UPMSP Intermediate Mathematics cheat sheet

A relation R from A to B is a subset of A×B; its domain is {a : (a,b)∈R} and range is {b : (a,b)∈R}.

R is a function if every element of its domain has exactly one image; equal ranges with repeated pre-images fail the vertical-line test.

f: A→B is one-one (injective) if f(x₁)=f(x₂) implies x₁=x₂; it is onto (surjective) if range f = B.

Bijection = one-one + onto; a bijection A→B exists iff |A|=|B| for finite sets, and only then is f⁻¹ a function.

Composition: (g∘f)(x)=g(f(x)); g∘f is defined only when range f ⊆ domain g, and composition is not commutative.

(f⁻¹)⁻¹=f, and (g∘f)⁻¹ = f⁻¹∘g⁻¹ (order reverses).

For f(x)=ax+b with a≠0, f⁻¹(x)=(x−b)/a; for f(x)=(ax+b)/(cx+d), invert by solving y(cx+d)=ax+b for x.

Number of relations from a set of m elements to a set of n elements is 2^(mn); number of functions is n^m.

Principal ranges: sin⁻¹:[−1,1]→[−π/2,π/2]; cos⁻¹:[−1,1]→[0,π]; tan⁻¹:ℝ→(−π/2,π/2).

cot⁻¹:ℝ→(0,π); sec⁻¹:(−∞,−1]∪[1,∞)→[0,π]{π/2}; cosec⁻¹:(−∞,−1]∪[1,∞)→[−π/2,π/2]{0}.

Reciprocal identities hold only for positive x: sin⁻¹x = cosec⁻¹(1/x), cos⁻¹x = sec⁻¹(1/x), tan⁻¹x = cot⁻¹(1/x) for x>0.

Complement rules: sin⁻¹x + cos⁻¹x = π/2, tan⁻¹x + cot⁻¹x = π/2, sec⁻¹x + cosec⁻¹x = π/2, on the respective domains.

Negative arguments: sin⁻¹(−x)=−sin⁻¹x, tan⁻¹(−x)=−tan⁻¹x, cosec⁻¹(−x)=−cosec⁻¹x, but cos⁻¹(−x)=π−cos⁻¹x and cot⁻¹(−x)=π−cot⁻¹x.

Addition formula: tan⁻¹x + tan⁻¹y = tan⁻¹((x+y)/(1−xy)) when xy<1; add π if xy>1 and x,y>0; subtract π if xy>1 and x,y<0.

Double angle: 2tan⁻¹x = tan⁻¹(2x/(1−x²)) for |x|<1; 2tan⁻¹x = cos⁻¹((1−x²)/(1+x²)) for x≥0.

Conversions: sin⁻¹x = tan⁻¹(x/√(1−x²)) for |x|<1; cos⁻¹x = tan⁻¹(√(1−x²)/x) with sign handled by the principal range.

(AB)ᵀ = BᵀAᵀ and (AB)⁻¹ = B⁻¹A⁻¹; matrix multiplication is associative but not commutative.

A is symmetric if Aᵀ=A, skew-symmetric if Aᵀ=−A; every square matrix is the sum of a symmetric and a skew-symmetric matrix.

For square A of order n: |Aᵀ|=|A|, |kA|=kⁿ|A|, |AB|=|A||B|, |adj A|=|A|^(n−1), and A(adj A)=(adj A)A=|A|I.

A⁻¹ = adj A/|A| exists iff |A|≠0; for 2×2 A=[[a,b],[c,d]], A⁻¹ = (1/(ad−bc))[[d,−b],[−c,a]].

Cramer/system form: AX=B has a unique solution X=A⁻¹B when |A|≠0; no solution or infinitely many when |A|=0.

Row/column operations change |A| predictably: swapping two rows negates it, multiplying a row by k multiplies it by k, adding a multiple of one row to another leaves it unchanged.

Area of a triangle with vertices (x₁,y₁),(x₂,y₂),(x₃,y₃) equals ½|det[[x₁,y₁,1],[x₂,y₂,1],[x₃,y₃,1]]|; zero determinant means collinear points.

If A is n×n, then adj(adj A)=|A|^(n−2)A (for n≥2) and |adj(adj A)|=|A|^((n−1)²).

f is continuous at a if lim(x→a)f(x)=f(a); both one-sided limits must exist, agree, and equal f(a).

Differentiability implies continuity, but continuity does not imply differentiability (|x| at 0).

For piecewise functions, check continuity and left/right derivatives at the join point separately.

Standard derivatives: d/dx(sin⁻¹x)=1/√(1−x²), d/dx(tan⁻¹x)=1/(1+x²), d/dx(sec⁻¹x)=1/(|x|√(x²−1)).

Chain rule: d/dx f(g(x)) = f′(g(x))·g′(x); for parametric forms dy/dx=(dy/dt)/(dx/dt).

Rolle's theorem needs continuity on [a,b], differentiability on (a,b), and f(a)=f(b); then f′(c)=0 for some c∈(a,b).

