[Calculus] Math in Theory

    The third Action Project of this term is not a core class but one of my two new workshops! If you don't know, workshops at GCE Lab school are similar to electives in most other schools. The first workshop I chose for this term is Calculus. Calculus is based on the core understanding that some things are infinite but since infinity can never come to be in real life, we never actually get to see these values. This is why it is math in theory. Using Calculus, we can estimate certain properties at any point on a graph even if the graph is infinite. In this first unit, we have gone over the basics, talking about limits, or when continuing to infinity is pointless because the number we have is so close to where it is going it is basically indistinguishable. We also talked about the derivative. The derivative is attempting to find the slope of a curved line. This is, of course, impossible since the slope changes. The goal of the derivative is to be able to find the slope at any given place on the curve by comparing two very close points. This then makes a Tangent line or a line that matches the slope of the curve at one point and therefore touches the curve for the slightest moment. 

    This is where this Action Project will end up, and I will explain how we get there. It is kind of scary.

    For our Action Project, we have been tasked with using some randomly generated numbers to create an interesting graph. The numbers I was given are 2 and 3 (I got very lucky) and the equation I made is f(x)=cos(3x^2). This makes a line that looks like this:

    In my opinion, this is a very interesting graph as the lines get closer and closer together to infinity while never actually touching. It may appear that they are touching but if you zoom in enough you can see the space in between. We were then tasked with evaluating the function at an x-coordinate of our choosing. I chose 0.001. When plugged into the equation, it is 0.999999999996, or about 1. This means that when the line is at 0.001 on the x-axis, it is at about 1 on the y-axis. 

    The next step of this process is to estimate the instantaneous slope, or the slope between two points very very close to a center point. This is to estimate the slope at that center point. For the center point of (0.001,1), I chose the exterior points (0,1) and (0.002,1).

    From here we can plug these numbers into the slope formula which leads to the slope being 0. This is because it is a horizontal line and the slope of those is always 0. This is backed up by plugging it into the derivative. The derivative is most often found by using the power rule, which essentially means to multiply the coefficient of x by x's exponent and then subtract the exponent by one. This equation uses cos so it is slightly more difficult to find the derivative. This equation's derivative is f’(x)=(-6x)sin(3x^2).  We get that using the chain rule. That looks like this:

By plugging in the x value to the derivative we back up our answer from before. Using this we can create a tangent line by plugging all of this into the basic slope equation y=mx+b. This creates y=0x+1 which looks like this:

In conclusion, calculus is very fun! It is very confusing and it makes my brain hurt but that is a good thing. I thought this Action Project was very cool because it allowed us to mess around with what we had learned in calculus. If I were to have done one thing differently, it probably would have been choosing an easier equation because I don't fully understand the chain rule. Other than that, this was a very fun Action Project and Unit!