AP Intermediate Mathematics (BIEAP) cheat sheet

Quadratic ax²+bx+c=0: roots x = (−b ± √(b²−4ac))/(2a); sum = −b/a, product = c/a.

Discriminant D = b²−4ac: D>0 real distinct, D=0 real equal, D<0 complex conjugate pair (rational only if D is a perfect square).

Remainder Theorem: remainder of p(x) divided by (x−a) is p(a); (x−a) is a factor exactly when p(a)=0.

Factor Theorem plus repeated division gives the multiplicity of a root; a degree-n polynomial has exactly n roots counted with multiplicity over ℂ.

Arithmetic progression: tₙ = a+(n−1)d, Sₙ = (n/2)[2a+(n−1)d]. Geometric: tₙ = arⁿ⁻¹, Sₙ = a(rⁿ−1)/(r−1), S∞ = a/(1−r) for |r|<1.

AM–GM: for positive reals, (a₁+…+aₙ)/n ≥ (a₁…aₙ)^(1/n), equality iff all equal.

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

Binomial theorem: (a+b)ⁿ = Σ C(n,r) aⁿ⁻ʳ bʳ; general term C(n,r) aⁿ⁻ʳ bʳ; middle term exists only when n is even.

Partial fractions: proper rational functions split by distinct linear, repeated linear, and irreducible quadratic factors; degree of numerator must be less than denominator first.

Pythagorean identities: sin²θ+cos²θ=1, 1+tan²θ=sec²θ, 1+cot²θ=csc²θ.

Compound angles: sin(A±B)=sinA cosB ± cosA sinB; cos(A±B)=cosA cosB ∓ sinA sinB; tan(A±B)=(tanA±tanB)/(1∓tanA tanB).

Double angle: sin2A=2sinA cosA, cos2A=cos²A−sin²A=1−2sin²A=2cos²A−1, tan2A=2tanA/(1−tan²A).

Triple angle: sin3A=3sinA−4sin³A, cos3A=4cos³A−3cosA.

Product-to-sum: 2sinA cosB=sin(A+B)+sin(A−B); 2cosA cosB=cos(A+B)+cos(A−B); 2sinA sinB=cos(A−B)−cos(A+B).

General solutions: sinθ=sinα ⇒ θ=nπ+(−1)ⁿα; cosθ=cosα ⇒ θ=2nπ±α; tanθ=tanα ⇒ θ=nπ+α.

Sine rule a/sinA=b/sinB=c/sinC=2R; cosine rule a²=b²+c²−2bc cosA; area = ½ab sinC.

Inverse functions: sin⁻¹x has range [−π/2, π/2], cos⁻¹x range [0, π], tan⁻¹x range (−π/2, π/2).

For a triangle, tan((A−B)/2) = ((a−b)/(a+b)) cot(C/2) (tangent rule); projections: a = b cosC + c cosB.

Distance between points = √((x₂−x₁)²+(y₂−y₁)²); midpoint = ((x₁+x₂)/2, (y₁+y₂)/2).

Section formula: point dividing AB in ratio m:n internally is ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)); external uses m−n.

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

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

Distance from point (x₁,y₁) to ax+by+c=0 is |ax₁+by₁+c|/√(a²+b²).

Angle between lines: tanθ = |(m₁−m₂)/(1+m₁m₂)|.

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

Tangent to x²+y²=r² at (x₁,y₁) is xx₁+yy₁=r²; condition for y=mx+c to touch the circle is c²=r²(1+m²).

Conics: parabola y²=4ax has focus (a,0) and directrix x=−a; ellipse x²/a²+y²/b²=1 has e=√(1−b²/a²); hyperbola x²/a²−y²/b²=1 has e=√(1+b²/a²).

Limit exists iff left and right limits agree; L'Hôpital applies to 0/0 and ∞/∞ forms only.

Standard limits: lim(x→0) sinx/x = 1, lim(x→0)(1−cosx)/x² = 1/2, lim(x→0)(eˣ−1)/x = 1, lim(x→0) ln(1+x)/x = 1.

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

Derivatives: d/dx(xⁿ)=nxⁿ⁻¹, d/dx(sinx)=cosx, d/dx(tanx)=sec²x, d/dx(lnx)=1/x, d/dx(aˣ)=aˣ ln a.

