Theorem Jeopardy

Geometry

100

This theorem states that the sum of the squares of the legs of a triangle is equal to the square of the hypotenuse

Pythagorean's Theorem

200

This formula states the following:

sin(a)/a = sin(b)/b = sin(c)/c​​​​

Law of Sines

300

This theorem states that the angles of the interior of a triangle always add up to 180*.

Triangle Angle Sum Theorem

400

For any triangle with side lengths a,b, and c with angle t opposite of side c:

c²=a²+b²-2ab(cos(t))

Law of Cosines

500

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Triangle Inequality Theorem

Algebra

100

This formula is the solution to the polynomial of the form:

ax²+bx+c=0.

The Quadratic Formula

200

This theorem states if f(x) is a polynomial, then x-a is a factor of f(x) if and only if f(a) = 0

The Factor Theorem

300

This theorem states that a degree-n polynomial has exactly n roots.

The Fundamental Theorem of Algebra

400

This theorem gives the expansion for a sum raised to a power:

(x+y)ⁿ

The Binomial Theorem

500

This process allows the breaking down of a complex rational expression into a sum of simpler fractions. 

Partial Fraction Decomposition

Calculus I & 2

100

This theorem states that integration and differentiation are inverse operations.

The Fundamental Theorem of Calculus

200

This theorem states that a function on a closed interval will hit every value in between its value at the start and at the end.

The Intermediate Value Theorem

300

This theorem states that a continuous function over an interval will have a point on that interval such that: the instantaneous rate of change is equal to the average rate of change over the interval

The Mean Value Theorem

400

This theorem states that you can represent a function as an infinite power series of derivatives.

Taylor's Theorem

500

If a functions is continuous on an interval, then it is guaranteed to have at least one absolute maximum value and one absolute minimum value within that interval.

The Extreme Value Theorem

Linear Algebra

100

This value is a fundamental building block of linear algebra and is often written as ac-bd for a 2x2 matrix.

The Determinant

200

These values give insight into the behavior of a matrix when associated with a linear transformation. 

They can be found by solving the equation:

Av = λv

Eigenvalues

300

A real square matrix whose columns and rows are ortho-normal vectors.

An Orthogonal Matrix

400

This theorem states that a matrix A with n columns will satisfy the following,

rank(A) + nullity(A) = n

Rank-Nullity Theorem

500

This theorem gives a list of statements that are all equivalent and is a major theorem in linear algebra:

For a nxn matrix A to have an inverse the following conditions must be met;

1. A is invertible

2. A has n pivots

3. Null(A) = 0

4. The columns of A are linearly independent

5. The columns of A span Rⁿ

6. Ax=b has unique solution for each b in Rⁿ

7. The equation Ax=0 has only the trivial solution

The Invertible Matrix Theorem

Multi-Variable Calculus

100

The term J is needed when performing change of variables in an integral.

J is called the...

The Jacobian

200

To find the extrema of a function subject to a constraint g you need to solve this equation:

∇f(x,y,z)=λ∇g(x,y,z) and g(x,y,z)=k.

The scalar λ is known as...

Lagrange Multiplier

300

This theorem states that if f(x,y) is continous on R=[a,b]x[c,d], then the orders of integration of each variable(integrating x then y, or integrating y then x) are equivalent.

Fubini's Theorem

400

This theorem states that if f has continuous second-order partial derivatives at every point in an open region in R, then the order of differentiation does not matter. 

For example: f_xy=f_yx

Clairaut's Theorem

500

This theorem states that the surface integral of a vector field over a closed surface (flux through the surface) is equal to the volume integral of the divergence over the region enclosed by the surface.

Divergence Theorem (Gauss's Theorem)