SCHOLASTIC BOWL MATHEMATICS

A limit

a value that a function approaches as the input approaches another value, even if the function is not equal to that output value. The definition of a ____ is formalized in a way traditionally written with the Greek letters epsilon and delta, a technique developed by **Augustin-Louis Cauchy**.

Continuity

the ability of a function’s graph to be drawn without lifting the pen. Properly, a function is ____ at a given point if the function has a limit at that point and is equal to its limit there.

The Derivative

An operation that takes a function and results in another function that gives the original function’s rate of change. Taking a __ __is__ called differentiation and is carried out with respect to the input variable that is changing. The ____ is calculated as the limit of a tangent line between two points on the function’s graph as the two points get arbitrarily close together.

differentiable

A function is called ___ if its derivative can be evaluated. This can be about a specific point or the whole function for continuity. The absolute value function is an example of a function that is ____ in some places but not others: it is differentiable everywhere except at an input of 0, where its graph has a sharp corner.

Definite Integration

an operation that can be interpreted as giving the area under a curve, or more precisely, the signed area between the curve and the *x*-axis (signed area meaning that area below the axis counts as negative). An ____ is denoted by a “long S” symbol, ∫.

Riemann sums

a way to formalize the definition of an integral. A ____ ____ approximates the area under a curve by partitioning the area into narrow rectangles that extend between the *x*-axis and the function’s graph, finding the areas of the rectangles, and adding those up.

The fundamental theorem of calculus

roughly states that integration and differentiation are opposite operations. This means that most basic rules for integration can be found by reversing the basic rules for differentiation given above. The formal statement of the theorem is usually given in two parts. The first part states that integrating a function gives an anti-derivative of the function.

The chain rule

is used to find the derivative of the composition of two functions, such as sin(*x*²). It states that the derivative of *f*(*g*(*x*)) is *f*′(*g*(*x*)) · *g*′(*x*). In Leibniz notation, it is written as d*f*/d*x* = d*f*/d*g* d*g*/d*x* and thus resembles canceling two fractions. Integrating both sides of this result gives a corresponding rule for integration by substitution.

The product rule

is used to find the derivative of the product of two functions, such as *x* sin *x*. It states that the derivative of *f*(*x*) *g*(*x*) is *f*′(*x*) *g*(*x*) + *f*(*x*) *g*′(*x*). Integrating both sides of this result gives a corresponding rule for integrating the product of two functions, called integration by parts. It can be combined with the chain rule to find the quotient rule (the way to differentiate a function written as one function divided by another).

Taylor series

is a way of approximating a differentiable function using an infinite sum of monomials (which can be truncated to get a polynomial). Coefficients of Taylor series are found using *n*th derivatives and the factorial function. Taylor series can be based around any point of a function, and when the chosen point is *x* = 0, it is called a Maclaurin series.

Differential equations

equations that relate a function to its derivatives, or even multiple derivatives (e.g., a function to its first and second derivatives). They are widely used throughout the sciences to model behavior like radioactive decay, population growth, predator-prey relations, waves, fluid flow, and more.

Isaac Newton

He is famous for generalizing the binomial theorem to non-integer exponents, doing the first rigorous manipulation with power series, and creating _____’s method for finding roots of differentiable functions. He is best known for a lengthy feud between British and Continental mathematicians over whether he or Gottfried Leibniz invented calculus. It is now generally accepted that they both did, independently.

Euclid

principally known for the *Elements*, a textbook on geometry and number theory, that has been used for over 2,000 years and which grounds essentially all of what is taught in modern high school geometry classes. The *Elements* also includes proof that there are infinitely many prime numbers.

Carl Friedrich Gauss

considered the “Prince of Mathematicians” for his extraordinary contributions to every major branch of mathematics. Author of *Disquisitiones Arithmeticae*. He proved the fundamental theorem of algebra (every non-constant polynomial has at least one root in the complex numbers), though that proof is not considered rigorous enough for modern standards.

Archimedes

______ is best known for his “eureka” moment, in which he realized he could use density considerations to determine the purity of a gold crown; nonetheless, he was the preeminent mathematician of ancient Greece. He found the ratios between the surface areas and volumes of a sphere and a circumscribed cylinder, accurately estimated pi, and developed a calculus-like technique to find the area of a circle, his method of exhaustion.

Gottfried Leibniz

is known for his independent invention of calculus and the ensuing priority dispute with Isaac Newton. Most modern calculus notation, including the integral sign and the use of *d* to indicate a differential, originated with _____. He also did work with the binary number system and did fundamental work in establishing boolean algebra and symbolic logic.

Pierre de Fermat

is remembered for his contributions to number theory including his little theorem, which states that if *p* is a prime number and *a* is any number at all, then *ap* – *a* will be divisible by *p*. Probably most famous for his “last theorem,” which he wrote in the margin of *Arithmetica* by the ancient Greek mathematician Diophantus with a note that “I have discovered a marvelous proof of this theorem that this margin is too small to contain.”

Leonhard Euler

He invented graph theory by solving the Seven Bridges of Königsberg problem, which asked whether there was a way to travel a particular arrangement of bridges so that you would cross each bridge exactly once. (He proved that it was impsosible to do so.) Euler introduced the modern notation for *e*, an irrational number about equal to 2.718, which is now called ____’s number in his honor.

Kurt Gödel

a logician best known for his two incompleteness theorems, which state that if a formal logical system is powerful enough to express ordinary arithmetic, it must contain statements that are true yet unprovable. ____ developed paranoia late in life and eventually refused to eat because he feared his food had been poisoned; he died of starvation.

Andrew Wiles

best known for proving the Taniyama-Shimura conjecture that all rational semi-stable elliptic curves are modular forms. When combined with work already done by other mathematicians, this immediately implied Fermat’s last theorem.

William Rowan Hamilton

is known for a four-dimensional extension of complex numbers, with six square roots of –1 (±*i*, ±*j*, and ±*k*), called the quaternions.

Pythagorean triples

Sets of small integers that satisfy the equation of the Pythagorean theorem, *a*2 + *b*2 = *c*2, and could therefore be the side lengths of a right triangle.

Matrices

Every team should be able to add, subtract, multiply, take the determinant of, transpose, and invert matrices, particularly 2×2 ones.

Vectors

Every team should be able to find the magnitude (length) of a _____, and add, subtract, find the angle between, find the dot product of, and find the cross product of two ____.

Solids

Teams should be able to calculate the volume and surface area of simple geometric figures including the sphere, cone, cylinder, pyramid, hemisphere, prism, and parallelepiped.

Combinatorics

Teams should be able to compute the number of permutations and ______ of *n* objects taken *m* at a time. They should also have memorized the first six (or so) values of the factorial function to make this easier.

Polynomials

Teams should be able to quickly add, subtract, multiply, divide, factor, and find the roots of low-degree ____, and understand how the degree behaves under the first four operations.