50 Fun Facts About Calculus

The word calculus comes from the Latin for which small object, once moved around Roman counting boards?

Pebble is the literal meaning: Romans did their sums by sliding little stones along a board. Doctors still use the same word for a kidney stone.

The fourth, fifth and sixth derivatives of position are jokingly named after the mascots of which breakfast cereal?

Rice Krispies supplied the names, borrowed from the trio of cartoon characters on the box. The joke is said to have been started by a NASA engineer working on the Apollo programme.

Zeno's paradox, later untangled by the calculus of infinite series, has swift Achilles forever chasing which animal?

Tortoise is the answer: given a head start, Zeno argued, it can never be caught, because Achilles must first reach where it was. Endless shrinking steps can still add up to a finite distance.

Isaac Newton developed much of his calculus at home in 1665–66 while which disaster kept Cambridge University closed?

The Great Plague shut the university just after Newton took his degree. In two years at the family farm in Woolsthorpe he also built his theories of optics and gravitation.

Dentists use the word calculus for which tooth problem, scraped away during a professional cleaning?

Tartar is plaque that has hardened with minerals from saliva. Once it has formed it is too firmly attached for a toothbrush to shift, which is why a hygienist uses a scaler.

Which 1988 film stars Edward James Olmos as a teacher coaching East Los Angeles students through AP Calculus?

Stand and Deliver tells the true story of Jaime Escalante, whose 18 students at Garfield High passed the exam in 1982. Olmos earned a Best Actor Oscar nomination for the part.

Gottfried Leibniz, co-inventor of calculus, also championed which number system that now runs every digital computer?

Binary, which writes every number using only two digits, was a system Leibniz kept returning to throughout his career. The English astronomer Thomas Harriot had devised it decades earlier.

Germany's Bahlsen company has sold which snack, named after calculus pioneer Gottfried Leibniz, since 1891?

A butter biscuit with 52 'teeth' around its edge carries his name. The only link is that Leibniz was one of the more famous residents of Hanover, where Bahlsen is based.

Professor Calculus is the hard-of-hearing inventor who builds a Moon rocket in which Belgian comic series?

Tintin is the series: the absent-minded professor first appeared in Red Rackham's Treasure in 1943. In Hergé's original French he is Professeur Tournesol, which means sunflower.

The squeeze theorem for limits in calculus is also nicknamed after which lunchtime food?

Sandwich theorem is the other name, because a function trapped between two others must share their limit. It is also known as the two policemen and a drunk theorem.

The derivative of a function at a point equals the slope of which line drawn to its curve there?

Tangent is the line: it grazes the curve at a single point and shares its direction there. Its name comes from the Latin word for touching.

A basic Riemann sum in calculus estimates the area under a curve by adding up thin strips of which shape?

Rectangles do the job, which is why the technique is also called the rectangle rule. The thinner the strips, the smaller the error, and an integral is what the total settles on.

In calculus, the second derivative of an object's position with respect to time gives which physical quantity?

Acceleration is how quickly velocity changes, and velocity is itself how quickly position changes. Differentiating twice is the reason its unit carries a 'squared'.

The third derivative of position over time, which engineers minimise on roller coasters, has which abrupt name?

Jerk measures how suddenly a push comes on or eases off. Railway curves and roller-coaster loops are laid out as gradual spirals to keep it low, and lift designers work hard to tame it.

Calculus students famously lose marks for forgetting which letter, added to the end of every indefinite integral?

C stands for the constant of integration. Differentiating wipes out any fixed number added to a function, so working backwards you can never tell which number was there.

Using the power rule of calculus, what is the derivative of x squared, the function whose graph is a parabola?

2x is the result: the power rule brings the exponent down in front and lowers it by one. So the parabola's slope is 6 where x is 3, and it is flat at the very bottom.

In calculus, taking the derivative of the sine function gives which other trigonometric function?

cos x is the derivative of the sine, because a sine wave climbs fastest exactly where the cosine is at its peak. Differentiate four times and you are back where you started.

