Mathematics, and in particular calculus, is treated as arcane knowledge known only to the very smart or professional mathematicians, and opaque/useless to everyone else.
In Real Life, calculus is the opposite of arcane in STEM; it is actually only a basic freshman-level class in most undergraduate STEM degrees. For undergraduate engineering and physics students in particular, it is the beginning of a 4 or 5-year-long journey to much more advanced mathematical concepts such as ordinary and partial differential equations, vector calculus and complex analysis, and these topics are in turn the base of branches of science such as statistics, electromagnetism, quantum mechanics, cosmology, chemistry and signal analysis. However, Writers Cannot Do Math. Most writers are people who felt much more attracted to the warm, flexible, humane and passionate arts and humanities rather than the cold, hard, mechanical and stoic STEM. Other writers had the misfortune of having had bad math teachers who left them hating exact sciences forever, others just plain fail at handling higher mathematics. Whatever the reason, this translates to most writers having made sure to take as little math in basic school as possible. Per the standard American high school curriculum, mathematics become an optional subject around the time when calculus is taught, so for anyone who quit the subject at the first opportunity, it's simply the hardest topic they're familiar with.
Because of this, in media works "calculus" is often used as shorthand for "brain-hurty smart people stuff". This is somewhat of an exaggeration—beginning calculus is not particularly difficult as mathematics goes, and is the foundation for much of the true higher maths and physics. And everyone's brains use the processes represented by calculus in their everyday lives (the derivative is the rate of growth or decrease, the integral is the cumulative total); what makes it difficult is substituting number-crunching for intuition. However, there are different kinds of intelligence, and mathematics that seems simple to one person can be entirely opaque to another.
Compare E = MC Hammer, which is when mathematical equations (often gibberish) are displayed to imply that math is occurring, and Mouthful of Pi, where brilliant mathematicians know pi to a ridiculous number of places. See also Formulaic Magic.
Should you want to see the "math path" that leads to calculus from basic math, check out the Useful Notes page for Math Curriculum.
Examples
- Inverted in the Itazura Na Kiss episode, "The Crucial Moment! Class F's Winter Battle", where every high school student tutored by Irie over a very short time period, even the usually dimwitted Kotoko, manages to pass their Calculus section of the college entrance exam. Everyone, save Kinnosuke, who was too proud to get tutored.
- In the English language version of Tintin, we have Professor Calculus, apparently named so to indicate just how brilliant he is, as the direct translation "Professor Sunflower" just wouldn't have the same ring.
- In the [1] fic what good is a star that has lost its light?, struggling with his calculus homework is the first sign that Yukio's mind isn't as sharp as it used to be.
- Joke: Two professors are arguing in a restaurant about whether calculus was obscure or well-known. The "obscure" professor went to the men's room. The "well-known" professor asks the waitress to help him play a joke on him — when he came back, he would call her over, and ask her a question, and she should answer, "One half x squared." He comes back, the professor asks him if the waitress knowing calculus would prove it, and the "obscure" professor agrees. He calls over the waitress and asks for the anti-derivative of "x". She answers, "One half x squared — plus a constant."^{note }
- Justified in Neal Stephenson's The Baroque Cycle — it takes place somewhere around the 18th century, when calculus was arcane knowledge and the most advanced topic in mathematics. As it actually happened, two of the main characters^{note } are actively engaged in inventing it separately.
- In Relativity, this trope is averted: Yes, technically, Michael (college student) is helping Kelly (platonic friend, another college student) with calculus. But rather than just say "he's helping her with calculus," the narrative specifically mentions that he is helping her find the line integral of a vector field.
- Alan Sokal's famous Stealth Parody paper Transgressing the Boundaries: Towards a Transformative Hermeneutics of Quantum Gravity makes a Running Gag about nonlinear mathematics showing the way to a new postmodern consciousness. A footnote commenting on a rather confused passage by Robert Markey incorrectly describes complex number theory as "a new and still quite speculative branch of mathematical physics", while other footnotes buffoonishly read a "pro-nuclear-energy worldview" into a book on Radon measures and liberal social attitudes into the equality and choice axioms of Zermelo-Fraenkel set theory.
