FY12CE Mathematics cheat sheet

Fundamental Theorem of Calculus (part 1): if f is continuous on [a,b] and F(x)=∫_a^x f(t)dt, then F'(x)=f(x); part 2 gives ∫_a^b f(x)dx = F(b) − F(a) for any antiderivative F.

L'Hôpital's rule applies only to 0/0 or ∞/∞ indeterminate forms; differentiate numerator and denominator separately, and re-check the form after each application before reapplying.

Taylor's theorem with Lagrange remainder: f(x)=Σ_{k=0}^{n} f^(k)(a)(x−a)^k/k! + f^(n+1)(ξ)(x−a)^(n+1)/(n+1)! for some ξ between a and x.

Second derivative test: at a critical point with f''>0 the point is a local minimum, f''<0 a local maximum, and f''=0 is inconclusive-fall back to the first derivative sign change.

Convergence tests: p-series Σ1/n^p converges iff p>1; geometric Σr^n converges iff |r|<1; ratio test is inconclusive when the limit equals 1.

Leibniz integral rule: d/dx ∫{a(x)}^{b(x)} f(x,t)dt = f(x,b(x))b'(x) − f(x,a(x))a'(x) + ∫{a(x)}^{b(x)} ∂f/∂x dt.

Absolute convergence implies convergence, but conditional convergence (e.g. Σ(−1)^n/n) does not permit arbitrary rearrangement of terms without changing the sum.

For a curve r(t), arc length is ∫|r'(t)|dt, curvature is κ=|r'×r''|/|r'|^3 in space, and the unit tangent is T=r'/|r'|.

A square matrix A is invertible iff det(A)≠0 iff its columns are linearly independent iff rank(A)=n iff 0 is not an eigenvalue.

Eigenvalues solve det(A−λI)=0; eigenvectors for λ are nonzero solutions of (A−λI)x=0, and algebraic multiplicity ≥ geometric multiplicity always.

Cramer's rule: for Ax=b with det(A)≠0, x_i = det(A_i)/det(A), where A_i replaces column i of A with b.

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

Projection of u onto v is (u·v/|v|^2)v; the distance from point P to plane n·x=d is |n·P − d|/|n|.

Gram–Schmidt: v_1=u_1, and v_k = u_k − Σ_{j<k} (u_k·v_j/|v_j|^2)v_j; the resulting set is orthogonal with the same span.

A real symmetric matrix has real eigenvalues and an orthonormal eigenbasis; positive definiteness is equivalent to all eigenvalues positive and all leading principal minors positive.

First-order linear y' + P(x)y = Q(x) has integrating factor μ(x)=exp(∫P dx), giving (μy)' = μQ and y = (1/μ)∫μQ dx.

Separable equations solve by ∫g(y)dy = ∫f(x)dx; check for constant solutions lost when dividing by g(y)=0.

Second-order linear homogeneous equations with constant coefficients use the characteristic equation ar^2+br+c=0: distinct real roots give e^{r1x}, e^{r2x}; repeated root gives e^{rx}, xe^{rx}; complex roots α±βi give e^{αx}cos βx, e^{αx}sin βx.

Method of undetermined coefficients: for forcing e^{αx} times a polynomial, guess the same form multiplied by x^s, where s is the multiplicity of α as a characteristic root (0, 1, or 2).

Existence–uniqueness (Picard–Lindelöf): y'=f(x,y) with y(x0)=y0 has a unique local solution if f and ∂f/∂y are continuous near (x0,y0).

A first-order autonomous equation y'=f(y) has equilibrium points where f(y)=0; an equilibrium is stable when f'(y)<0 there and unstable when f'(y)>0.

Laplace transform rules: L{y'}=sY−y(0), L{y''}=s^2Y−sy(0)−y'(0), L{e^{at}}=1/(s−a), L{t^n}=n!/s^{n+1}; use partial fractions before inverting.

Bayes' theorem: P(A|B)=P(B|A)P(A)/P(B), with P(B)=ΣP(B|A_i)P(A_i) over a partition of the sample space.