In calculus, a function whose derivative is zero everywhere has a graph of which simple shape?

Horizontal line is right: a derivative measures steepness, and a flat graph has none. That is why differentiating a constant such as 7 leaves nothing behind.

Which technique in calculus differentiates a function nested inside another, working from the outside in?

Chain rule is the name: differentiate the outer function, then multiply by the derivative of the inner one. In Leibniz's notation the dy and dx terms look as if they cancel like fractions.

Which function is famous in calculus for being its own derivative, with a slope always equal to its height?

e^x is the one: apart from constant multiples of it, no other function equals its own derivative. That makes it the model for anything that grows in proportion to its size.

Isaac Newton never called his invention calculus; English schools long knew it as the science of what?

Fluxions was Newton's word for rates of change, and the name lasted in English schools into the 19th century. His treatise on the method, finished in 1671, was not published until 1736.

Gottfried Leibniz drew his integral sign ∫ as a stretched letter S, standing for which Latin word?

Summa was the word he had in mind, because he pictured an integral as endlessly many tiny pieces added together. The shape is the old-fashioned long s once used in print.

Which ancient Greek from Syracuse found the area under a parabola many centuries before calculus was invented?

Archimedes cut the region into endlessly many triangles whose areas shrink in a fixed ratio, then added them all up. His method of exhaustion is integration in all but name.

Which London body ruled for Isaac Newton in the calculus priority dispute, in a report Newton wrote himself?

The Royal Society was hardly neutral ground: Newton was its president at the time, and the committee it set up never asked Leibniz for his version of events.

Which Irish bishop mocked the vanishing quantities of early calculus as ghosts in his 1734 attack The Analyst?

George Berkeley addressed his pamphlet to an 'infidel mathematician', and the critique went on to influence how mathematics developed. The Californian city and its university are named after him.

Which Swiss mathematician began using the letter e for the constant 2.718… that runs all through calculus?

Leonhard Euler first wrote it in the late 1720s, in an unpublished paper on explosive forces in cannons. Nobody knows why he chose that particular letter.

Isaac Newton's 1687 book, which used calculus-like methods to explain planetary orbits, goes by which short Latin title?

Principia keeps its calculus out of sight: Newton set out his arguments as geometric propositions about vanishingly small shapes instead of using the new methods openly.

Isaac Newton's notation for a derivative, still common in physics, places which small mark above a letter?

A dot goes over the changing quantity, and a second one means differentiate again. The style has been rudely nicknamed flyspeck notation.

The Kerala school found key infinite series two centuries before European calculus in which present-day country?

India is where Madhava of Sangamagrama founded the school in the 14th century. Its results were set down in Sanskrit verse, with the proofs supplied later in a work written in Malayalam.

A mistranslation of Maria Agnesi's 1748 calculus textbook gave one of her curves which spooky English name?

Witch of Agnesi is what the curve is still called. Her book used the Italian name versiera, a word about turning, but translator John Colson read it as avversiera, a woman who is against God.

Which shape in calculus has a finite volume but an infinite surface, so paint could fill it yet never coat it?

Gabriel's horn is named for the archangel said to blow a trumpet on Judgment Day. Evangelista Torricelli first studied the figure in the 17th century.

Which calculus result guarantees that a car averaging 60 mph on a trip was doing exactly 60 mph at some moment?

Mean value theorem is the one: on any smooth journey the speedometer must at some instant match the trip's average. Cover a stretch too quickly and you were certainly speeding somewhere.

Which result in calculus shows that differentiation and integration are inverse operations that undo each other?

Fundamental theorem of calculus is its full name. It turns the hard problem of measuring an area into the easier one of running a derivative backwards.

Now a business buzzword, which calculus term names the spot where a curve switches from bending up to bending down?

Inflection point is the term: the curve's steepness stops increasing there and starts decreasing, or the reverse. On a rising sales chart it marks the moment growth begins to slow.

In calculus, the area under the curve y = 1/x, measured from 1 to a, defines which function of a?

