Pancyprian Advanced Math flashcards

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

Injective function

A function f with f(a)=f(b) implying a=b, so distinct inputs always give distinct outputs. Only injective functions can be inverted on their range.

Vertex of a parabola

For y=ax²+bx+c with a≠0, the turning point occurs at x=−b/(2a), with y-value c−b²/(4a). It is a minimum when a>0 and a maximum when a<0.

Factor theorem

A polynomial P(x) has (x−a) as a factor exactly when P(a)=0. Equivalently, the remainder on dividing by (x−a) is P(a).

Logarithm laws

For positive x and y and base a>0, a≠1: log_a(xy)=log_a x+log_a y, log_a(x/y)=log_a x−log_a y, and log_a(x^r)=r log_a x. Also log_a a=1 and log_a 1=0.

Limit of a function

lim_{x→a} f(x)=L means f(x) can be made arbitrarily close to L by taking x close to a but distinct from a. The value f(a) need not exist.

Continuity at a point

f is continuous at a when lim_{x→a} f(x)=f(a). This requires the limit to exist, f(a) to be defined, and the two values to coincide.

L'Hôpital's rule

If lim f/g has indeterminate form 0/0 or ∞/∞, then lim f(x)/g(x)=lim f′(x)/g′(x), provided the latter limit exists. Differentiate numerator and denominator separately.

Chain rule

The derivative of a composite function is (f(g(x)))′=f′(g(x))·g′(x). Differentiate the outer function at the inner value, then multiply by the inner derivative.

Fundamental Theorem of Calculus

If f is continuous on [a,b] and F′=f, then ∫_a^b f(x)dx=F(b)−F(a). Differentiation and definite integration are inverse processes in this sense.

Integration by parts

The rule ∫u dv=uv−∫v du, derived from the product rule. It suits products such as x e^x or x ln x, where choosing u wisely simplifies the remaining integral.

Volume by discs

Rotating y=f(x) about the x-axis from a to b gives V=π∫_a^b [f(x)]²dx. Each cross-section perpendicular to the axis is a disc of radius |f(x)|.

Trapezium rule

With h=(b−a)/n and ordinates y_0 to y_n, ∫_a^b f(x)dx ≈ (h/2)[y_0+y_n+2(y_1+…+y_{n−1})]. It overestimates for curves concave upward.

Sum of geometric series

For a geometric series with first term a and ratio r, S_n=a(1−r^n)/(1−r) for r≠1. If |r|<1 the infinite sum converges to a/(1−r).

Binomial theorem

(a+b)^n=Σ_{k=0}^{n} C(n,k)a^{n−k}b^k, where C(n,k)=n!/(k!(n−k)!). The term in a^{n−k}b^k has coefficient C(n,k).

Ratio test

For a series Σa_n, compute L=lim|a_{n+1}/a_n|. The series converges absolutely if L<1 and diverges if L>1; L=1 gives no conclusion.

Convergence of series

Σa_n converges only if a_n→0. The converse fails: the harmonic series Σ1/n diverges even though 1/n tends to zero.

Modulus and argument

For z=a+bi, modulus |z|=√(a²+b²) is its distance from the origin, and argument θ is the angle from the positive real axis, with tan θ=b/a in the correct quadrant.

De Moivre's theorem

[r(cos θ+i sin θ)]^n=r^n(cos nθ+i sin nθ) for integer n. It converts powers and roots of complex numbers into operations on modulus and argument.

Complex conjugate

The conjugate of z=a+bi is z̄=a−bi. Then z z̄=a²+b²=|z|², and z is real exactly when z=z̄. Conjugates rationalise complex denominators.

nth roots of unity

The solutions of z^n=1 are e^{2πik/n} for k=0,1,…,n−1. They lie equally spaced on the unit circle and their sum is zero.

Dot product

a·b=a_1b_1+a_2b_2+a_3b_3=|a||b|cos θ. It is zero exactly when nonzero a and b are perpendicular, and it gives the angle between the vectors.

Cross product

a×b is perpendicular to both a and b, with magnitude |a||b|sin θ equal to the parallelogram area. Its direction follows the right-hand rule.

Vector equation of a plane

The plane through point A with normal n is (r−a)·n=0, equivalently n·r=n·a. The normal vector is perpendicular to every direction lying in the plane.

Skew lines

Two lines in space are skew when their direction vectors are not parallel and the equations a+td=b+se have no solution. They are non-parallel and never meet.

Distance point to line

For point P and line r=a+td, the perpendicular distance is |(p−a)×d|/|d|. The cross product gives the area of the parallelogram, divided by the base length.

Circle equation

Centre (h,k), radius r: (x−h)²+(y−k)²=r². Expanded form x²+y²+2gx+2fy+c=0 has centre (−g,−f) and radius √(g²+f²−c).

Ellipse foci and eccentricity

For x²/a²+y²/b²=1 with a>b>0, foci are (±c,0) where c²=a²−b², and eccentricity e=c/a<1. The major axis has length 2a.

Hyperbola asymptotes

For x²/a²−y²/b²=1 the asymptotes are y=±(b/a)x and the foci are (±c,0) with c²=a²+b². Eccentricity e=c/a exceeds 1.

Distance point to line formula

The distance from (x_0,y_0) to Ax+By+C=0 is |Ax_0+By_0+C|/√(A²+B²). The same formula gives the gap between parallel lines from their constants.

Conditional probability

P(A|B)=P(A∩B)/P(B) for P(B)>0. It rescales the sample space to B, and independence means P(A|B)=P(A).

Binomial distribution mean

If X counts successes in n independent trials each with probability p, then E(X)=np and Var(X)=np(1−p). Its probabilities are C(n,k)p^k(1−p)^{n−k}.

Standardising a normal variable

If X~N(μ,σ²), then Z=(X−μ)/σ has the standard normal distribution with mean 0 and variance 1. Tables of Φ give P(Z≤z).