Ehime Entrance Mathematics cheat sheet

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); the two one-sided limits must both equal L.

f'(a)=lim_{h→0}[f(a+h)−f(a)]/h=lim_{x→a}[f(x)−f(a)]/(x−a); differentiability at a implies continuity at a, but continuity does not imply differentiability.

(uv)'=u'v+uv', (u/v)'=(u'v−uv')/v^2 for v≠0, and {f(g(x))}'=f'(g(x))g'(x).

(x^n)'=nx^{n−1}, (sin x)'=cos x, (cos x)'=−sin x, (tan x)'=sec^2 x, (e^x)'=e^x, and (ln x)'=1/x for x>0.

Tangent at x=a: y−f(a)=f'(a)(x−a); normal when f'(a)≠0: y−f(a)=−1/f'(a).

On an interval, f'>0 implies f is increasing and f'<0 implies f is decreasing; a sign change of f' at c gives a local extremum, while f'(c)=0 alone is inconclusive.

f''>0 means concave up (graph above its tangents) and f''<0 means concave down; an inflection point requires a change of concavity, not merely f''=0.

For continuous f on [a,b], compare values at endpoints and at all critical points in (a,b) to find global extrema; vertical asymptotes occur where the function diverges, horizontal ones from limits as x→±∞.

If F'=f on [a,b], then ∫_a^b f(x)dx=F(b)−F(a); also d/dx∫_a^x f(t)dt=f(x) for continuous f.

Substitution: ∫ f(g(x))g'(x)dx=∫ f(u)du with u=g(x); for definite integrals change the limits to g(a) and g(b).

Integration by parts: ∫ u dv=uv−∫ v du; for definite integrals, ∫_a^b u v' dx=[uv]_a^b−∫_a^b u'v dx.

Area between curves: if f≥g on [a,b], A=∫_a^b[f(x)−g(x)]dx; if they cross, integrate |f−g| or split at the intersection points.

Volume about the x-axis: V=π∫_a^b y^2 dx for y=f(x)≥0; washers: V=π∫_a^b(R^2−r^2)dx; shells about the y-axis: V=2π∫_a^b x f(x)dx when applicable.

Cross-section volume: V=∫_a^b A(x)dx, where A(x) is the area of the slice perpendicular to the x-axis.

∫_α^β (x−α)(β−x)dx=(β−α)^3/6; this shortcut is useful for parabolic areas and many entrance-exam integrals.

For symmetric intervals, ∫_{−a}^a f=0 if f is odd and 2∫_0^a f if f is even; improper integrals are evaluated as limits of proper integrals.

Arithmetic sequence: a_n=a+(n−1)d; sum S_n=n[2a+(n−1)d]/2=n(a+l)/2, where l is the last term.

Geometric sequence: a_n=ar^{n−1}; for r≠1, S_n=a(1−r^n)/(1−r), and for |r|<1 the infinite sum is a/(1−r).

Σ_{k=1}^n k=n(n+1)/2, Σ_{k=1}^n k^2=n(n+1)(2n+1)/6, and Σ_{k=1}^n k^3=[n(n+1)/2]^2.

For a_{n+2}=p a_{n+1}+q a_n, try a_n=r^n; the characteristic equation r^2−pr−q=0 gives a_n=Ar1^n+Br2^n for distinct roots, and (A+Bn)r^n for a double root.

Induction: prove the base case, assume the statement for n=k, and derive it for n=k+1; the induction hypothesis must be used explicitly.

An infinite geometric series converges iff |r|<1, with sum a/(1−r); if |r|≥1 the terms do not tend to 0, so the series diverges.

If S_n=a_1+...+a_n, then a_n=S_n−S_{n−1} for n≥2 and a_1=S_1; use this to recover a sequence from its partial sums.

Squeeze theorem: if a_n≤b_n≤c_n eventually and a_n,c_n→L, then b_n→L; bounded monotone sequences converge.

Dot product: a·b=a1b1+a2b2+a3b3=|a||b|cosθ; a·a=|a|^2; a·b=0 exactly when nonzero a and b are perpendicular.

Cross product in space: a×b=(a2b3−a3b2, a3b1−a1b3, a1b2−a2b1); a×b is perpendicular to both a and b.

|a×b|=|a||b|sinθ is the area of the parallelogram spanned by a and b; triangle area=½|a×b|; parallelepiped volume=|a·(b×c)|.

Line through point p with direction d: x=p+td; in space this gives (x,y,z)=(p1,p2,p3)+t(d1,d2,d3).

Plane through p with normal n: n·(x−p)=0, i.e. n1(x−p1)+n2(y−p2)+n3(z−p3)=0.

Distance from point q to plane n·x=d is |n·q−d|/|n|; distance from q to line p+td is |(q−p)×d|/|d|.

Section formula: the point dividing AB internally in ratio m:n is (nA+mB)/(m+n); the centroid of a triangle with position vectors a,b,c is (a+b+c)/3.

Sphere with center c and radius r: |x−c|=r, i.e. (x−c1)^2+(y−c2)^2+(z−c3)^2=r^2; a plane meets a sphere in a circle, a point, or not at all.

sin^2θ+cos^2θ=1, tanθ=sinθ/cosθ, 1+tan^2θ=sec^2θ; sin(α±β)=sinαcosβ±cosαsinβ and cos(α±β)=cosαcosβ∓sinαsinβ.

Double-angle identities: sin2θ=2sinθcosθ and cos2θ=cos^2θ−sin^2θ=1−2sin^2θ=2cos^2θ−1.

