 Image from John Atkinson at Wrong Hands

Every May, the College Board offers four advanced placement exams for high school physics students: Physics 1, Physics 2, Physics C: Mechanics, and Physics C: Electricity and Magnetism. Physics C tests on proficiency in calculus-based physics; Physics 1 and 2, in algebra-based.

Sir Isaac Newton originally studied the concepts that would become calculus to solve problems related to physics, so it makes sense to begin a discussion about applications of calculus with physics.

One could solve physics using just algebra and trigonometry, but that track requires many more lines of calculations (and a good deal more frustration) than using calculus. Image from SlidePlayer

Take the image above, for example. With a perfect right triangle slope, given the mass of the box, the constant of friction, and the magnitude and direction of the slope, one could use simple algebra and trigonometry to solve for the amount of force required for the boy – we’ll call him Henry – to push the box to the top of the slope. Here, we assume that the slope and force expended does not change

With the second, undulated slope, one could apply algebra and trigonometry again, but it would yield, at best, an under-approximation, because the slope changes, so the amount of force Henry needs to expend changes. The derivative, which expresses such rate of change, greatly simplifies the process of solving for the force required of Henry. Both physicists and engineers use calculus-based physics in their calculations. Calculus misconceptions (Art by Bill Watterson)

Economists and businessmen may also use calculus to determine maximum profit by graphing the curves of marginal profit and marginal cost, finding the points where they intersect, and solving for the absolute maximum value.

These optimization calculations also apply to construction. Given a certain length of fence material, one could determine the arrangement of a rectangular fence that yields the most area. (I, for one, have seen a problem or two in the math sections of the SAT and ACT that ask students to solve for just that, though not necessarily with calculus.)

Biochemistry employs differentiation and integration in kinetic rate problems for chemical reactions, similar to kinematic physics, and biology in measuring the rate of growth of a bacteria culture in different conditions. In the development of medication, this can help researchers develop the right treatment to combat an illness. Mike from Pixar’s Monsters Inc. (Image from Pixar)

Calculus has applications outside of scientific fields, too. Graphic artists need calculus to manipulate the new parabolic functions that create the digital world. Calculus calculates the necessary shape and function for a character or image to appear real and helps graphic artists predict the movement of computerized constructs under differing conditions.

Whether one likes calculus or not, its application impacts the modern world in numerous, diverse fields. From Newton’s first application of calculus to kinematic physics problems in the 1700s to computer technology in the 21st century, calculus has demonstrated its flexibility and multifaceted usefulness through the ages.

Thank you for joining me on this short exploration on the history of this joyous mathematics study.

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