Pancyprian Advanced Math cheat sheet

A function f is injective on its domain if f(a)=f(b) implies a=b; it is surjective onto its codomain if every y in the codomain equals f(x) for some x.

A function is invertible exactly when it is injective; then f⁻¹(f(x))=x on the domain of f and f(f⁻¹(y))=y on the range of f.

The graph of y=f(x−a)+b is the graph of y=f(x) translated a units right and b units up; y=f(−x) is a reflection in the y-axis, y=−f(x) in the x-axis.

For a quadratic ax²+bx+c with a≠0, the vertex is at x=−b/(2a), the discriminant Δ=b²−4ac gives two real roots if Δ>0, one if Δ=0, none if Δ<0.

The remainder of dividing polynomial P(x) by (x−a) is P(a); (x−a) is a factor of P(x) exactly when P(a)=0.

If a polynomial with real coefficients has root a+bi with b≠0, then a−bi is also a root; complex non-real roots occur in conjugate pairs.

The exponential function a^x is increasing for a>1, decreasing for 0<a<1, and its inverse is log_a x; log_a(xy)=log_a x+log_a y and log_a(x^r)=r log_a x.

An asymptote of a rational function P(x)/Q(x) is vertical where Q(x)=0 and P(x)≠0; the horizontal asymptote is the ratio of leading coefficients when deg P=deg Q.

lim_{x→a} f(x)=L means f(x) can be made arbitrarily close to L by taking x sufficiently close to a but not equal to a; the value f(a) is irrelevant to the limit.

A function f is continuous at a if lim_{x→a} f(x)=f(a); continuity requires the limit to exist, the value to exist, and the two to agree.

Standard limits: lim_{x→0} (sin x)/x=1, lim_{x→0} (e^x−1)/x=1, lim_{x→0} (ln(1+x))/x=1, and lim_{x→∞} (1+1/x)^x=e.

L'Hôpital's rule: if a limit has form 0/0 or ∞/∞ and the derivatives' quotient has a limit, then lim f/g = lim f′/g′.

Derivative definition: f′(a)=lim_{h→0} (f(a+h)−f(a))/h; it equals the slope of the tangent line at (a,f(a)).

Differentiability at a implies continuity at a, but continuity does not imply differentiability; |x| is continuous but not differentiable at 0.

Product, quotient and chain rules: (uv)′=u′v+uv′, (u/v)′=(u′v−uv′)/v², and (f(g(x)))′=f′(g(x))·g′(x).

Derivatives: (x^n)′=nx^{n−1}, (sin x)′=cos x, (cos x)′=−sin x, (tan x)′=sec²x, (e^x)′=e^x, (ln x)′=1/x.

A differentiable f is increasing on an interval where f′>0 and decreasing where f′<0; stationary points with f′ changing sign are local extrema.

The Fundamental Theorem of Calculus: if F′=f and f is continuous on [a,b], then ∫_a^b f(x)dx=F(b)−F(a).

A definite integral is signed area: ∫_a^b f(x)dx equals the area above the x-axis minus the area below it; total area uses ∫|f(x)|dx.

The area between curves y=f(x) and y=g(x) from a to b is ∫_a^b |f(x)−g(x)|dx; find intersection points to split intervals where the order changes.

Volume of revolution about the x-axis: V=π∫_a^b [f(x)]²dx (disc method); about the y-axis with shells: V=2π∫_a^b x f(x)dx.

Integration by substitution reverses the chain rule: ∫ f(g(x))g′(x)dx=∫ f(u)du with u=g(x), and limits must be converted when the integral is definite.

Integration by parts: ∫ u dv=uv−∫ v du; choose u so that du is simpler, as in ∫ x e^x dx with u=x.

Standard antiderivatives: ∫x^n dx=x^{n+1}/(n+1)+C for n≠−1, ∫(1/x)dx=ln|x|+C, ∫e^x dx=e^x+C, ∫cos x dx=sin x+C, ∫sin x dx=−cos x+C.

The average value of f on [a,b] is (1/(b−a))∫_a^b f(x)dx; the trapezium rule with n strips approximates ∫a^b f ≈ (h/2)[y_0+2(y_1+…+y{n−1})+y_n], h=(b−a)/n.

Arithmetic sequence: a_n=a_1+(n−1)d; sum S_n=(n/2)(a_1+a_n)=(n/2)[2a_1+(n−1)d].

Geometric sequence: a_n=a_1 r^{n−1}; sum S_n=a_1(1−r^n)/(1−r) for r≠1, and S_∞=a_1/(1−r) when |r|<1.

A series converges only if its terms tend to zero; the harmonic series Σ1/n diverges although 1/n→0, so the test is necessary but not sufficient.

The 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 general term is C(n,k)a^{n−k}b^k.

Pascal's identity C(n,k)=C(n−1,k−1)+C(n−1,k), and the row sums satisfy Σ_{k=0}^{n} C(n,k)=2^n.

The Maclaurin expansions e^x=Σx^n/n!, sin x=Σ(−1)^k x^{2k+1}/(2k+1)!, cos x=Σ(−1)^k x^{2k}/(2k)! converge for all real x.