Inverse trig derivatives: d/dx(sin⁻¹x)=1/√(1−x²), d/dx(tan⁻¹x)=1/(1+x²).

Parametric: dy/dx = (dy/dt)/(dx/dt). Implicit differentiation differentiates both sides treating y as a function of x.

Logarithmic differentiation suits products/quotients of powers; take ln of both sides then differentiate.

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

Mean Value Theorem: if f is continuous on [a,b] and differentiable on (a,b), some c has f'(c)=(f(b)−f(a))/(b−a).

Fundamental Theorem: ∫ₐᵇ f(x)dx = F(b)−F(a) where F'=f.

Standard integrals: ∫xⁿdx = xⁿ⁺¹/(n+1)+C (n≠−1), ∫dx/x = ln|x|+C, ∫eˣdx=eˣ+C, ∫sinx dx=−cosx+C, ∫sec²x dx=tanx+C.

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

Integration by parts: ∫u dv = uv − ∫v du; choose u by the LIATE order (log, inverse trig, algebraic, trig, exponential).

Properties: ∫ₐᵇ = −∫ᵦᵃ; ∫ₐᵇ = ∫ₐᶜ + ∫꜀ᵇ; ∫ₐᵇ f(x)dx = ∫ₐᵇ f(a+b−x)dx (king property).

Even/odd: ∫₋ₐᵃ f = 2∫₀ᵃ f if f is even, 0 if f is odd.

Area under curve = ∫ₐᵇ y dx; area between curves = ∫ₐᵇ (upper − lower) dx.

Volume of revolution about x-axis: V = π∫ₐᵇ y² dx; about y-axis: V = π∫ x² dy.

Reduction formula example: ∫sinⁿx dx = −(sinⁿ⁻¹x cosx)/n + ((n−1)/n)∫sinⁿ⁻²x dx.

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

Variable separable: write dy/dx = f(x)g(y), separate as dy/g(y) = f(x)dx, then integrate both sides.

Homogeneous equation dy/dx = F(y/x): substitute y = vx so dy/dx = v + x dv/dx, giving a separable equation in v.

Linear first-order dy/dx + Py = Q: integrating factor IF = e^∫P dx; solution y·IF = ∫Q·IF dx + C.

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

Second-order linear homogeneous ay''+by'+cy=0: solve am²+bm+c=0; distinct roots give C₁e^{m₁x}+C₂e^{m₂x}, repeated roots give (C₁+C₂x)e^{mx}.

Complex roots m = α±iβ give y = e^{αx}(C₁cosβx + C₂sinβx).

For non-homogeneous equations, particular integral by undetermined coefficients matching the form of the forcing term; general solution = CF + PI.

Growth/decay model dy/dt = ky gives y = y₀e^{kt}; Newton cooling dT/dt = −k(T−Tₛ) gives T−Tₛ = (T₀−Tₛ)e^{−kt}.

Magnitude of a = a₁i+a₂j+a₃k is |a| = √(a₁²+a₂²+a₃²); unit vector = a/|a|.

Dot product a·b = |a||b|cosθ = a₁b₁+a₂b₂+a₃b₃; a·b = 0 iff a and b are perpendicular.

Cross product a×b = |a||b|sinθ n̂; component form is the determinant of i, j, k rows. a×b = 0 iff a and b are parallel.

Scalar triple product [a b c] = a·(b×c) equals the volume of the parallelepiped; zero iff the three vectors are coplanar.

Vector triple product a×(b×c) = (a·c)b − (a·b)c.

Projection of a on b is (a·b/|b|²)b; scalar projection is a·b/|b|.

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

Direction cosines l, m, n satisfy l²+m²+n²=1; direction ratios are proportional to the direction cosines.

Line through point a parallel to b: r = a + tb; angle between lines uses the angle between their direction vectors.

P(A∪B) = P(A)+P(B)−P(A∩B); P(A') = 1−P(A); conditional P(A|B) = P(A∩B)/P(B).

Independent events satisfy P(A∩B)=P(A)P(B). Bayes' theorem: P(Aᵢ|B) = P(Aᵢ)P(B|Aᵢ) / Σ P(Aⱼ)P(B|Aⱼ).

