AP Intermediate Mathematics (BIEAP) flashcards

33 cards. Read the term and the definition here. The deck at the top of this page is optional.

Quadratic formula

For ax²+bx+c=0 with a≠0, the roots are x = (−b ± √(b²−4ac))/(2a). The expression b²−4ac is the discriminant and determines whether roots are real, equal, or complex.

Remainder theorem

When a polynomial p(x) is divided by (x−a), the remainder equals p(a). Consequently (x−a) is a factor of p(x) exactly when p(a)=0, which is the factor theorem.

Sum of geometric series

For first term a and ratio r, the sum of n terms is a(rⁿ−1)/(r−1) when r≠1. If |r|<1 the infinite sum converges to a/(1−r).

Binomial general term

In the expansion of (a+b)ⁿ, the term containing bʳ is C(n,r)·aⁿ⁻ʳ·bʳ. Setting the exponent of a or b to a required value locates a specific term.

AM–GM inequality

For n positive real numbers, the arithmetic mean is at least the geometric mean: (a₁+…+aₙ)/n ≥ (a₁…aₙ)^(1/n). Equality holds only when all numbers are equal.

Compound angle formula for sine

sin(A±B) = sinA·cosB ± cosA·sinB. The sign matches the sign inside the bracket, unlike the cosine formula, where the sign reverses.

Double angle for cosine

cos2A = cos²A − sin²A = 1 − 2sin²A = 2cos²A − 1. The last two forms are useful for converting between powers and multiple angles in integration.

General solution of sinθ=sinα

The complete solution set is θ = nπ + (−1)ⁿα for integer n. The (−1)ⁿ factor accounts for the two symmetric positions where sine takes the same value.

Cosine rule

In any triangle, a² = b²+c²−2bc·cosA. It finds a side from two sides and the included angle, or an angle from all three sides.

Distance from point to line

For point (x₁,y₁) and line ax+by+c=0, the perpendicular distance is |ax₁+by₁+c|/√(a²+b²). The absolute value ensures a non-negative result.

Circle centre and radius

The circle x²+y²+2gx+2fy+c=0 has centre (−g,−f) and radius √(g²+f²−c). The radius is real only when g²+f²−c is positive.

Eccentricity of an ellipse

For x²/a²+y²/b²=1 with a>b, eccentricity e = √(1−b²/a²), which lies between 0 and 1. Foci are at (±ae,0) and directrices are x=±a/e.

Section formula

A point dividing the segment joining (x₁,y₁) and (x₂,y₂) internally in ratio m:n has coordinates ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)). External division replaces m+n with m−n.

Chain rule

If y depends on u and u depends on x, then dy/dx = (dy/du)·(du/dx). It is the standard tool for differentiating composite functions.

Derivative of inverse sine

d/dx(sin⁻¹x) = 1/√(1−x²) for |x|<1. Similarly d/dx(tan⁻¹x) = 1/(1+x²), valid for all real x.

Second derivative test

At a stationary point where f'(c)=0, f''(c)<0 indicates a local maximum and f''(c)>0 a local minimum. If f''(c)=0 the test is inconclusive.

L'Hôpital's rule

For limits giving 0/0 or ∞/∞, lim f/g = lim f'/g', provided the latter limit exists. It applies only to these indeterminate forms, not to finite ratios.

Integration by parts

∫u dv = uv − ∫v du. Choose u following LIATE: logarithmic, inverse trigonometric, algebraic, trigonometric, exponential. This order usually simplifies the remaining integral.

King property of definite integrals

∫ₐᵇ f(x)dx = ∫ₐᵇ f(a+b−x)dx. Applying it often converts an awkward integrand into one that cancels or simplifies with the original.

Area between two curves

If the upper curve is y=f(x) and the lower is y=g(x) on [a,b], the enclosed area is ∫ₐᵇ [f(x)−g(x)]dx. Intersection points usually supply the limits.

Volume of revolution about x-axis

Rotating y=f(x) about the x-axis from x=a to x=b sweeps a solid of volume V = π∫ₐᵇ y² dx. The disc cross-section has radius y.

Integrating factor for linear ODE

For dy/dx + Py = Q, the integrating factor is e^∫P dx. Multiplying through makes the left side the derivative of (y·IF), so y·IF = ∫Q·IF dx + C.

Order and degree of ODE

Order is the highest derivative appearing in the equation. Degree is the exponent of that highest derivative once the equation is cleared of radicals and fractions in derivatives.

Homogeneous ODE substitution

For dy/dx = F(y/x), set y = vx, giving dy/dx = v + x·dv/dx. The transformed equation is separable in v and x, then v is replaced by y/x.

Characteristic equation for second-order ODE

For ay''+by'+cy=0, solve am²+bm+c=0. Real distinct roots give exponentials, a repeated root gives (C₁+C₂x)e^{mx}, and complex roots give damped sine and cosine terms.

Dot product

a·b = |a||b|cosθ = a₁b₁+a₂b₂+a₃b₃. It is zero exactly when the vectors are perpendicular, and it yields the scalar projection a·b/|b|.

Cross product

a×b has magnitude |a||b|sinθ and direction normal to both vectors by the right-hand rule. It vanishes when the vectors are parallel, and its magnitude gives parallelogram area.

Scalar triple product

[a b c] = a·(b×c) equals the signed volume of the parallelepiped formed by the three vectors. It is zero exactly when the vectors are coplanar.

Conditional probability

P(A|B) = P(A∩B)/P(B) for P(B)>0. Rearranging gives the multiplication rule P(A∩B) = P(B)·P(A|B), which under independence reduces to P(A)P(B).

Binomial distribution mean and variance

For X~B(n,p) with q=1−p, the mean is np and the variance is npq. The probability of exactly r successes is C(n,r)pʳqⁿ⁻ʳ.

Poisson distribution

P(X=r) = e^{−λ}λʳ/r! for r=0,1,2,… The parameter λ is both the mean and the variance, so the standard deviation is √λ.

Plane equation and normal vector

A plane ax+by+cz+d=0 has normal vector (a,b,c). The distance from (x₁,y₁,z₁) to it is |ax₁+by₁+cz₁+d|/√(a²+b²+c²).

Angle between line and plane

If b is the line direction and n the plane normal, then sinθ = |b·n|/(|b||n|). The angle is measured from the plane, not from the normal.