2022-12-20

Reflections on Calculus

Simple reflections

When you study Algebra, you study the rules of equations and polynomials. Algebra is used when there is some unknown variable or number in an equation.

Geometry deals with the study of shapes like triangles, squares, hexagons, and polygons.

Calculus, while using techniques taught in both algebra and geometry, focuses on the rates of change of functions and the area and slope of curves.

For example, say you have an equation that, when graphed, is a straight line y = 3x - 5.

Using the slope intercept formula taught in algebra tells you the slope of that function is 3.

So what do you do if you need to calculate the slope of a parabolic function, like y = x^2 + x - 3? This is where calculus comes in.

First, you take the derivative of the function. The derivative of y = x^2 + x - 3 is f’(x) = 2x + 1. Once you have the derivative function, you take the point you want to find the slope of and plug it in. If the point is (1, 3), you’d calculate f’(3) = 2(1) + 1 = 3 (This function is saying “when x is 3, what is y?. So at the point (1, 3), the slope of our function is 3.

Calculus also becomes useful when you want to find the area under a curve. A classic problem is where you have a plot of land that borders a river on one side, and a straight road on all the others. Since this isn’t a proper square, the formula for calculating the area of a square won’t work, but if you have the function of the curve of the river (which I know is unlikely to come on the deed of the property), you can calculate the area.

Another part of calculus deals with the instantaneous rate of change of an object. Let’s say you were going on a road trip and you were traveling at a constant speed of 60mph for the entire trip. If the trip took two hours, it’s easy to calculate that we went 120 miles.

But if your speed wasn’t constant and instead averaged at 60mph, which meant you were going 70mph at some points and 50 mph at others, you can use calculus to determine how fast you were going at any point in time on your trip. Pretty cool!

Reflections on how I learned it

I took Math 104 (Calculus 1) on Study.com as a prerequisite to a Computer Science program I’d like to attend. I was accepted, but just needed to get this course out of the way. It was…daunting…to say the least. I hadn’t taken a proper math class in about a year and the last one I took was more of a logic-math, not pure math.

Surprisingly, I really enjoyed it. I told myself all through high school that I was bad at math (Algebra was the only class I got a C in) and had always been more of a words guy. Turns out I was wrong. I simply didn’t have the desire to get good at math, nor have the resources available to help me when I was stuck. Though I had YouTube, I didn’t really know how to use YouTube for help at the time. The main limiting factor, though, was my desire: there wasn’t any. I had no need for algebra in my life and didn’t see how it would be useful for my career as a video editor/graphic designer. But now I have the desire.

As my curiosity has grown over the last few years, I’ve become more interested in science and especially physics. Most of physics is math. I also am pursuing a degree in Computer Science, which is a math heavy degree. Therefore, in an effort to understand armchair physics better and graduate with a degree in Computer Science, my desire to get good at math sprouted.

Study.com was helpful because after each video I had to take a 5 question quiz. If I got more than one answer wrong, I was forced to retake it in order to get actual credit for the course. This rapid learn-then-test-yourself environment prevents you from fooling yourself by saying you understand something when you really don’t.

The course had 124 quizzes, which I had to get an 80% or better on. When I finished the quizzes, I had to take a proctored exam. The exam was easier than I expected, thanks in no small part to the unlimited practice exams I could take. To study for the exam, I took a practice exam, saw what types of questions I didn’t understand well, study those, retake the exam, and repeat until I felt ready to take the final.

The downside of taking the exam on Study.com is that it takes two weeks to get your official results back because proctors watch the video back to ensure there were no shenanigans. It makes sense, but I feel like they could give you the official score and then come back if there are any problems. I also feel like you could easily train an AI model to exam the videos and flag any odd behavior for a human to double-check.

In conclusion

I’m excited I did it. I’m excited I learned. I’m excited for the rest of my learning journey. Thanks for following along.

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