Random variable mean E(X)=Σx P(x); variance Var(X)=E(X²)−[E(X)]²; SD = √Var.

Binomial: P(X=r)=C(n,r)pʳqⁿ⁻ʳ with q=1−p; mean np, variance npq.

Poisson: P(X=r)=e^{−λ}λʳ/r!; mean = variance = λ.

Normal distribution: z = (x−μ)/σ; standard normal has mean 0, SD 1; area under the curve is 1.

Mean of grouped data = Σfᵢxᵢ/Σfᵢ; median from cumulative frequency locates the (N/2)th observation; mode is the most frequent value.

Standard deviation of a sample = √(Σ(xᵢ−x̄)²/n) (population form) or with n−1 for sample SD; variance is its square.

Correlation coefficient r lies in [−1,1]; regression line of y on x is y−ȳ = b(x−x̄) with b = Σ(x−x̄)(y−ȳ)/Σ(x−x̄)².

Distance between points = √((x₂−x₁)²+(y₂−y₁)²+(z₂−z₁)²).

Direction ratios of the line through two points are (x₂−x₁, y₂−y₁, z₂−z₁).

Line in symmetric form: (x−x₁)/a = (y−y₁)/b = (z−z₁)/c; vector form r = a + t b.

Plane equation: ax+by+cz=d with normal (a,b,c); plane through point with given normal: a(x−x₁)+b(y−y₁)+c(z−z₁)=0.

Distance from point (x₁,y₁,z₁) to plane ax+by+cz+d=0 is |ax₁+by₁+cz₁+d|/√(a²+b²+c²).

Angle between planes is the angle between their normals: cosθ = (n₁·n₂)/(|n₁||n₂|).

Angle between a line and a plane: sinθ = |b·n|/(|b||n|), where b is the line direction and n is the plane normal.

Two lines are coplanar iff (a₂−a₁)·(b₁×b₂)=0; skew lines fail this condition.

Shortest distance between skew lines = |(a₂−a₁)·(b₁×b₂)|/|b₁×b₂|.

Quadratic ax²+bx+c=0: roots x = (−b ± √(b²−4ac))/(2a); sum = −b/a, product = c/a.

Discriminant D = b²−4ac: D>0 real distinct, D=0 real equal, D<0 complex conjugate pair (rational only if D is a perfect square).

Remainder Theorem: remainder of p(x) divided by (x−a) is p(a); (x−a) is a factor exactly when p(a)=0.

Factor Theorem plus repeated division gives the multiplicity of a root; a degree-n polynomial has exactly n roots counted with multiplicity over ℂ.

Arithmetic progression: tₙ = a+(n−1)d, Sₙ = (n/2)[2a+(n−1)d]. Geometric: tₙ = arⁿ⁻¹, Sₙ = a(rⁿ−1)/(r−1), S∞ = a/(1−r) for |r|<1.

AM–GM: for positive reals, (a₁+…+aₙ)/n ≥ (a₁…aₙ)^(1/n), equality iff all equal.

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

Binomial theorem: (a+b)ⁿ = Σ C(n,r) aⁿ⁻ʳ bʳ; general term C(n,r) aⁿ⁻ʳ bʳ; middle term exists only when n is even.

Partial fractions: proper rational functions split by distinct linear, repeated linear, and irreducible quadratic factors; degree of numerator must be less than denominator first.

Pythagorean identities: sin²θ+cos²θ=1, 1+tan²θ=sec²θ, 1+cot²θ=csc²θ.

Compound angles: sin(A±B)=sinA cosB ± cosA sinB; cos(A±B)=cosA cosB ∓ sinA sinB; tan(A±B)=(tanA±tanB)/(1∓tanA tanB).

Double angle: sin2A=2sinA cosA, cos2A=cos²A−sin²A=1−2sin²A=2cos²A−1, tan2A=2tanA/(1−tan²A).

Triple angle: sin3A=3sinA−4sin³A, cos3A=4cos³A−3cosA.

Product-to-sum: 2sinA cosB=sin(A+B)+sin(A−B); 2cosA cosB=cos(A+B)+cos(A−B); 2sinA sinB=cos(A−B)−cos(A+B).

