FY12CE Mathematics flashcards

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

Fundamental Theorem of Calculus

For continuous f on [a,b] with antiderivative F, the definite integral of f from a to b equals F(b) minus F(a). The derivative of the accumulation function recovers f.

L'Hôpital's rule conditions

Applies only to 0/0 or infinity/infinity indeterminate forms with differentiable numerator and denominator. Differentiate each separately, then recheck whether the new limit is still indeterminate before repeating.

Taylor series remainder

The Lagrange remainder after n terms is f^(n+1)(ξ)(x−a)^(n+1)/(n+1)! for some ξ between a and x, bounding truncation error when the derivative is bounded.

Second derivative test

At a stationary point, positive second derivative gives a local minimum, negative gives a local maximum, and zero is inconclusive, requiring a sign-change check of the first derivative.

p-series convergence

The series of 1/n^p converges exactly when p exceeds 1 and diverges otherwise. The harmonic case p=1 diverges despite terms tending to zero.

Absolute versus conditional convergence

Absolute convergence means the series of absolute values converges, which implies ordinary convergence. Conditional convergence allows rearrangement to change the sum, so term order matters.

Matrix invertibility equivalences

A square matrix is invertible exactly when its determinant is nonzero, its columns are linearly independent, its rank equals its size, and zero is not an eigenvalue.

Eigenvalue multiplicity inequality

For any eigenvalue, the geometric multiplicity (dimension of its eigenspace) is at least one and never exceeds the algebraic multiplicity, the root multiplicity in the characteristic polynomial.

Cramer's rule

For a square system with nonzero determinant, each unknown equals the determinant of the matrix with that column replaced by the right-hand side, divided by the original determinant.

Scalar triple product

The dot product of one vector with the cross product of two others gives the signed parallelepiped volume. It vanishes exactly when the three vectors lie in a common plane.

Point-to-plane distance

For a plane with normal n and equation n·x=d, the perpendicular distance from point P is the absolute value of n·P minus d, divided by the length of n.

Gram–Schmidt process

Builds an orthogonal basis from independent vectors by subtracting from each new vector its projections onto all previously constructed orthogonal vectors, preserving the span at every stage.

Integrating factor method

For a first-order linear equation, multiply through by exp of the integral of the coefficient function. The left side becomes an exact derivative, so integrate both sides directly.

Characteristic equation cases

For constant-coefficient second-order homogeneous equations, distinct real roots give two exponentials, a repeated root gives an extra factor of x, and complex roots give damped sine and cosine terms.

Undetermined coefficients multiplier

When the forcing term matches a homogeneous solution, multiply the trial guess by x raised to the multiplicity of that forcing exponent as a characteristic root.

Picard–Lindelöf theorem

An initial value problem has a unique solution near the initial point when the right-hand side and its partial derivative with respect to the dependent variable are continuous there.

Autonomous equilibrium stability

Equilibria of an autonomous first-order equation occur where the rate function vanishes. The derivative there being negative signals stability; positive signals instability.

Laplace transform of derivatives

The transform of the first derivative is s times the transform minus the initial value; the second derivative adds an s-squared factor and subtracts both initial conditions.

Bayes' theorem

The posterior probability of a hypothesis given evidence equals the likelihood times the prior, divided by the total probability of the evidence over all hypotheses.

Binomial versus Poisson variance

A binomial count with n trials and success probability p has mean np and variance np(1−p). A Poisson variable has equal mean and variance, both equal to lambda.

Central Limit Theorem

For independent identically distributed variables with finite variance, the standardized sample mean approaches a standard normal distribution as sample size grows, regardless of the underlying shape.

Confidence interval width scaling

Interval width is proportional to the standard error, which shrinks as the inverse square root of sample size. Halving the width therefore requires quadrupling the sample.

Type I and Type II errors

A Type I error rejects a true null hypothesis with probability alpha. A Type II error fails to reject a false null with probability beta, and power equals one minus beta.

Contrapositive equivalence

A conditional statement is logically equivalent to its contrapositive, formed by negating and swapping antecedent and consequent. It is not equivalent to its converse.

Equivalence relation definition

A relation that is reflexive, symmetric, and transitive. Such a relation partitions the underlying set into pairwise disjoint equivalence classes whose union is the whole set.

Pigeonhole principle

If more objects than containers are distributed, at least one container receives at least two objects. The generalized form gives a ceiling bound for repeated occupancy.

Euler's planar graph formula

For a connected planar graph drawn without crossings, the number of vertices minus edges plus faces equals two, counting the unbounded outer region as a face.

Tree edge count

A tree on n vertices has exactly n minus one edges. Any connected graph with that many edges is a tree, and adding any missing edge creates exactly one cycle.

Newton's method convergence

The iteration subtracts the function value divided by its derivative. Convergence is quadratic near a simple root but can fail at stationary points or cycle for poor starting values.

Bisection error bound

Halving an interval that brackets a sign change each step guarantees the root lies within the interval. After n steps the error is at most the original length divided by two to the n.

Simpson's rule accuracy

Composite Simpson's rule needs an even number of subintervals and achieves fourth-order error for smooth integrands, outperforming the trapezoidal rule's second-order error.

Matrix condition number

The product of a matrix norm and the norm of its inverse bounds relative error amplification. For a normal matrix in the two-norm it equals the largest eigenvalue magnitude over the smallest.

Logistic growth solution

Population follows a sigmoid approaching carrying capacity K, with growth rate r and inflection at half of K. The closed form involves an exponential decay of the initial deficit ratio.

Basic reproduction number

In the SIR model it equals the transmission rate divided by the recovery rate. An outbreak grows only when this ratio exceeds one, and herd immunity thresholds derive from it.

Buckingham Pi theorem

A physically consistent relationship among n dimensional variables involving k independent base dimensions can be rewritten as a relationship among n minus k dimensionless groups.

Least squares slope

In simple linear regression the estimated slope equals the covariance of predictor and response divided by the predictor variance; the intercept makes the fitted line pass through the means.