This week, we plan on explaining what the Fundamental Theorem of Calculus is and what it means, including why it is important, by using a specific example from baseball to illustrate our point.
As the name suggests, the Fundamental Theorem of Calculus is, well, fundamental to calculus and its applications. Naturally then this theorem was discovered and proved before calculus was even invented, by a famous Isaac, but not the one you think.
Interestingly, the Fundamental Theorem of Calculus was first proved by Isaac Burrow, a professor of Isaac Newton, in his work studying how infinitesimal quantities can be used to calculate tangent lines. Isaac Newton, and Gottfried Leibniz as well (not to go into the whole who invented calculus debate), built upon this groundwork to fully develop and invent what we know now as calculus.
As a last bit of history, although Riemann sums are usually taught in calculus classes before actual integration, Riemann sums were invented almost two hundred years after the Fundamental Theorem of Calculus. Riemann sums, named after Bernhard Riemann, were developed as a rigorous formation of integrals, because apparently Riemann loved making things more difficult and unsuitable for theoretical purposes. Riemann also was a contemporary of other famous mathematicians studying complex analyses at that time, such as Gauss and Cauchy whom you may recognize from other mathematical concepts.
The Fundamental Theorem of Calculus in essence codifies the relationship between a derivative and an integral. It comes in two parts, so not only does it describe how integration and differentiation are connected, but it also defines exactly what a definite integral is and does. The first part regards the derivative. It states that for a continuous real-valued function f, over a closed domain f: [a,b] where (-infinity < a < b < infinity) then for each x in [a,b] the following is true:
Which basically states that the derivative of the integral of f at x is just f(x). This part states that the derivative of the anti-derivative cancels each other out. This allows us the ability to convert one to the other by integration or differentiation respectively.
The corollary, or the second part of the theorem, states that for a continuous real-valued function f, over a closed domain f: [a,b] where (-infinity < a < b < infinity) for which an anti-derivative F is known, then the following is true:
Which basically states that the integral of a function over a closed domain is equal to the anti-derivative of the greater endpoint minus the anti-derivative of the lower endpoint. Additionally, this proves that when an anti-derivative F does exist, then there are infinitely many anti-derivatives for f, which is obtained by adding an arbitrary constant to F. Also, by the first part of the theorem described above, anti-derivatives of f always exist when f is continuous.
In essence, the Fundamental Theorem of Calculus simply asserts that the sum of infinitesimal changes in a quantity over time, or over some equivalent variable, adds up to the net change in the quantity over the original domain. Most examples of this use speed over time to calculate distance. With spring training finally starting, we are going celebrate by applying the FTC to a typical baseball pitch. This analogy will take an almost out-of-body experience.
Imagine you are relaxing on top of a baseball. Since you are sufficiently tiny to perch on such an inanimate object, you are unaware of how far away you are from home plate. However, you are equipped with a tiny stopwatch and a tiny speedometer. Naively, you think that stopping both the stopwatch and the speedometer right before you hit the glove after a pitch would help you calculate the distance. Unfortunately, your reflexes are pretty bad at this, since you always seem to be screaming and not paying attention as you hurtle rapidly towards the glove.
Luckily, you can apply the Fundamental Theorem of Calculus to calculate distance traveled. Additionally, you know that distance is simply the anti-derivative of velocity with respect to time. In other words, the distance traveled equals:
As such, on the next fastball that Jake Arrieta throws towards home, you record your consequent speed from the speedometer at every dt moment of time. Even though you cringe and forget to record the final dt speed right before you hit the glove, you can add up the velocity multiplied by the dt for each dt recorded and calculate an estimate for the total distance travel. In this instance, you calculate the distance at about 60ft 6in, which in fact is essentially the distance you did travel!
Isaac Newton, back in the middle of the 17th century, would never have guessed that his invention of calculus would lead to analysis of sports, let alone provide analysts of sports teams with a competitive advantage. Also, full disclosure, we are sure Isaac Burrow though did have a hunch that his star pupil Isaac Newton would surpass him in both fame and grandiosity. Yet in fact, the Fundamental Theorem of Calculus probably still has far reaching applications, even to baseball, that will not be discovered until far into the future.
The SaberSmart Team