General solutions: sinθ=sinα ⇒ θ=nπ+(−1)ⁿα; cosθ=cosα ⇒ θ=2nπ±α; tanθ=tanα ⇒ θ=nπ+α.

Sine rule a/sinA=b/sinB=c/sinC=2R; cosine rule a²=b²+c²−2bc cosA; area = ½ab sinC.

Inverse functions: sin⁻¹x has range [−π/2, π/2], cos⁻¹x range [0, π], tan⁻¹x range (−π/2, π/2).

For a triangle, tan((A−B)/2) = ((a−b)/(a+b)) cot(C/2) (tangent rule); projections: a = b cosC + c cosB.

Distance between points = √((x₂−x₁)²+(y₂−y₁)²); midpoint = ((x₁+x₂)/2, (y₁+y₂)/2).

Section formula: point dividing AB in ratio m:n internally is ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)); external uses m−n.

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

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

Distance from point (x₁,y₁) to ax+by+c=0 is |ax₁+by₁+c|/√(a²+b²).

Angle between lines: tanθ = |(m₁−m₂)/(1+m₁m₂)|.

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

Tangent to x²+y²=r² at (x₁,y₁) is xx₁+yy₁=r²; condition for y=mx+c to touch the circle is c²=r²(1+m²).

Conics: parabola y²=4ax has focus (a,0) and directrix x=−a; ellipse x²/a²+y²/b²=1 has e=√(1−b²/a²); hyperbola x²/a²−y²/b²=1 has e=√(1+b²/a²).

Limit exists iff left and right limits agree; L'Hôpital applies to 0/0 and ∞/∞ forms only.

Standard limits: lim(x→0) sinx/x = 1, lim(x→0)(1−cosx)/x² = 1/2, lim(x→0)(eˣ−1)/x = 1, lim(x→0) ln(1+x)/x = 1.

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

Derivatives: d/dx(xⁿ)=nxⁿ⁻¹, d/dx(sinx)=cosx, d/dx(tanx)=sec²x, d/dx(lnx)=1/x, d/dx(aˣ)=aˣ ln a.

Inverse trig derivatives: d/dx(sin⁻¹x)=1/√(1−x²), d/dx(tan⁻¹x)=1/(1+x²).

Parametric: dy/dx = (dy/dt)/(dx/dt). Implicit differentiation differentiates both sides treating y as a function of x.

Logarithmic differentiation suits products/quotients of powers; take ln of both sides then differentiate.

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

Mean Value Theorem: if f is continuous on [a,b] and differentiable on (a,b), some c has f'(c)=(f(b)−f(a))/(b−a).

Fundamental Theorem: ∫ₐᵇ f(x)dx = F(b)−F(a) where F'=f.

Standard integrals: ∫xⁿdx = xⁿ⁺¹/(n+1)+C (n≠−1), ∫dx/x = ln|x|+C, ∫eˣdx=eˣ+C, ∫sinx dx=−cosx+C, ∫sec²x dx=tanx+C.

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

Integration by parts: ∫u dv = uv − ∫v du; choose u by the LIATE order (log, inverse trig, algebraic, trig, exponential).

Properties: ∫ₐᵇ = −∫ᵦᵃ; ∫ₐᵇ = ∫ₐᶜ + ∫꜀ᵇ; ∫ₐᵇ f(x)dx = ∫ₐᵇ f(a+b−x)dx (king property).

Even/odd: ∫₋ₐᵃ f = 2∫₀ᵃ f if f is even, 0 if f is odd.

Area under curve = ∫ₐᵇ y dx; area between curves = ∫ₐᵇ (upper − lower) dx.

Volume of revolution about x-axis: V = π∫ₐᵇ y² dx; about y-axis: V = π∫ x² dy.

Reduction formula example: ∫sinⁿx dx = −(sinⁿ⁻¹x cosx)/n + ((n−1)/n)∫sinⁿ⁻²x dx.

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

Variable separable: write dy/dx = f(x)g(y), separate as dy/g(y) = f(x)dx, then integrate both sides.

Homogeneous equation dy/dx = F(y/x): substitute y = vx so dy/dx = v + x dv/dx, giving a separable equation in v.

