AP Intermediate Mathematics (BIEAP) key terms

A formula that expands a power of a sum, (a + b)^n, as a sum of terms involving binomial coefficients; each term has the form C(n, r) a^(n-r) b^r.

The number C(n, r), read as n choose r, equal to n! divided by r!(n - r)!. It counts subsets of size r from n objects and appears as a coefficient in the binomial expansion.

In the expansion of (a + b)^n, the term indexed by r, written T(r+1) = C(n, r) a^(n-r) b^r. Setting it equal to a required form lets you solve for r.

The process of writing a rational function as a sum of simpler fractions whose denominators are the factors of the original denominator. It is used before integrating or expanding such functions.

The statement that for non-negative numbers, the arithmetic mean is at least the geometric mean, with equality only when all numbers are equal. It is used to prove inequalities and find extreme values.

A sequence in which each term after the first is obtained by multiplying the previous term by a fixed constant called the common ratio. Its nth term is a r^(n-1) and its finite sum is a(r^n - 1)/(r - 1).

A sequence in which the difference between consecutive terms is constant. The nth term is a + (n - 1)d and the sum of n terms is (n/2)(2a + (n - 1)d).

An equation of the form ax^2 + bx + c = 0 with a not zero. Its roots are found by factoring, completing the square, or the quadratic formula, and its discriminant b^2 - 4ac describes the nature of the roots.

The quantity b^2 - 4ac for a quadratic ax^2 + bx + c = 0. Its sign tells whether the roots are real and distinct, real and equal, or complex conjugates.

The values of x that satisfy ax^2 + bx + c = 0. Their sum equals -b/a and their product equals c/a, relations used to build equations from given roots.

An arrangement of objects in which order matters. The number of permutations of r objects chosen from n distinct objects is n!/(n - r)!, written as nPr.

A selection of objects in which order does not matter. The number of ways to choose r objects from n distinct objects is C(n, r) = n!/[r!(n - r)!].

The product of all positive integers up to a given positive integer n, written n!. By convention 0! equals 1, and factorials appear in permutation and combination formulas.

The probability that event A occurs given that event B has occurred, written P(A|B) and equal to P(A and B)/P(B) when P(B) is not zero.

Two events whose occurrence does not affect each other's probability, so that P(A and B) = P(A) times P(B). This is equivalent to P(A|B) = P(A) when P(B) is not zero.

A rule that reverses a conditional probability: P(A|B) = P(B|A)P(A)/P(B). It updates a prior probability using evidence and is often applied to partitioned sample spaces.

A function that assigns a numerical value to each outcome of a random experiment. It may be discrete, taking separate values, or continuous, taking any value in an interval.

The long-run average of a random variable, computed as the sum of each value times its probability for discrete variables. It is also called the mean of the distribution.

A measure of how spread out a random variable's values are, equal to the expected value of the squared deviation from the mean. Its square root is the standard deviation.

The probability distribution of the number of successes in n independent trials, each with the same success probability p. Its probabilities are given by C(n, r) p^r (1 - p)^(n-r).

The positive square root of the variance, expressed in the same units as the data. It measures the typical distance of observations from the mean.

A continuous, symmetric, bell-shaped probability distribution defined by its mean and standard deviation. Many natural measurements approximate it, and probabilities are found from standard normal tables after standardizing.

An identity expressing a trigonometric function of a sum or difference, such as sin(A + B) = sin A cos B + cos A sin B. These formulas underpin double-angle and sum-to-product identities.

An identity giving a trigonometric ratio of twice an angle in terms of ratios of the angle, for example sin 2A = 2 sin A cos A and cos 2A = cos^2 A - sin^2 A.

A function that returns an angle whose trigonometric ratio is a given value, such as arcsin or arctan. Each is defined on a restricted principal range so that it remains single-valued.

The complete set of angles satisfying a trigonometric equation, expressed with an integer parameter. For example, sin x = 0 has general solution x = n times pi.

The value that a function's output approaches as the input approaches a specified point. A limit may exist even when the function is not defined at that point.

A property of a function at a point meaning the function is defined there, its limit exists there, and the limit equals the function's value. A function continuous on an interval has no breaks there.

The instantaneous rate of change of a function with respect to its variable, defined as the limit of a difference quotient. Geometrically it gives the slope of the tangent line.

The property that a function has a derivative at a point. Differentiability at a point implies continuity there, but continuity does not guarantee differentiability.

A rule for differentiating a composite function: the derivative of f(g(x)) is f'(g(x)) times g'(x). It is applied repeatedly for functions nested several levels deep.

The tangent at a point on a curve is the line whose slope equals the derivative there; the normal is the perpendicular line through the same point, with slope the negative reciprocal.

The largest or smallest values of a function, locally or over an interval. They are located using critical points where the derivative is zero or undefined, tested by sign changes or second derivatives.

The limit of a sum of areas of thin rectangles under a curve between two limits, written with integral signs and bounds. It gives a signed area and is evaluated using an antiderivative.

The result linking differentiation and integration: the definite integral of a continuous function over an interval equals the difference of an antiderivative evaluated at the endpoints.

A function whose derivative is the given function. Any two antiderivatives of the same function on an interval differ only by a constant.

A technique derived from the product rule, written as the integral of u dv equals uv minus the integral of v du. It suits products of unlike functions such as polynomials and exponentials.

A method for integrating a rational function by first splitting it into simpler fractions with linear or quadratic denominators, then integrating each piece separately.

The order is the highest derivative appearing in the equation; the degree is the power of that highest derivative after the equation is cleared of radicals and fractions in derivatives.

A function that satisfies the equation when substituted together with its derivatives. A general solution contains arbitrary constants equal in number to the order; a particular solution fixes them.

An equation in which the unknown function and its derivatives appear only to the first power and are not multiplied together. First-order linear equations are solved with an integrating factor.

For three vectors a, b, c, the dot product of a with the cross product of b and c. Its absolute value equals the volume of the parallelepiped they span, and it is zero when they are coplanar.

The dot product of two vectors, equal to the product of their magnitudes and the cosine of the angle between them. It is zero for perpendicular vectors and gives the projection of one on another.

The cross product of two vectors, a vector perpendicular to both with magnitude equal to the product of their magnitudes and the sine of the angle between them. It is used to find areas and normals.

The cosines of the angles a line or vector makes with the positive coordinate axes. For a vector, they are its components divided by its magnitude, and their squares sum to one.

Any three numbers proportional to the direction cosines of a line. They describe a line's direction without requiring unit length, and are convenient for writing equations of lines and planes.

A flat surface determined by a point and a normal direction, with equation lx + my + nz = d. The angle between planes and the distance from a point to a plane follow from the normal vectors.

Vectors that lie parallel to a single plane. Three vectors are coplanar exactly when their scalar triple product is zero.

A curve obtained by intersecting a cone with a plane: circle, parabola, ellipse, or hyperbola. Each has a standard equation with eccentricity, focus, and directrix characterizing its shape.

A number describing how far a conic deviates from a circle, equal to the distance from a point on the conic to the focus divided by its distance to the directrix. It is less than one for an ellipse, one for a parabola, and greater than one for a hyperbola.

A line that a curve approaches ever more closely as it extends to infinity, without necessarily meeting it. Hyperbolas have two asymptotes through their centre.