JC Mathematics flashcards

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

Injective function

A function f where f(a)=f(b) implies a=b; distinct inputs always produce distinct outputs, so no horizontal line meets the graph more than once.

Inverse function condition

A function has an inverse iff it is bijective; the inverse reverses the mapping, and its graph is the reflection of the original graph in the line y=x.

Even and odd functions

f is even if f(−x)=f(x) for all x in its domain; f is odd if f(−x)=−f(x); even graphs are symmetric about the y-axis, odd graphs about the origin.

Graph translation rule

Replacing x by x−a shifts the graph a units to the right; adding b outside the function shifts it b units up; both changes affect every point of the graph equally.

Horizontal asymptote rule

For a rational function p(x)/q(x), compare degrees: numerator lower gives y=0; equal degrees give the ratio of leading coefficients; numerator higher by one gives an oblique asymptote.

Amplitude and period

For y=a sin(bx+c)+d, amplitude is |a| and period is 2π/|b|; the phase shift is −c/b and the vertical shift is d.

Arithmetic series sum

The sum of the first n terms of an arithmetic sequence is Sₙ=(n/2)(2u₁+(n−1)d), equivalently n times the average of the first and last terms.

Geometric series convergence

An infinite geometric series with first term u₁ and ratio r converges to u₁/(1−r) exactly when |r|<1; otherwise the partial sums diverge.

Method of differences

If each term can be written as f(n)−f(n+1), the sum telescopes to f(1)−f(n+1); this evaluates many rational and factorial series exactly.

Mathematical induction steps

Prove the base case P(1); assume P(k) and prove P(k+1); then conclude P(n) holds for all positive integers n by the principle of induction.

Dot product formula

a·b=a₁b₁+a₂b₂+a₃b₃=|a||b|cosθ; the dot product is zero exactly when nonzero vectors a and b are perpendicular.

Cross product properties

a×b is perpendicular to both a and b, has magnitude |a||b|sinθ, and equals the area of the parallelogram spanned by a and b.

Plane equation

A plane with normal vector n passing through point A has equation n·(r−A)=0, equivalently n·r=n·A; the normal is perpendicular to every line in the plane.

Distance point to plane

For plane n·r=d and point P, the perpendicular distance is |n·P−d|/|n|; the absolute value ensures a nonnegative result.

Euler form of complex number

z=r e^{iθ} where r=|z| and θ=arg z; multiplication multiplies moduli and adds arguments, while division divides moduli and subtracts arguments.

De Moivre's theorem

(cosθ+i sinθ)ⁿ=cos(nθ)+i sin(nθ) for integer n; it gives powers of complex numbers and underlies the formula for nth roots.

nth roots of complex number

The n distinct nth roots of r e^{iθ} are r^{1/n} e^{i(θ+2πk)/n} for k=0,1,…,n−1; they lie equally spaced on a circle centred at the origin.

Chain rule

For a composite function y=f(g(x)), dy/dx=f′(g(x))·g′(x); the derivative of the outer function is evaluated at the inner function and multiplied by the inner derivative.

Second derivative test

At a stationary point where f′(x)=0, f″(x)>0 indicates a local minimum, f″(x)<0 a local maximum, and f″(x)=0 is inconclusive.

Implicit differentiation

Differentiate both sides of a relation in x and y with respect to x, applying the chain rule to y, then solve algebraically for dy/dx.

Integration by parts

∫u dv=uv−∫v du; choose u to simplify when differentiated and dv to be easy to integrate, often following the LIATE order.

Volume of revolution

Rotating y=f(x) about the x-axis from a to b gives volume V=π∫ₐᵇ y²dx; the integrand is the area of a typical circular cross-section.

Integration by substitution

Set u=g(x) so du=g′(x)dx; the integral ∫f(g(x))g′(x)dx becomes ∫f(u)du; for definite integrals also change the limits.

Separable differential equation

An equation dy/dx=f(x)g(y) is solved by separating variables: ∫dy/g(y)=∫f(x)dx, then applying the initial condition to fix the constant.

Integrating factor method

For dy/dx+P(x)y=Q(x), multiply by I(x)=e^{∫P(x)dx}; the left side becomes (Iy)′, so y=I⁻¹∫IQ dx.

Logistic differential equation

dP/dt=kP(1−P/L) models bounded growth; the solution P=L/(1+Ae^{−kt}) tends to the carrying capacity L as t increases.

Conditional probability

P(A|B)=P(A∩B)/P(B) for P(B)>0; it gives the probability of A given that B has occurred, restricting the sample space to B.

Binomial distribution

X~B(n,p) counts successes in n independent trials with success probability p; P(X=k)=C(n,k)pᵏ(1−p)ⁿ⁻ᵏ, with mean np and variance np(1−p).

Normal standardisation

If X~N(μ,σ²), then Z=(X−μ)/σ follows N(0,1); standardisation allows probabilities to be read from standard normal tables.

Combinations formula

C(n,r)=n!/(r!(n−r)!) counts subsets of size r chosen from n distinct objects without regard to order; it equals the binomial coefficient in (a+b)ⁿ.

Pigeonhole principle

If n objects are placed into m containers and n>m, at least one container holds at least ⌈n/m⌉ objects; used to prove existence without construction.

Matrix inverse 2x2

For A=(a b; c d) with det A=ad−bc≠0, A⁻¹=(1/det A)(d −b; −c a); if det A=0 the matrix is singular and has no inverse.

Eigenvalue equation

λ is an eigenvalue of A if det(A−λI)=0; a corresponding eigenvector v≠0 satisfies Av=λv, so the transformation scales v by λ.

Proof by contradiction

Assume the statement is false, derive a logical contradiction, and conclude the statement must be true; used for results such as the irrationality of √2.