Binomial mean and variance are np and np(1−p); Poisson mean and variance are both λ; exponential mean is 1/λ and variance 1/λ^2.

Central Limit Theorem: for i.i.d. variables with mean μ and finite variance σ^2, (X̄−μ)/(σ/√n) converges in distribution to standard normal.

A 95% confidence interval for a mean with known σ is x̄ ± 1.96σ/√n; the width shrinks with √n, so quadrupling precision requires 16 times the sample size.

Type I error is rejecting a true null hypothesis (probability α); Type II error is failing to reject a false null (probability β), and power is 1−β.

Covariance Cov(X,Y)=E[XY]−E[X]E[Y]; Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y), and independence implies zero covariance but not conversely.

Law of total probability and expectation: E[X]=Σ E[X|A_i]P(A_i) over a partition; conditional expectation is linear and satisfies the tower property.

De Morgan's laws: ¬(P∧Q) ≡ ¬P∨¬Q and ¬(P∨Q) ≡ ¬P∧¬Q; a conditional P→Q is equivalent to ¬P∨Q and to its contrapositive ¬Q→¬P.

A relation is an equivalence relation iff it is reflexive, symmetric, and transitive; it partitions the set into disjoint equivalence classes.

Pigeonhole principle: placing n+1 objects into n boxes forces at least one box to hold two or more objects.

Euler's formula for connected planar graphs: V−E+F=2; a graph has an Euler circuit iff it is connected and every vertex has even degree.

Inclusion–exclusion: |A∪B∪C| = Σ|A| − Σ|A∩B| + |A∩B∩C|, alternating signs for larger unions.

A tree on n vertices has exactly n−1 edges; a connected graph on n vertices is a tree iff adding any missing edge creates exactly one cycle.

Strong induction assumes all cases below n to prove case n; structural induction proves a property for all recursively defined objects by proving it for base cases and constructors.

Newton's method iterates x_{n+1}=x_n−f(x_n)/f'(x_n) and converges quadratically near a simple root, but fails or diverges when f'(x_n)≈0.

Bisection converges linearly and always works when f is continuous and changes sign on [a,b]; the error after n steps is at most (b−a)/2^n.

Trapezoidal rule error is O(h^2); Simpson's rule requires an even number of subintervals and has error O(h^4) for smooth integrands.

Condition number κ(A)=||A||·||A^{-1}|| bounds error amplification; for the 2-norm of a normal matrix it equals |λ_max|/|λ_min|.

LU decomposition solves Ax=b via forward and back substitution; partial pivoting is required for stability when a pivot is small or zero.

Floating-point machine epsilon ε satisfies fl(1+ε)≠1; rounding error in summation grows with n, and Kahan summation reduces it.

For the linear system Ax=b, Jacobi and Gauss–Seidel iteration converge when A is strictly diagonally dominant; Gauss–Seidel typically converges about twice as fast.

Logistic growth dP/dt = rP(1−P/K) has carrying capacity K, inflection at P=K/2, and explicit solution P(t)=K/(1+((K−P_0)/P_0)e^{−rt}).

SIR epidemic model: dS/dt=−βSI/N, dI/dt=βSI/N−γI, dR/dt=γI; the basic reproduction number is R_0=β/γ, and an outbreak grows only if R_0>1.

Exponential decay model N(t)=N_0 e^{−kt} has half-life t_{1/2}=ln2/k, independent of the initial amount.

Dimensional analysis: a physically meaningful equation must have consistent units on both sides; the Buckingham Pi theorem reduces n variables with k base dimensions to n−k dimensionless groups.

Linear regression least squares estimates are β̂ = Cov(x,y)/Var(x) and α̂ = ȳ − β̂x̄; residuals sum to zero when an intercept is included.

A Markov chain's long-run behavior is given by the stationary distribution π satisfying πP=π with Σπ_i=1; it is unique when the chain is irreducible and aperiodic.

Optimization models require identifying the objective, decision variables, and constraints; a linear program attains its optimum at a vertex of the feasible region when the region is bounded and nonempty.