Natural logarithm is the answer, which is why integrating 1/x breaks the usual power rule. It was first met as the hyperbolic logarithm, because xy = 1 is the equation of a hyperbola.

Integration by parts, first published by Brook Taylor in 1715, mirrors which technique for differentiating?

Product rule run in reverse gives the method, which trades one integral for an easier one. Students often choose which part to differentiate with a rule of thumb called LIATE.

Which Greek letter stands for the change in a quantity in the slope formulas of calculus?

Delta is the first letter of the Greek word for difference, which is why it labels the rise and the run of a slope. River deltas take their name from the capital letter's outline.

Adding 1/2 + 1/4 + 1/8 + 1/16 and so on forever, as calculus allows, gives a total of exactly what?

One is the exact total: each new term covers half of whatever gap is left, so the running sum creeps as close as you like without overshooting. Picture shading half a square, then half the rest.

The endless sum 1 − 1/3 + 1/5 − 1/7 + …, named after Gottfried Leibniz, equals a quarter of which constant?

Pi divided by four is where the sum is heading, but it crawls: more than five billion terms are needed for ten correct decimal places. James Gregory found the series two years before Leibniz.

A classic related-rates problem in calculus asks how fast the top of which leaning object slides down a wall?

A ladder is the textbook favourite: as its foot is pulled away at a steady speed, the top drops faster and faster. Pythagoras links the two speeds, since ladder, wall and ground make a right triangle.

Which French lawyer, famous for a 'last theorem' scribbled in a margin, explored key ideas of calculus before Newton?

Pierre de Fermat was a magistrate by profession. His method for finding maxima and minima, circulated in the 1630s, came decades before Newton or Leibniz published anything.

Differentiating under the integral sign is nicknamed after which American physicist, who praised it in his memoir?

Richard Feynman learned it in high school from an old textbook, Advanced Calculus by Frederick Woods. At Princeton it let him solve integrals that were otherwise hard to crack.

Johannes Kepler anticipated integral calculus in a 1615 book about measuring the volume of which containers?

Wine barrels caught his eye in 1613, when he bought wine for his household and saw a cask gauged with a single diagonal rod. In German, Simpson's rule is still known as Kepler's barrel rule.

Calculus textbook author James Stewart spent his royalties on Integral House, a curved mansion in which Canadian city?

Toronto's Rosedale neighbourhood is home to the house, named after the long curly sign used in calculus. Stewart asked his architects for curves and a large, flexible space for performances.

Calculus proved that a rope hung between two points takes which curve, also the ideal shape for a freestanding arch?

Catenary is the curve, which looks like a parabola but is not one. Robert Hooke studied it in the 1670s, and Leibniz and Huygens were among those who derived its equation in 1691.

L'Hôpital's rule in calculus was really discovered by which Swiss tutor, whom the French marquis paid for lessons?

Johann Bernoulli signed a contract that let l'Hôpital use his discoveries as he pleased in return for an annual allowance. The rule then appeared in the marquis's 1696 textbook.

Which German mathematician published a curve in 1872 that is continuous everywhere yet has no slope at any point?

Karl Weierstrass built it to challenge the belief that a continuous function must be smooth almost everywhere. He had spent years as a schoolteacher, teaching botany and gymnastics as well as maths.

Which English mathematician, a contemporary of Isaac Newton, introduced the ∞ symbol for infinity in 1655?

John Wallis used the sign in his treatise on conic sections. He also served for decades as chief cryptographer, first for Parliament and later for the king's court.

Which curve gives a sliding bead the fastest trip between two points, a 1696 puzzle Isaac Newton solved overnight?

Cycloid is the answer, the path traced by a point on the rim of a rolling wheel. Galileo had guessed an arc of a circle; Newton posted his solution anonymously the morning after staying up to find it.

Which mathematician first met calculus as a girl because her bedroom was papered with old lecture notes?

Sofia Kovalevskaya was 11 when the family ran short of wallpaper and used her father's student notes instead. In 1874 she became the first woman to earn a modern doctorate in mathematics.