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AP Precalculus
46 flashcards
A sequence is an ordered list of numbers. A series is the sum of the terms in a sequence.
A function is a relation between two variables where each input value has a unique output value.
The domain is the set of input values for which the function is defined. The range is the set of output values that the function produces.
An explicit function is represented by a single algebraic equation that defines y in terms of x. An implicit function is represented by an equation involving both x and y without y being explicitly solved for.
Common types include linear, quadratic, polynomial, rational, exponential, logarithmic, and trigonometric functions.
The inverse of a function f(x) is a new function f^-1(x) such that f(f^-1(x)) = x for all x in the range of f, and f^-1(f(x)) = x for all x in the domain of f.
A one-to-one function has distinct output values for each input value. A many-to-one function can have multiple input values that map to the same output value.
An even function satisfies f(-x) = f(x) for all x. An odd function satisfies f(-x) = -f(x) for all x.
The composition of two functions f(x) and g(x), denoted f(g(x)), is the result of substituting g(x) into the function f.
A polynomial function is a function that can be written in the form f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where the coefficients a_i are constants and n is a non-negative integer called the degree.
A rational function is a function that can be written as the ratio of two polynomial functions, f(x) = p(x)/q(x), where p(x) and q(x) are polynomials and q(x) ≠ 0.
A logarithmic function is the inverse of an exponential function. It is defined as log_b(x) = y if and only if b^y = x, where b > 0 and b ≠ 1.
Trigonometric functions relate the angles of a triangle to the ratios of the sides. They include sine, cosine, tangent, cotangent, secant, and cosecant.
The unit circle is a circle with radius 1 centered at the origin, used to define trigonometric functions in terms of x and y coordinates.
Exponential functions have the form f(x) = a^x, where a > 0 and a ≠ 1. Their graphs are either increasing or decreasing and have a horizontal asymptote.
A piecewise function is a function defined by multiple sub-functions, each representing the function over a different range of input values.
An arithmetic sequence is a sequence where the difference between any two consecutive terms is constant.
A geometric sequence is a sequence where the ratio between any two consecutive terms is constant.
A recursive sequence is defined by a rule that expresses each term as a function of one or more preceding terms.
The limit of a function f(x) as x approaches a value a is the value that f(x) gets arbitrarily close to as x gets closer and closer to a.
A continuous function is a function whose graph can be drawn without lifting the pencil from the paper.
The main types are jump discontinuities, infinite discontinuities, and removable discontinuities.
A relative maximum is the largest value in an open interval around a point. An absolute maximum is the largest value over the entire domain.
The derivative of a function is the rate of change of the function at a point. Differentiation is the process of finding the derivative.
Common rules include the power rule, product rule, quotient rule, chain rule, and rules for trigonometric, exponential, and logarithmic functions.
The derivative represents the slope of the tangent line to the curve at a given point.
The second derivative is the derivative of the derivative. It represents the rate of change of the rate of change, or the concavity of the curve.
An antiderivative is a function whose derivative is the given function. It is the inverse operation of differentiation.
The Fundamental Theorem of Calculus relates differentiation and integration, allowing one to be computed from the other.
A definite integral has upper and lower limits and represents the area under the curve. An indefinite integral has no limits and represents an antiderivative.
Common techniques include substitution, integration by parts, trigonometric integrals, and integration of rational functions.
A parametric equation represents a relation between two or more variables, each expressed as a separate function of a third variable called the parameter.
A polar equation is an equation that expresses the position of a point in terms of its distance from the origin (radius) and angle from the positive x-axis.
A vector is a quantity having both magnitude and direction, often represented by an arrow.
Common operations include vector addition, scalar multiplication, and dot and cross products.
A conic section is a curve formed by the intersection of a cone and a plane. The main types are circles, ellipses, parabolas, and hyperbolas.
A circle is the set of points equidistant from a fixed point (the center). Its key properties involve the radius, diameter, circumference, and area.
An ellipse is a closed curve with two focal points. Its key properties involve the semi-major axis, semi-minor axis, eccentricity, and equations.
A parabola is an open curve that is symmetric about its axis. It is the locus of points equidistant from a fixed point (the focus) and a fixed line (the directrix).
A hyperbola is an open curve with two branches that are mirror images of each other. It has two focal points and two asymptotes.
An algebraic function involves only algebraic operations. A transcendental function involves operations like exponents, logarithms, and trigonometric functions.
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit satisfying i^2 = -1.
For rational exponents m/n, (x^m)^(1/n) = x^(m/n), x^(m/n) * x^(p/q) = x^((m*q + n*p)/nq), and (x^m)^(p/q) = x^(mp/q).
Mathematical induction is a method of mathematical proof used to establish that a given statement holds for all positive integers.
A sequence is an ordered list of real numbers. A series is the sum of the terms in an infinite sequence.
The binomial theorem provides an explicit formula for expanding binomials raised to a power, expressed in terms of combinations of the binomial coefficients.