The ratio test: if |a_{n+1}/a_n|→L<1 the series Σa_n converges absolutely; if L>1 it diverges; if L=1 the test is inconclusive.

Method of differences: if a_n=b_n−b_{n+1}, then Σ_{n=1}^{N} a_n=b_1−b_{N+1}, so the sum telescopes to a limit when b_N converges.

For z=a+bi, the modulus is |z|=√(a²+b²) and the argument arg z is the angle θ with tan θ=b/a, chosen in the correct quadrant.

Polar form: z=r(cos θ+i sin θ)=r e^{iθ}, where r=|z| and θ=arg z; multiplication multiplies moduli and adds arguments.

De Moivre's theorem: [r(cos θ+i sin θ)]^n=r^n(cos nθ+i sin nθ); this gives the nth roots z_k=r^{1/n}(cos((θ+2kπ)/n)+i sin((θ+2kπ)/n)) for k=0,…,n−1.

The complex conjugate of z=a+bi is z̄=a−bi; z z̄=|z|², z+z̄=2Re z, and z is real exactly when z=z̄.

Division is done by multiplying numerator and denominator by the conjugate of the denominator: (a+bi)/(c+di)=((a+bi)(c−di))/(c²+d²).

Euler's formula e^{iθ}=cos θ+i sin θ yields e^{iπ}+1=0 and the identities cos θ=(e^{iθ}+e^{−iθ})/2, sin θ=(e^{iθ}−e^{−iθ})/(2i).

The n solutions of z^n=1 are the nth roots of unity e^{2πik/n}, k=0,1,…,n−1; they form a geometric progression summing to zero.

Loci: |z−z_0|=r is a circle centre z_0 radius r; |z−z_1|=|z−z_2| is the perpendicular bisector of the segment joining z_1 and z_2.

The dot product in 3D is a·b=a_1b_1+a_2b_2+a_3b_3=|a||b|cos θ; a·b=0 exactly when a and b are perpendicular (for nonzero vectors).

The cross product a×b=(a_2b_3−a_3b_2, a_3b_1−a_1b_3, a_1b_2−a_2b_1) is perpendicular to both a and b, with |a×b|=|a||b|sin θ equal to the area of the parallelogram they span.

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

The projection of a onto b is (a·b/|b|²)b, and the component of a along b is a·b/|b|.

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

The distance from point P to the plane n·r=k is |n·p−k|/|n|; the distance from P to the line r=a+td is |(p−a)×d|/|d|.

The angle between two planes equals the angle between their normals; the angle between a line and a plane φ satisfies sin φ=|d·n|/(|d||n|).

Two lines r=a+td and r=b+se intersect exactly when a+td=b+se has a solution (t,s); skew lines have no solution and non-parallel directions.

Line forms: slope m=(y_2−y_1)/(x_2−x_1); point-slope y−y_1=m(x−x_1); general form Ax+By+C=0 has normal vector (A,B) and slope −A/B.

Distance from point (x_0,y_0) to line Ax+By+C=0 is |Ax_0+By_0+C|/√(A²+B²); parallel lines Ax+By+C_1=0 and Ax+By+C_2=0 are |C_1−C_2|/√(A²+B²) apart.

Two lines are perpendicular when m_1m_2=−1 (or A_1A_2+B_1B_2=0); they are parallel when their direction vectors are scalar multiples.

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

Parabola y²=4ax has focus (a,0), directrix x=−a, and every point on it is equidistant from focus and directrix; x²=4ay opens upward with focus (0,a).

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

Hyperbola x²/a²−y²/b²=1 has foci (±c,0) with c²=a²+b², eccentricity e=c/a>1, and asymptotes y=±(b/a)x.

For a conic, the tangent at (x_1,y_1) on x²/a²+y²/b²=1 is xx_1/a²+yy_1/b²=1; the chord of contact from an external point follows the same substitution rule.

Conditional probability: P(A|B)=P(A∩B)/P(B) for P(B)>0; A and B are independent exactly when P(A∩B)=P(A)P(B), equivalently P(A|B)=P(A).

Addition rule: P(A∪B)=P(A)+P(B)−P(A∩B); for mutually exclusive events the last term is zero.

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 distribution: P(X=k)=C(n,k)p^k(1−p)^{n−k}, mean np, variance np(1−p); it models n independent trials with constant success probability p.

For a discrete random variable, E(X)=ΣxP(X=x) and Var(X)=E(X²)−[E(X)]²; for continuous X replace sums by ∫x f(x)dx.

Normal distribution: if XN(μ,σ²) then Z=(X−μ)/σN(0,1); approximately 68% of values lie within μ±σ and 95% within μ±2σ.

Sample mean of n independent observations has standard error σ/√n; the central limit theorem makes the distribution of the sample mean approximately normal for large n.

Correlation coefficient r=Σ(x_i−x̄)(y_i−ȳ)/√(Σ(x_i−x̄)²Σ(y_i−ȳ)²) lies in [−1,1]; the least-squares line is y−ȳ=(S_xy/S_xx)(x−x̄).