In high school, I loved math. I enjoyed everything I learned in all of my math courses over the four year period. I rarely had a hard time with the subject, and it seemed to come somewhat naturally to me. My senior year, I took Advanced Placement Statistics rather than Pre-Calculus, as I love the Stats teacher, and heard that it was a great class. I did well in the class, but the problems began after, when I went into college Math a semester or even a year behind most students in my Pre-Calculus class.
Pre-Calculus and Calculus were difficult. Right from the beginning, I had a hard time on the exams, and found myself getting behind. It wasn’t that I disliked the material, or didn’t understand it, it was mostly that I could not get into it. For whatever reason, Calculus just bored me. I found it very difficult to do well in a course that I could not relate to in the slightest.
Derivatives were a huge part of Calculus, and for the most part, I could not do them at all. I would listen in class and feel that I understood, however when I went to do the homework later that night, it turned out that I had no clue what I was doing at all. And this became apparent on my tests and quizzes. I struggled to understand derivatives as we learned about them each and every day, adding more and more to the list of things I did not understand such as the chain rule, quotient rule, and product rule.
Finally, in chapter 3, came the section where I could slightly relate. It went from being, in my mind, just a bunch of numbers and variables, to real life situations. Problems about a ladder leaning against a wall and bacteria multiplying showed me that these derivatives were not just math problems which I had no clue how to solve, but they could actually be put to use in the real world. All it took was this section of the book on applying the skills from prior lessons to get me ‘into’ Calculus.
One of the first problems we did in our notes that dealt with applying these math skills to real life was :
“A ladder 10 feet long rests against a vertical wall. Suppose the bottom slides away from the wall at a rate of 1 ft./s. How fast is the angle between the ladder and the ground changing when the bottom of the ladder is 6 feet from the wall.”
In this problem, we were to draw a triangle, label each side, and solve for dx/dt. In order to do so, it was necessary to implicitly differentiate with respect to ‘t’. This is where the derivatives and implicit differentiation came in. It was during this problem that I realized just what this kind of math was useful for.
Upon seeing a problem like this, it became clear to me that derivatives CAN in fact be useful for something other than Mr. Rohal’s exams. It was in about section 3.7 or 3.8 of the book that I actually began to understand Calculus.