Differential Calculus Introduction: Simple Polynomial Equations

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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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