Calculus Using Algebra: Part 1

Teaching a month's worth of pre-calculus using only algebra

Calculus has a reputation as one of the hardest math topics in the world, a topic so difficult and complex that is has the 2nd-highest fail rate of 40% out of all math courses. This post will take this difficult course and transform it's introduction into a simple and speedy 5 minute read!

Calculus's introduction covers two of the most fundamental algebra notations: the derivative and the integral. Let's take a look at the derivative now, and Integrals in part 2!

The Derivative

Imagine you are riding a bike with a speedometer attached to it. As you speed down a never ending hill, your speed accelerates in a quadratic manner. Plotting x=time after start and y=speed, you realize that speed=time^2, meaning a graph of your bike would be y=x^2!

Here's a question. At the exact moment, J seconds after you've taken off, what is your speed at that exact time? Let's look at how to solve this, splitting it into 2 parts!

The Geometric Part

Obviously, you can't calculate the slope of a quadratic. Or can you? As we zoom into the "curve" of the parabola, we get an infinite number of infinitely small straight lines, each connected and turning, making up the curve. Looking at the infinitesimal line at the x = J mark, and taking the slope of that infinitesimally small line, we get the derivative. But as it is hard to calculate the slope of an infinitesimal line, how do we do it? Well...

The Algebraic Part

Let's call the length E, or Epsilon. Since the length of the line is infinitesimal, Epsilon is also infinitesimal. Calling the starting X of the line a, the ending x is (a+E). Since this is a quadratic, the starting Y is a^2 and ending is a^2+2ae+e^2. In calc, we ignore any terms with an epsilon power >1 since as E approaches 0 E^2 is too small to be counted. Therefore, the change in Y is 2ae, and 2ae/e=2a, so that is our derivative!

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