Lagrange's Mean Value Theorem: f′(c)=(f(b)−f(a))/(b−a) for some c∈(a,b), under continuity on [a,b] and differentiability on (a,b).

Logarithmic differentiation: for y=u^v (u>0), take ln y = v ln u, then dy/dx = u^v(v·u′/u + v′ ln u).

f is increasing on an interval where f′(x)>0 and decreasing where f′(x)<0; f′(x)=0 gives stationary points.

First derivative test: sign change + to − gives a local maximum; − to + gives a local minimum; no sign change gives an inflection point.

Second derivative test: f′(c)=0 and f″(c)<0 gives local maximum; f″(c)>0 gives local minimum; f″(c)=0 is inconclusive.

Absolute extrema on [a,b] occur at critical points in (a,b) or at the endpoints a, b - evaluate and compare.

Rate of change: if y=f(x) and x changes at rate dx/dt, then dy/dt = f′(x)·dx/dt (related rates).

Marginal cost/revenue is the derivative of the total cost/revenue function; marginal revenue equals price plus quantity times dp/dq.

Tangent at (x₁,y₁) on y=f(x): y−y₁=f′(x₁)(x−x₁); normal slope is −1/f′(x₁) when f′(x₁)≠0.

Approximation: f(x+Δx) ≈ f(x) + f′(x)Δx; used for error estimates with Δx = (Δx/x)·x.

∫xⁿdx = x^(n+1)/(n+1)+C for n≠−1; ∫(1/x)dx = ln|x|+C; ∫eˣdx=eˣ+C; ∫aˣdx=aˣ/ln a+C.

∫tan x dx = ln|sec x|+C; ∫sec x dx = ln|sec x+tan x|+C; ∫dx/(x²+a²) = (1/a)tan⁻¹(x/a)+C.

∫dx/√(a²−x²)=sin⁻¹(x/a)+C; ∫dx/(x²−a²)=(1/2a)ln|(x−a)/(x+a)|+C; ∫dx/√(x²+a²)=ln|x+√(x²+a²)|+C.

Integration by parts: ∫u v′dx = uv − ∫u′v dx; choose u by the ILATE order (Inverse, Log, Algebraic, Trig, Exponential).

Fundamental theorem: ∫ₐᵇ f(x)dx = F(b)−F(a) where F′=f; swapping limits negates the value.

Properties: ∫ₐᵇf = ∫ₐᶜf + ∫ᶜᵇf; ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a−x)dx; ∫₋ₐᵃ f = 0 if f is odd, 2∫₀ᵃf if f is even.

Area between y=f(x) above and y=g(x) below from x=a to b is ∫ₐᵇ(f−g)dx; take |f−g| if curves cross.

Area in polar/parametric or with y-axis rotation uses the appropriate radius: area of region bounded by x=f(y) is ∫|f(y)|dy over the stated y-interval.

Order = highest derivative present; degree = power of the highest-order derivative after the equation is polynomial in derivatives.

Variable separable: dy/dx = f(x)g(y) ⇒ ∫dy/g(y) = ∫f(x)dx + C.

Homogeneous equation dy/dx = F(y/x): substitute y=vx so dy/dx = v + x(dv/dx), then separate.

Linear equation dy/dx + Py = Q: integrating factor IF = e^(∫P dx), and solution is y·IF = ∫Q·IF dx + C.

Exact form M dx + N dy = 0 with ∂M/∂y = ∂N/∂x; solution is ∫M dx + (terms of N free of x) = C.

Solution of dy/dx = f(ax+by+c): substitute u = ax+by+c to reduce it to a separable equation.

Family of curves: y=mx+c has one parameter (order 1); a circle x²+y²+2gx+2fy+c=0 has three parameters (order 3).

Growth/decay model: dN/dt = kN gives N = N₀e^(kt); Newton cooling dT/dt = −k(T−Tₛ) gives T−Tₛ = (T₀−Tₛ)e^(−kt).

a·b = |a||b|cosθ = a₁b₁+a₂b₂+a₃b₃; a⊥b iff a·b=0.

a×b is perpendicular to both a and b, with |a×b|=|a||b|sinθ; a∥b iff a×b=0.

Determinant form: a×b = det[[i,j,k],[a₁,a₂,a₃],[b₁,b₂,b₃]]; note b×a = −(a×b).

Scalar triple product [a b c]=a·(b×c) gives the parallelepiped volume; zero value means coplanar vectors.

Area of a triangle with sides a, b = ½|a×b|; area of a parallelogram = |a×b|.

Line through point a with direction b: r = a + λb; Cartesian form (x−x₁)/l = (y−y₁)/m = (z−z₁)/n.

Plane with normal n through point a: n·(r−a)=0, i.e. lx+my+nz=d; distance from (x₁,y₁,z₁) is |lx₁+my₁+nz₁−d|/√(l²+m²+n²).

