GACE Mathematics (522) flashcards

30 cards. Read the term and the definition here. The deck at the top of this page is optional.

Limit

The value a function approaches as its input approaches a chosen point, whether or not the function is defined there. It is the foundation for continuity, derivatives, and integrals.

Continuity

A property of a function whose value at a point equals the limit of the function there. Informally, the graph has no break, jump, or hole at that point.

Derivative

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.

Differentiability

The property that a function has a well-defined derivative at a point. Differentiability at a point implies continuity there, but continuity does not imply differentiability.

Critical Point

An interior point of a function's domain where the derivative is zero or fails to exist. Such points are candidates for local maxima, minima, or inflection behavior.

Concavity

The way a curve bends, determined by the sign of the second derivative. Positive second derivative means concave up; negative means concave down; a sign change marks an inflection point.

Definite Integral

The signed area between a function's graph and the horizontal axis over an interval, defined as a limit of Riemann sums. It yields a number, unlike the indefinite integral.

Fundamental Theorem of Calculus

The result linking differentiation and integration: differentiating an accumulated area function recovers the integrand, and a definite integral equals the change in an antiderivative.

Improper Integral

An integral over an unbounded interval or of a function with an infinite discontinuity. It is defined by a limit and either converges to a finite value or diverges.

Taylor Series

An infinite power series expansion of a smooth function about a point, built from the function's derivatives there. Truncating it gives polynomial approximations with controllable error.

Convergence

The property of a sequence or series whose terms approach a finite limit or whose partial sums approach a finite sum. Divergence means no such finite limit exists.

Partial Derivative

The derivative of a multivariable function with respect to one variable while holding the others fixed. It measures sensitivity to that single input direction.

Gradient

The vector of all first partial derivatives of a scalar function. It points in the direction of steepest increase and is perpendicular to level curves or surfaces.

Jacobian Matrix

The matrix of first partial derivatives of a vector-valued function, describing its best linear approximation. Its determinant measures local volume scaling under the transformation.

Vector Space

A set closed under addition and scalar multiplication, satisfying axioms such as associativity, commutativity, and existence of a zero vector and additive inverses.

Linear Independence

A property of a set of vectors where no nontrivial linear combination equals the zero vector. Equivalently, no vector in the set is a combination of the others.

Basis

A linearly independent set of vectors that spans a vector space, so every element is expressed uniquely as a linear combination of them. The number of basis vectors is the dimension.

Rank

The dimension of the column space of a matrix, equal to the number of linearly independent rows or columns. It determines whether a linear system has unique, many, or no solutions.

Determinant

A scalar computed from a square matrix that is nonzero exactly when the matrix is invertible. It also gives the signed volume scaling factor of the associated linear map.

Eigenvalue

A scalar for which some nonzero vector is mapped to a scalar multiple of itself by a linear operator. Eigenvalues characterize stability, vibration modes, and diagonalizability.

Diagonalization

Writing a matrix as a product involving a diagonal matrix and a matrix of eigenvectors. It is possible exactly when there are enough linearly independent eigenvectors.

Dot Product

A scalar operation on two vectors equal to the sum of products of corresponding components. It relates to the cosine of the angle between them and tests orthogonality.

Cross Product

A vector operation defined in three dimensions, producing a vector perpendicular to both inputs with magnitude equal to the area of the spanned parallelogram.

Orthogonal Projection

The component of one vector along the direction of another, computed using the dot product. It underlies least squares fitting and Gram-Schmidt orthogonalization.

Random Variable

A function assigning a numerical value to each outcome of a random experiment. It may be discrete, taking countable values, or continuous, described by a density.

Expectation

The probability-weighted average value of a random variable, also called its mean. For continuous variables it is computed by integrating value times density.

Variance

The expected squared deviation of a random variable from its mean, measuring spread. Its square root is the standard deviation, expressed in the original units.

Probability Density Function

A nonnegative function whose integral over an interval gives the probability that a continuous random variable falls in that interval. It integrates to one over the whole line.

Cumulative Distribution Function

A function giving the probability that a random variable is at most a given value. It is nondecreasing, ranges from zero to one, and its derivative is the density when continuous.

Central Limit Theorem

For independent identically distributed variables with finite nonzero variance, the suitably centered and scaled sample mean approaches a standard normal distribution as sample size grows; finite-sample accuracy depends on the population distribution.