MTEL Mathematics (63) flashcards

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

Limit definition of derivative

The derivative at a point is the limit as h approaches zero of [f(x+h) − f(x)] divided by h; it gives the instantaneous rate of change or slope of the tangent line.

Chain rule

For a composite function f(g(x)), the derivative equals f'(g(x)) multiplied by g'(x); differentiate the outer function, keep the inner unchanged, then multiply by the inner derivative.

Fundamental Theorem of Calculus

The derivative of the integral from a to x of f(t) dt equals f(x) when f is continuous; also, the definite integral equals F(b) − F(a) for any antiderivative F.

Second derivative test

At a critical point c where f'(c) = 0, a positive second derivative means a local minimum, a negative second derivative means a local maximum, and zero is inconclusive.

Geometric series convergence

The series formed by summing a times r to the n converges exactly when the absolute value of r is less than one, and its sum equals a divided by (1 − r).

p-series test

The series summing 1 over n to the p converges if and only if p is greater than one; when p equals one it is the harmonic series, which diverges.

L'Hôpital's rule

For a limit producing the indeterminate form 0/0 or infinity/infinity, replace the quotient of functions by the quotient of their derivatives, provided that resulting limit exists.

Taylor series formula

A function is expanded about a as the sum over n of its nth derivative at a times (x − a) to the n, divided by n factorial, valid where the series converges.

Dot product

The dot product of vectors u and v equals the product of their magnitudes times the cosine of the angle between them; it is a scalar and is zero when the vectors are perpendicular.

Matrix invertibility condition

A square matrix has an inverse if and only if its determinant is nonzero; the determinant of a product equals the product of the determinants of the factors.

Eigenvalue equation

A nonzero vector v is an eigenvector of A with eigenvalue lambda when Av equals lambda times v; eigenvalues are found by solving the characteristic equation det(A − lambda I) = 0.

Conditional probability formula

The probability of A given B equals the probability of both A and B occurring divided by the probability of B, defined only when the probability of B is positive.

Combinations versus permutations

Combinations count selections where order does not matter, computed as n! divided by r!(n−r)!; permutations count ordered arrangements and equal n! divided by (n−r)!.

Binomial distribution mean and variance

For n independent trials each with success probability p, the number of successes has mean np and variance np(1 − p), with probability of k successes given by the binomial coefficient times p to the k times (1−p) to the n−k.

Variance identity

The variance of a random variable equals the expected value of its square minus the square of its expected value; for independent variables, variances of a sum add.

Margin of error scaling

The margin of error of a sample mean is z* times the standard deviation divided by the square root of sample size, so quadrupling the sample size halves the margin.

p-value interpretation

The p-value is the probability, computed assuming the null hypothesis is true, of obtaining data at least as extreme as observed; it is not the probability that the null hypothesis is true.

Type I versus Type II error

A Type I error rejects a true null hypothesis and occurs with probability alpha; a Type II error fails to reject a false null hypothesis and occurs with probability beta.

Euclidean algorithm

The greatest common divisor of a and b equals the greatest common divisor of b and the remainder when a is divided by b; repeat until the remainder is zero.

Fermat's little theorem

If p is prime and a is not divisible by p, then a raised to the p minus one power is congruent to one modulo p; this underlies modular inverses in prime fields.

Field axioms summary

A field is a commutative ring with identity in which every nonzero element has a multiplicative inverse; the integers modulo n form a field exactly when n is prime.

Lagrange's theorem

In a finite group, the order of any subgroup divides the order of the group; consequently the order of any element divides the group order.

Rational root theorem

For a polynomial with integer coefficients, any rational root in lowest terms has a numerator dividing the constant term and a denominator dividing the leading coefficient.

Inscribed angle theorem

An angle inscribed in a circle measures half the intercepted arc; an angle inscribed in a semicircle is a right angle, and angles subtending the same arc are equal.

Similar figure scaling

When two similar figures have lengths in ratio k, their perimeters are in ratio k, their areas in ratio k squared, and their volumes in ratio k cubed.

Polygon interior angle sum

The interior angles of a polygon with n sides sum to (n − 2) times 180 degrees; the exterior angles of any convex polygon sum to 360 degrees.

Quadratic discriminant

For ax² + bx + c = 0, the discriminant b² − 4ac determines the roots: positive gives two distinct real roots, zero gives one repeated real root, negative gives two complex conjugate roots.

Logarithm change of base

The logarithm of x base b equals the natural logarithm of x divided by the natural logarithm of b; this allows evaluation with any available logarithm function.

Rational function asymptotes

Vertical asymptotes occur where the denominator is zero and the numerator is not; horizontal asymptotes depend on comparing numerator and denominator degrees, using the ratio of leading coefficients when equal.

Geometric sequence sum

A geometric sequence has nth term a₁ times r to the n−1; the sum of the first n terms is a₁(1 − rⁿ)/(1 − r) when r is not one.

Inverse function condition

A function has an inverse only if it is one-to-one; the graphs of a function and its inverse are reflections across the line y = x, and domains and ranges swap.

Complex number division

To divide by a complex number, multiply numerator and denominator by the conjugate of the denominator; the modulus of a + bi equals the square root of a² + b².