I've taken a calculus class for a semester at university as a mandatory part of my curriculum. In this class, we were taught formal definitions of what are derivatives and integrals, how to calculate them, and shown some of their generic applications (locating the local extrema of functions, finding the area of a shape enclosed between two curves etc.) However, despite finishing that course and passing the exam, I feel that I'm still missing the most important thing to know - when to use it. After the calculus class came a physics class, and whenever the lecturer told us that a problem is best solved by applying an integral, I simply took his word for it, being unable to understand why this was.

What I'm interested is not examples of common uses of calculus (I can easily find plenty with a simple web search). Rather, what I would like to know is how to tell if a solving a particular problem will require using calculus, when it isn't one of the commonplace scenarios that I brought up in the first paragraph. In the case of algebra, there are certain keywords that are helpful in figuring out what to do in a task (add, sum, increase imply addition; times, product imply multiplication, and so on). Are there similar keywords that hint that a task is to be solved by calculus? And if not, can this be deduced from the problem's description in other ways?

I am in awareness that after taking a calculus class and solving different tasks that are representative cases of their usage, I should be able to recognise other situations where it is necessary. Unfortunately, this simply went over my head, and now upon being told that a problem is best solved with calculus, I sit pondering how to come to such a conclusion, to no avail.

  • $\begingroup$ At the risk of sounding insensitive, the main places I use calculus are situations where I need to do something already listed (locating the local extreme of functions, etc.). For example, if a particle moves with speed $v(t)$ at time $t\in[0,1],$ then I might visualize this as a graph with $v$ on the $y$-axis and $t$ on the $x$-axis. The answer is then the area under the graph, i.e., $\int_{0}^{1}v(t)\,\mathrm{d}t.$ Similar reasoning applies to momentum and impulse, current and charge. Could you give an example problem, in the answer to which calculus is used to do something not listed? $\endgroup$ – Will R Aug 20 '16 at 10:06

The two words that most indicate that a word problem is to be solved by calculus are minimum and maximum. Examples of these problems include maximising the area in a box or minimising the packaging that box uses – usually this is a non-trivial function of some variable, and that requires calculus.

Another type of problem which almost certainly requires calculus is finding the area of a curve swept out by some object in the plane and bounded by other planes, or the volume of a solid generated by rotating a curve around some axis. The volume of Gabriel's Horn is an example of the latter (it works out to $\pi$) and a problem about a rotating pentagon in a square I solved on this site is an example of the former.

In general, anywhere something changes and we have to find overall properties about that change, calculus is likely to step in.

In my own experience, the reason why a suggestion of "use the Calculus" gets echoed and accepted without pause across the room is because it is almost too powerful for the high school or college-level problems it is meant to solve. Yet its uses outside the classroom are far more profound.

Calculus underpins almost all mathematics developed since the 18th century, having a first great blossoming in the latter half of the 19th century when Weierstrass and others developed real and complex analysis. Interest dipped for a while with the set theory craze of the 1900s and 1910s, but then analytic number theory was developed in the 1920s by Hardy, Littlewood and Ramanujan, and integrals were thrust back into the spotlight.

Given this history, it might have seemed then that the techniques were going to be confined in the doors of academia. But numerical methods to evaluate otherwise intractable integrals and differential equations were developed in the 1950s and 1960s, and with the development of the first computers happening around that time, suddenly everybody could harness the power of higher mathematics! News agencies got better weather predictions, buildings were built taller and in more unusual shapes. We sent humans to the moon, developed accurate models of ourselves and so on and so forth – and all these require calculus. Naturally the need to educate people about the burgeoning uses appeared, and it was included into curricula like yours.

As someone who's got a 5 in AP Calculus BC and Statistics and Chemistry, I acknowledge the ubiquity of this branch of mathematics in our modern world. But I still chuckle a bit when someone says that calculus is needed for a problem that clearly doesn't require it, like "What's the volume of this apple?" (To which we fill up a pail and throw the apple in.)

  • 2
    $\begingroup$ Better fill the pail with a liquid lighter than water, because apples float. :-) $\endgroup$ – David K Aug 20 '16 at 12:04
  • $\begingroup$ Thank you for the answer, as well as the informative digression on the background of calculus. I'll be on a lookout for those words and analyse the pentagon problem you have pointed me to. I assume the best way to develop this intuition is to just solve more problems that need calculus, and with time it will work out into a "sixth sense" of sorts for the less explicit problems. $\endgroup$ – cobalt_blue Aug 20 '16 at 14:58
  1. When you try to "aggregate" something (like calculate the potential of a charge distribution on spheres. More examples include Fourier transform, expectation of a random variable, and stochastic integral)
  2. When you do optimization (in economics, engineering, or applied math), you use derivatives.
  3. When you want to "manipulate" a large family of some objects that are "abundant and continuous" (I'm being very loose here), like functional calculus.

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