JC Mathematics cheat sheet

A function f: A → B is injective if f(a₁)=f(a₂) implies a₁=a₂; surjective if for every b∈B there exists a∈A with f(a)=b; bijective if both.

The composition (g∘f)(x)=g(f(x)) is defined only when the codomain of f is contained in the domain of g; composition is associative but not generally commutative.

The inverse f⁻¹ exists iff f is bijective; then f⁻¹∘f = id_A and f∘f⁻¹ = id_B, and (f⁻¹)⁻¹ = f.

For f(x)=ax+b with a≠0, f⁻¹(x)=(x−b)/a; for f(x)=x² restricted to x≥0, f⁻¹(x)=√x.

The graph of y=f(x−a)+b is the graph of y=f(x) translated a units right and b units up.

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

The domain of f/g excludes points where g(x)=0; the domain of √(h(x)) requires h(x)≥0; the domain of ln(h(x)) requires h(x)>0.

If f is strictly increasing on an interval then f is injective there, so it has an inverse on that interval.

For y=a·f(b(x−h))+k with a,b>0: horizontal stretch by factor 1/b, vertical stretch by factor a, translation h right and k up.

Order matters: apply stretches/reflections before translations when building from the parent graph, because f(b(x−h)) ≠ f(bx−h) in general.

Reflection in the x-axis: y=−f(x); reflection in the y-axis: y=f(−x); reflection in y=x: swap x and y to get x=f(y).

For y=f(x), replacing x by |x| keeps the right half and reflects it across the y-axis; replacing y by |y| keeps the upper half and reflects it across the x-axis.

A rational function p(x)/q(x) has a vertical asymptote at x=c when q(c)=0 and p(c)≠0; the horizontal asymptote is determined by comparing degrees of p and q.

If deg(p)<deg(q), horizontal asymptote y=0; if deg(p)=deg(q), y=leading coefficient ratio; if deg(p)=deg(q)+1, there is an oblique asymptote found by division.

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

The graph of y=|f(x)| is obtained by reflecting any part of y=f(x) below the x-axis into the upper half-plane.

Arithmetic sequence: uₙ=u₁+(n−1)d; sum Sₙ=(n/2)(2u₁+(n−1)d)=(n/2)(u₁+uₙ).

Geometric sequence: uₙ=u₁rⁿ⁻¹; sum Sₙ=u₁(1−rⁿ)/(1−r) for r≠1; infinite sum converges to u₁/(1−r) iff |r|<1.

The method of differences: if uₙ=f(n)−f(n+1), then Σₖ₌₁ⁿ uₖ=f(1)−f(n+1); many rational sums telescope this way.

Principle of mathematical induction: base case P(1) true; inductive step assume P(k) true and prove P(k+1) true; conclude P(n) true for all n≥1.

Strong induction assumes P(1),…,P(k) to prove P(k+1); it is needed for recurrences such as the Fibonacci sequence.

Standard sums: Σₖ₌₁ⁿ k=n(n+1)/2; Σₖ₌₁ⁿ k²=n(n+1)(2n+1)/6; Σₖ₌₁ⁿ k³=[n(n+1)/2]².

For a recurrence uₙ₊₁=a uₙ+b, the fixed point L satisfies L=aL+b, and uₙ−L=(u₁−L)aⁿ⁻¹ when a≠1.

A series Σuₙ converges only if uₙ→0; the converse fails (harmonic series diverges although 1/n→0).

Dot product: a·b=|a||b|cosθ=a₁b₁+a₂b₂+a₃b₃; a·b=0 iff a and b are perpendicular (for nonzero vectors).

Cross product: |a×b|=|a||b|sinθ; a×b is perpendicular to both a and b, and its direction follows the right-hand rule.

In component form, a×b=(a₂b₃−a₃b₂, a₃b₁−a₁b₃, a₁b₂−a₂b₁).

Line through point A with direction d: r=A+td; plane through A with normal n: n·(r−A)=0, i.e. n·r=n·A.

Distance from point P to plane n·r=d is |n·P−d|/|n|; distance from P to line through A with direction d is |(P−A)×d|/|d|.

The angle between two planes is the acute angle between their normals; the angle between a line and a plane satisfies sinφ=|d·n|/(|d||n|).

Skew lines have no intersection and are not parallel; their shortest distance is |(A₂−A₁)·(d₁×d₂)|/|d₁×d₂|.

The scalar triple product a·(b×c) gives the signed volume of the parallelepiped spanned by a, b, c; it is zero iff the three vectors are coplanar.

Cartesian form z=x+iy; modulus |z|=√(x²+y²); argument arg z=atan2(y,x), defined modulo 2π.

