) = 2*a*x.
Derive the Slope of “f(x) = a*x^p” at Any One Point
Remember that “x^p” means “evaluate ‘x’ to the power ‘p’”. Since we have explicitly dealt with p={0, 1, 2, 3}, this section is restricted to p>3.
We have f(x) = a*x^p, and want to find f’(x) = limit of f(x + δ)/δ.
Substituting, we have f’(x) = limit of (a*(x + δ)^p – a*x^p)/δ. To expand the numerator requires a small feat of mathematical legerdemain.
The Binomial Theorem
The Binomial Theorem states that:
(a + b)^p = a^p + p*(a^(p-1))*b +… lesser powers of ‘a’ times higher powers of ‘b’ …+ p*(a*b^(p-1)) + b^p.
(This theorem alone deserves an article or three).
Therefore f’(x) = limit of (a*(x + δ)^p – a*x^p)/δ =…
= limit of a*((x^p + (p*x^(p-1))*δ + (lesser powers of ‘x’ times higher powers of ‘δ’) + δ^p – x^p )/δ.
The “(x^p) – (x^p)” will always cancel out.
So f’(x) = limit of a*((p*x^(p-1))*δ + (lesser powers of ‘x’ times higher powers of ‘δ’) + δ^p)/δ =…
= a*δ*((p*x^(p-1)) + (lesser powers of ‘x’ times (higher powers – 1) of ‘δ’) + δ^(p-1))/δ.
When ‘δ’ is cancelled from the top and bottom, we have f’(x) = limit of a*((p*x^(p-1)) + (lesser powers of ‘x’ times (higher powers – 2) of ‘δ’) + δ^(p-2)).
Therefore only the term “(p*x^(p-1)” will remain without any further multiples of ‘δ’. As ‘δ’ approaches zero, all the terms also go to zero except “(p*x^(p-1)”.
Finally, we can say that, for any p>3, the slope of f(x) = a*x^p is the function f’(x) = a*(p*x^(p-1)) = a*p*x^(p-1).
We also note this is the pattern we saw when y=a*x; y = a*x^2; and y = a*x^3. That means we used a variation of Mathematical Induction.
Differential Calculus for Simple Polynomial Equations
Any polynomial function “f(x) = a*x^p + … + b*x + c” has a derivative “f’(x) = a*p*x^(p-1) + … + b”. This derivative is the formula to calculate the slope of the tangent to that polynomial at any given point ‘x’.
References:
Stapel, E. The Binomial Theorem: Formulas. Purplemath. Accessed July 31, 2011.
Thomas, C. Introduction to Differential Calculus. University of Sydney. (1997). Accessed July 3, 2011.
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