Linear first-order dy/dx + Py = Q: integrating factor IF = e^∫P dx; solution y·IF = ∫Q·IF dx + C.

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

Second-order linear homogeneous ay''+by'+cy=0: solve am²+bm+c=0; distinct roots give C₁e^{m₁x}+C₂e^{m₂x}, repeated roots give (C₁+C₂x)e^{mx}.

Complex roots m = α±iβ give y = e^{αx}(C₁cosβx + C₂sinβx).

For non-homogeneous equations, particular integral by undetermined coefficients matching the form of the forcing term; general solution = CF + PI.

Growth/decay model dy/dt = ky gives y = y₀e^{kt}; Newton cooling dT/dt = −k(T−Tₛ) gives T−Tₛ = (T₀−Tₛ)e^{−kt}.

Magnitude of a = a₁i+a₂j+a₃k is |a| = √(a₁²+a₂²+a₃²); unit vector = a/|a|.

Dot product a·b = |a||b|cosθ = a₁b₁+a₂b₂+a₃b₃; a·b = 0 iff a and b are perpendicular.

Cross product a×b = |a||b|sinθ n̂; component form is the determinant of i, j, k rows. a×b = 0 iff a and b are parallel.

Scalar triple product [a b c] = a·(b×c) equals the volume of the parallelepiped; zero iff the three vectors are coplanar.

Vector triple product a×(b×c) = (a·c)b − (a·b)c.

Projection of a on b is (a·b/|b|²)b; scalar projection is a·b/|b|.

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

Direction cosines l, m, n satisfy l²+m²+n²=1; direction ratios are proportional to the direction cosines.

Line through point a parallel to b: r = a + tb; angle between lines uses the angle between their direction vectors.

P(A∪B) = P(A)+P(B)−P(A∩B); P(A') = 1−P(A); conditional P(A|B) = P(A∩B)/P(B).

Independent events satisfy P(A∩B)=P(A)P(B). Bayes' theorem: P(Aᵢ|B) = P(Aᵢ)P(B|Aᵢ) / Σ P(Aⱼ)P(B|Aⱼ).

Random variable mean E(X)=Σx P(x); variance Var(X)=E(X²)−[E(X)]²; SD = √Var.

Binomial: P(X=r)=C(n,r)pʳqⁿ⁻ʳ with q=1−p; mean np, variance npq.

Poisson: P(X=r)=e^{−λ}λʳ/r!; mean = variance = λ.

Normal distribution: z = (x−μ)/σ; standard normal has mean 0, SD 1; area under the curve is 1.

Mean of grouped data = Σfᵢxᵢ/Σfᵢ; median from cumulative frequency locates the (N/2)th observation; mode is the most frequent value.

Standard deviation of a sample = √(Σ(xᵢ−x̄)²/n) (population form) or with n−1 for sample SD; variance is its square.

Correlation coefficient r lies in [−1,1]; regression line of y on x is y−ȳ = b(x−x̄) with b = Σ(x−x̄)(y−ȳ)/Σ(x−x̄)².

Distance between points = √((x₂−x₁)²+(y₂−y₁)²+(z₂−z₁)²).

Direction ratios of the line through two points are (x₂−x₁, y₂−y₁, z₂−z₁).

Line in symmetric form: (x−x₁)/a = (y−y₁)/b = (z−z₁)/c; vector form r = a + t b.

Plane equation: ax+by+cz=d with normal (a,b,c); plane through point with given normal: a(x−x₁)+b(y−y₁)+c(z−z₁)=0.

Distance from point (x₁,y₁,z₁) to plane ax+by+cz+d=0 is |ax₁+by₁+cz₁+d|/√(a²+b²+c²).

Angle between planes is the angle between their normals: cosθ = (n₁·n₂)/(|n₁||n₂|).

Angle between a line and a plane: sinθ = |b·n|/(|b||n|), where b is the line direction and n is the plane normal.

Two lines are coplanar iff (a₂−a₁)·(b₁×b₂)=0; skew lines fail this condition.

Shortest distance between skew lines = |(a₂−a₁)·(b₁×b₂)|/|b₁×b₂|.