Euler form: z=r e^{iθ}=r(cosθ+i sinθ); multiplication multiplies moduli and adds arguments; division divides moduli and subtracts arguments.

De Moivre: (cosθ+i sinθ)ⁿ=cos(nθ)+i sin(nθ); used for powers and for finding nth roots.

The n distinct nth roots of z=r e^{iθ} are zₖ=r^{1/n} e^{i(θ+2πk)/n}, k=0,1,…,n−1; they form a regular n-gon centred at the origin.

Conjugate z̄=x−iy; z z̄=|z|²; z is real iff z=z̄; z is purely imaginary iff z=−z̄.

For a polynomial with real coefficients, non-real roots occur in conjugate pairs; this constrains factorisations over the reals.

Triangle inequality: |z₁+z₂|≤|z₁|+|z₂|, with equality iff z₁ and z₂ have the same argument (one is a nonnegative real multiple of the other).

Loci: |z−a|=r is a circle centre a radius r; |z−a|=|z−b| is the perpendicular bisector of the segment joining a and b.

Product rule: (uv)′=u′v+uv′; quotient rule: (u/v)′=(u′v−uv′)/v²; chain rule: dy/dx=dy/du·du/dx.

d/dx(xⁿ)=nxⁿ⁻¹; d/dx(eˣ)=eˣ; d/dx(ln x)=1/x; d/dx(sin x)=cos x; d/dx(cos x)=−sin x; d/dx(tan x)=sec²x.

d/dx(arcsin x)=1/√(1−x²); d/dx(arctan x)=1/(1+x²); d/dx(aˣ)=aˣ ln a.

Implicit differentiation: differentiate both sides with respect to x, treating y as a function of x, then solve for dy/dx.

Parametric curve (x(t),y(t)): dy/dx=(dy/dt)/(dx/dt); second derivative d²y/dx²=(d/dt(dy/dx))/(dx/dt).

Stationary points satisfy f′(x)=0; classify by sign change of f′ or by f″: f″>0 local minimum, f″<0 local maximum, f″=0 inconclusive.

For optimisation, check stationary points and endpoints of the stated interval; the global extremum is among these candidates.

L'Hôpital's rule applies to 0/0 or ∞/∞ indeterminate forms: lim f/g=lim f′/g′, provided the latter limit exists.

Fundamental Theorem: ∫ₐᵇ f(x)dx=F(b)−F(a) where F′=f; integration and differentiation are inverse operations up to a constant.

Standard integrals: ∫xⁿdx=xⁿ⁺¹/(n+1)+C (n≠−1); ∫(1/x)dx=ln|x|+C; ∫eˣdx=eˣ+C; ∫sin x dx=−cos x+C; ∫cos x dx=sin x+C.

Integration by substitution: ∫f(g(x))g′(x)dx=∫f(u)du with u=g(x); change the limits when the integral is definite.

Integration by parts: ∫u dv=uv−∫v du; choose u to simplify on differentiation (LIATE heuristic).

Partial fractions: decompose a proper rational function into linear and irreducible quadratic factors before integrating; divide first if the degree of the numerator is not lower.

Area between curves: ∫ₐᵇ|f(x)−g(x)|dx; volume of revolution about the x-axis: V=π∫ₐᵇ y²dx; about the y-axis (shells): V=2π∫ₐᵇ x f(x)dx.

Arc length: L=∫ₐᵇ√(1+(dy/dx)²)dx; for parametric curves L=∫√((dx/dt)²+(dy/dt)²)dt.

Improper integral ∫₁^∞ x⁻ᵖdx converges iff p>1; compare with a known convergent or divergent integral to decide convergence.

A first-order separable equation dy/dx=f(x)g(y) is solved by separating: ∫dy/g(y)=∫f(x)dx, then applying the initial condition.

A first-order linear equation dy/dx+P(x)y=Q(x) has integrating factor I(x)=e^{∫P(x)dx}; then (Iy)′=IQ and y=I⁻¹∫IQ dx.

The logistic equation dP/dt=kP(1−P/L) has solution P=L/(1+Ae^{−kt}); P→L as t→∞ for any positive initial population.

Exponential growth/decay: dP/dt=kP gives P=P₀e^{kt}; k>0 growth, k<0 decay; half-life t½=ln2/|k|.

Second-order linear homogeneous equation ay″+by′+cy=0: solve the characteristic equation ar²+br+c=0.

Characteristic roots: distinct real r₁,r₂ give y=Ae^{r₁x}+Be^{r₂x}; repeated root r gives y=(A+Bx)e^{rx}; complex α±iβ give y=e^{αx}(A cosβx+B sinβx).