General solutions: sinθ=s gives θ=α+2kπ or π−α+2kπ; cosθ=c gives θ=±α+2kπ; tanθ=t gives θ=α+kπ, where α is a principal solution.

Logs: log_c(ab)=log_c a+log_c b, log_c(a/b)=log_c a−log_c b, log_c a^k=k log_c a, and log_c a=log_b a/log_b c; arguments must be positive.

Exponential and log graphs: y=a^x with a>1 is increasing with horizontal asymptote y=0; y=log_a x is its inverse, defined for x>0 with vertical asymptote x=0.

Derivatives: (e^x)'=e^x, (a^x)'=a^x ln a, (ln x)'=1/x, and (log_a x)'=1/(x ln a); the chain rule applies in related rates.

Related rates: differentiate the relation in t, then substitute known values; for a ladder x^2+y^2=L^2, 2x dx/dt+2y dy/dt=0.

For trigonometric inequalities, solve on one period using the unit circle or graph, then add the period; check whether interval endpoints are included.

Addition principle: if choices are mutually exclusive, add counts; multiplication principle: for successive independent choices, multiply counts.

nPr=n!/(n−r)! for ordered r-selections; nCr=n!/[r!(n−r)!] for unordered selections; nCr=nC(n−r) and nCr=(n−1)C(r−1)+(n−1)Cr.

Circular permutations of n distinct objects: (n−1)!; permutations of n objects with identical groups: n!/(n1!n2!...).

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) for P(B)>0; multiplication rule P(A∩B)=P(A|B)P(B).

Independence: P(A∩B)=P(A)P(B), equivalently P(A|B)=P(A) when P(B)>0.

Expected value E[X]=Σ x_i p_i; variance V[X]=E[X^2]−(E[X])^2, and E[aX+b]=aE[X]+b.

Binomial probability: P(X=k)=nCk p^k(1−p)^{n−k} for k=0,1,...,n, when X counts successes in n independent trials with success probability p.

z=a+bi=r(cosθ+i sinθ)=re^{iθ}, where r=|z|=√(a^2+b^2) and θ=arg z satisfies cosθ=a/r and sinθ=b/r.

Conjugate z̄=a−bi; z z̄=|z|^2; 1/z=z̄/|z|^2 for z≠0.

Multiplication: moduli multiply and arguments add; division: moduli divide and arguments subtract.

De Moivre: [r(cosθ+i sinθ)]^n=r^n(cos nθ+i sin nθ) for integer n.

nth roots of w=ρ(cosφ+i sinφ): z_k=ρ^{1/n}[cos((φ+2kπ)/n)+i sin((φ+2kπ)/n)], k=0,...,n−1.

Locus |z−a|=r is a circle; |z−a|=|z−b| is the perpendicular bisector of ab; arg(z−a)=α is a ray from a.

Triangle inequality: ||z1|−|z2||≤|z1+z2|≤|z1|+|z2|; equality depends on the directions of z1 and z2.

Euler: e^{iθ}=cosθ+i sinθ; e^{iθ}+e^{−iθ}=2cosθ and e^{iθ}−e^{−iθ}=2i sinθ.

Remainder theorem: P(x) divided by x−a leaves remainder P(a); factor theorem: x−a is a factor iff P(a)=0.

Rational root theorem: for integer-coefficient P, any rational root p/q in lowest terms has p dividing the constant term and q dividing the leading coefficient.

Vieta: for ax^2+bx+c=0, sum of roots=−b/a and product=c/a; for ax^3+bx^2+cx+d=0, sum=−b/a, pairwise sum=c/a, product=−d/a.

Division algorithm: a=bq+r with 0≤r<|b|; the Euclidean algorithm replaces (a,b) by (b,r) until r=0, giving gcd(a,b).

gcd(a,b)·lcm(a,b)=|ab| for positive integers a,b; if gcd(a,b)=1, then a and b are coprime.

Base conversion: to convert integer N to base b, repeatedly divide by b and read remainders bottom-up; every digit d satisfies 0≤d<b.

Modular arithmetic: if a≡b (mod m) and c≡d (mod m), then a±c≡b±d and ac≡bd (mod m); division requires an inverse, which exists iff gcd(divisor,m)=1.

For integer equations, use divisibility and bounds: rearrange to a product such as (x−a)(y−b)=k, then list factor pairs of k.

Triangle: A+B+C=π; law of sines a/sin A=b/sin B=c/sin C=2R; law of cosines c^2=a^2+b^2−2ab cos C.

Area: S=½ab sin C=abc/(4R)=rs, where R is the circumradius, r the inradius, and s the semiperimeter.

Circle: an inscribed angle is half the central angle subtending the same arc; angles in the same segment are equal; an angle in a semicircle is 90°.

Tangent-chord theorem: the angle between a tangent and a chord equals the inscribed angle in the alternate segment; a tangent is perpendicular to the radius at the point of contact.

Power of a point: for secants PA·PB=PC·PD; for a tangent PT, PT^2=PA·PB.

Similar triangles have equal corresponding angles and equal side ratios; areas scale as the square of the ratio and volumes as the cube.

Coordinate geometry: distance √[(x2−x1)^2+(y2−y1)^2]; section formula; line through (x1,y1),(x2,y2): (y−y1)/(x−x1)=(y2−y1)/(x2−x1) when defined.

Volumes: cylinder πr^2h, cone (1/3)πr^2h, sphere (4/3)πr^3; sphere surface area 4πr^2; similar solids have volume ratio equal to the cube of the length ratio.