- In David Brin's Uplift series practically all alien races consider any human mathematics more complex than arithmetic to be "arcane wolfling superstitions", even the Tymbrimi-created AI on the Streaker says so. Since they were all uplifted from the pre-stone age to interstellar tech by an older race, they can simply use computers to brute-force any difficult mathematical problems.
- The Big Bang Theory. Four of the five main characters know advanced physics, and the math behind it. But most non-scientists on the show seem to have not even gotten to linear algebra... which makes sense, since linear algebra is traditionally learned after calculus and differential equations. Calculus would be preceded by college algebra and trigonometry, commonly taught as "pre-calculus".
- Family Matters also does this in one episode, where Laura complains about how difficult her calculus test was. And Urkel, of course, rants about how easy calculus is.
- One of the things that, canonically, Chief Engineer Commander Tucker cannot do in Star Trek: Enterprise.
- A case of Writers Cannot Do Math, as a lack of knowledge of calculus would make Tucker a very poor engineer. Another episode had him unable to solve a very simple Train Problem.
- A couple of centuries thence, even ten-year-old children are expected to have a basic knowledge of calculus, as an episode of Star Trek: The Next Generation shows.
- Young Sheldon: In "A Rival Prodigy and Sir Isaac Neutron", Paige asks Sheldon if he knows "how to differentiate under the integral sign?" Smart as he is, Sheldon hasn't learned calculus yet.
- From the song "White and Nerdy", by "Weird Al" Yankovic: "I do vector calculus, just for fun."
- Also, "Yo, I know pi to a thousand places!"
- During Weird Al's turn as Sir Isaac Newton on Epic Rap Battles of History, he stumps Bill Nye with the question "The integral sec y dy from zero to one-sixth of pi is log to base e of the square root of three times the sixty-fourth power of WHAT?"^{note }
- Paranoia supplement Acute Paranoia. In a section on mathematics tests, calculus is given as an example of "Very Complex Mathematics". The only category higher than that is "Impossible Mathematics", such as "What is the final digit of pi?"
- The Major General in The Pirates of Penzance cites calculus as part of his educated background in his opening song. Then again, the comedy of the song stems from the fact that pretty much everything the Major General brags about is either trivially easy or flat-out impossible.
- In Mass Effect, the biologist Mordin Solus uses a species' ability to perform calculus as a rule for ethical experimentation. Any species that can is off limits.
- In Puyo Puyo, Ringo applies this trope in quite a literal way, as her magic spells (save for her fourth spell in the English dub, which is instead called Algorithm, as well as her Counter spell, and all her chants in her Puyo Puyo Tetris 2 alternate voice set) are all mathematical concepts: Sine, Cosine, Tangent, Integral (Japanese-only) and Permutation (Calculus in the English dub). She even performs the first two spells by using the right-hand rule.
- "Look lively, people! Spot calculus check!◊" Bob the Angry Flower uses calculus as a measure of whether someone's smart enough to be worth his time (given how often he's been portrayed as a Mad Scientist, it shouldn't be surprising that Bob knows higher mathematics). Notice that, while other people seem to have half-forgotten their calculus, they still have studied it at some time in the past.
- In Questionable Content, while Raven is working out the topology of Marten's penis and jeans on the board, Faye mutters "is that calculus?"
- In Epic Rap Battles of History, Neil deGrasse Tyson shows off by solving a problem posed by Isaac Newton, and this is treated as an impressive achievement. The problem? To find the integral of a secant.
- In an episode of Dexter's Laboratory, Dexter quips, "this will be easier than calculus!"
- In an episode of Futurama, "The Duh-Vinci Code", Professor Farnsworth says something along the lines of, "Oooh, I'm going to go listen to that calculus lecture!"
- In Kim Possible, the Mathter, despite being a mathematics-themed villain, seems to possess no knowledge of any math above basic algebra. He is understandably beaten by Ron Stoppable's father, who is an actuary and thus actually has knowledge of math of calculus level and beyond.
- Total Drama Revenge of the Island has the Jersey Shore reject character, Anne Maria, mention that she's "no algebra whiz" when referring to a numbers code clue.