For non-homogeneous ay″+by′+cy=f(x), the general solution is complementary function plus a particular integral chosen to match the form of f.

Numerical methods such as Euler's method update yₙ₊₁=yₙ+h f(xₙ,yₙ); smaller step size h generally improves accuracy.

For events A and B: P(A∪B)=P(A)+P(B)−P(A∩B); P(A′)=1−P(A); A and B are independent iff P(A∩B)=P(A)P(B).

Conditional probability P(A|B)=P(A∩B)/P(B); Bayes' theorem P(A|B)=P(B|A)P(A)/P(B).

Binomial distribution X~B(n,p): P(X=k)=C(n,k)pᵏ(1−p)ⁿ⁻ᵏ, mean np, variance np(1−p).

Poisson distribution X~Po(λ): P(X=k)=e^{−λ}λᵏ/k!, mean λ, variance λ; used for counts of rare events in a fixed interval.

Normal distribution X~N(μ,σ²): standardise Z=(X−μ)/σ; approximately 68% of values lie within 1σ, 95% within 2σ, 99.7% within 3σ of the mean.

Sample mean X̄ has E(X̄)=μ and Var(X̄)=σ²/n; by the central limit theorem X̄ is approximately normal for large n regardless of the population distribution.

A 95% confidence interval for a mean with known σ is x̄±1.96σ/√n; the margin of error decreases as n increases.

Hypothesis test: compute a test statistic, compare with the critical value at the stated significance level, and reject H₀ when the statistic lies in the critical region.

Multiplication principle: if a task has stages with n₁,n₂,…,nₖ choices, the total number of outcomes is the product n₁n₂…nₖ.

Permutations: P(n,r)=n!/(n−r)!; combinations: C(n,r)=n!/(r!(n−r)!); C(n,r)=C(n,n−r).

The number of arrangements of n objects with repeated types is n!/(n₁!n₂!…nₖ!).

Pigeonhole principle: if n items are placed in m containers and n>m, at least one container holds at least ⌈n/m⌉ items.

Inclusion–exclusion: |A∪B|=|A|+|B|−|A∩B|; general form alternates sums over intersections of all subset sizes.

Binomial theorem: (a+b)ⁿ=Σₖ₌₀ⁿ C(n,k)aⁿ⁻ᵏbᵏ; the term independent of x is found by setting the exponent of x to zero.

A graph is Eulerian (has a closed trail using every edge once) iff every vertex has even degree; a Hamiltonian cycle visits every vertex exactly once.

Recurrence relations model counting problems; solve linear recurrences by finding the characteristic roots or by iteration and pattern spotting.

For A=(a b; c d), det A=ad−bc; A is invertible iff det A≠0, and A⁻¹=(1/det A)(d −b; −c a).

Matrix multiplication is associative and distributive but not commutative: AB≠BA in general; (AB)ᵀ=BᵀAᵀ and (AB)⁻¹=B⁻¹A⁻¹ when both inverses exist.

A system Ax=b has a unique solution iff det A≠0; if det A=0, it has either no solution or infinitely many solutions.

Cramer's rule: xᵢ=det(Aᵢ)/det(A), where Aᵢ is A with the i-th column replaced by b.

The determinant of a 3×3 matrix can be expanded along any row or column using cofactors: det A=Σⱼ aᵢⱼCᵢⱼ with Cᵢⱼ=(−1)^{i+j}Mᵢⱼ.

Eigenvalues satisfy det(A−λI)=0; for each eigenvalue, eigenvectors solve (A−λI)v=0.

A 2×2 matrix with distinct real eigenvalues λ₁,λ₂ is diagonalisable as PDP⁻¹, where the columns of P are eigenvectors.

Row reduction to echelon form gives the rank of a matrix; rank equals the number of nonzero rows in reduced echelon form.

Direct proof: assume the hypothesis and derive the conclusion by logical steps; state definitions precisely at the start.

Proof by contradiction: assume the negation of the statement and derive a contradiction; commonly used to prove irrationality of √2 and infinitude of primes.

Proof by contrapositive: proving ¬Q→¬P establishes P→Q; often easier when the conclusion is a negative statement.

Counterexample disproves a universal statement; a single valid instance is sufficient and must satisfy all hypotheses.

Mathematical modelling cycle: formulate assumptions, build a model, solve, interpret, validate against data, and refine.

When modelling, state units and domain restrictions explicitly; check that predictions are physically meaningful at boundary values.

Proof by exhaustion covers all finite cases; proof by induction is required when the statement is indexed by the natural numbers.

To prove an identity, start from one side and transform it into the other, or manipulate both sides to a common form; never assume the identity itself.