Angle between line (direction b) and plane (normal n) satisfies sinθ = |b·n|/(|b||n|); between two lines use cosθ = |b₁·b₂|/(|b₁||b₂|).

A linear programming problem has a linear objective Z=ax+by to maximise or minimise subject to linear constraints and non-negativity x,y≥0.

The feasible region is the intersection of the half-planes defined by the constraints; it is a convex polygon (possibly unbounded).

Corner point theorem: if an optimum exists, it is attained at a vertex of the feasible region (or on an entire edge between two optimal vertices).

Graphical method: plot each constraint as a line, shade the feasible side, find vertices, then evaluate Z at each vertex and compare.

Unbounded feasible region: if Z can increase without limit in the feasible direction, no maximum exists; check by testing whether the region extends in the improving direction.

Slope comparison: for Z=ax+by, the objective line has slope −a/b; the optimum vertex is where this slope is just leaving the feasible polygon.

Shadow price/optimal value sensitivity: changing a constraint constant shifts the boundary line parallel to itself, moving the optimal vertex along an edge.

A constraint like x+y≤5 with x,y≥0 creates a bounded triangle with vertices (0,0),(5,0),(0,5); corner evaluation of Z there is sufficient.

P(A∪B)=P(A)+P(B)−P(A∩B); for mutually exclusive events P(A∩B)=0.

Conditional probability P(A|B)=P(A∩B)/P(B), P(B)>0; A and B are independent iff P(A∩B)=P(A)P(B).

Bayes' theorem: P(Aᵢ|B)=P(Aᵢ)P(B|Aᵢ)/ΣⱼP(Aⱼ)P(B|Aⱼ) for a partition {Aⱼ}.

Total probability: P(B)=Σⱼ P(Aⱼ)P(B|Aⱼ) over a partition of the sample space.

Binomial distribution: P(X=r)=C(n,r)pʳ(1−p)^(n−r); mean=np, variance=np(1−p), standard deviation √(npq).

Random variable expectation E(X)=Σxᵢpᵢ and variance Var(X)=E(X²)−[E(X)]².

For a fair die, P(even)=3/6=1/2; the complement rule P(A′)=1−P(A) is often the fastest route for 'at least one' problems.

Bernoulli trials require fixed n, two outcomes per trial, constant p, and independence; otherwise the binomial model does not apply.

Slope m=(y₂−y₁)/(x₂−x₁); parallel lines have equal slopes, perpendicular lines have m₁m₂=−1.

Line forms: y−y₁=m(x−x₁) point-slope; x/a+y/b=1 intercept; ax+by+c=0 general with slope −a/b.

Distance from (x₁,y₁) to ax+by+c=0 is |ax₁+by₁+c|/√(a²+b²); distance between parallel lines ax+by+c₁=0 and ax+by+c₂=0 is |c₁−c₂|/√(a²+b²).

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

Parabola y²=4ax: focus (a,0), directrix x=−a, latus rectum length 4a; x²=4ay opens upward with focus (0,a).

Ellipse x²/a²+y²/b²=1 (a>b): e=√(1−b²/a²), foci (±ae,0), latus rectum 2b²/a; sum of focal distances = 2a.

Hyperbola x²/a²−y²/b²=1: e=√(1+b²/a²), foci (±ae,0), asymptotes y=±(b/a)x; conjugate hyperbola swaps a and b.

Tangency condition for y=mx+c to y²=4ax is c=a/m; for x²/a²+y²/b²=1 it is c²=a²m²+b².

Quadratic ax²+bx+c=0: roots (−b±√(b²−4ac))/(2a); sum = −b/a, product = c/a; discriminant D=b²−4ac decides real/equal/complex roots.

For real roots of ax²+bx+c=0 with a>0: D>0 gives two distinct; D=0 gives equal; D<0 gives none. Sign of a determines the parabola's opening.

Binomial theorem: (a+b)ⁿ=Σ C(n,r)a^(n−r)bʳ for r=0..n; general term T_(r+1)=C(n,r)a^(n−r)bʳ.

C(n,r)=C(n,n−r); C(n,0)=C(n,n)=1; number of terms in (a+b)ⁿ is n+1.

Limits: lim(x→0)(sin x)/x=1, lim(x→0)(tan x)/x=1, lim(x→0)(1−cos x)/x²=1/2, lim(x→0)(eˣ−1)/x=1, lim(x→0)ln(1+x)/x=1.

L'Hôpital's rule applies to 0/0 or ∞/∞ forms: differentiate numerator and denominator separately and re-evaluate.

AP: aₙ=a+(n−1)d, Sₙ=n/2[2a+(n−1)d]; GP: aₙ=ar^(n−1), Sₙ=a(1−rⁿ)/(1−r) for r≠1, S∞=a/(1−r) for |r|<1.

Logarithm rules: log(ab)=log a+log b, log(a/b)=log a−log b, log aⁿ=n log a; used to simplify exponential limits and derivatives.