UPMSP Intermediate Mathematics flashcards

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

Relation from A to B

Any subset of the Cartesian product A×B. Its domain is the set of first components appearing, and its range is the set of second components appearing.

Bijective function

A function that is both one-one and onto. Every element of the codomain has exactly one pre-image, so an inverse function exists on the codomain.

Principal value of sin⁻¹

The unique angle in [−π/2, π/2] whose sine equals the given value in [−1,1]. This restricted range makes sine invertible.

Complement identity for inverse trig

For x in the domain, sin⁻¹x + cos⁻¹x = π/2, tan⁻¹x + cot⁻¹x = π/2, and sec⁻¹x + cosec⁻¹x = π/2.

Skew-symmetric matrix

A square matrix A with Aᵀ = −A. All diagonal entries must be zero, and every square matrix splits into symmetric plus skew-symmetric parts.

Inverse of a square matrix

For a square matrix A with nonzero determinant, A⁻¹ = adj A / |A|. It satisfies AA⁻¹ = A⁻¹A = I.

Differentiability implies continuity

If f has a derivative at a point, then f is continuous there. The converse fails: |x| is continuous at 0 but not differentiable there.

Rolle's theorem

If f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then some c in (a,b) satisfies f′(c)=0.

Second derivative test

At a stationary point c with f′(c)=0: f″(c)<0 gives a local maximum, f″(c)>0 gives a local minimum, and f″(c)=0 is inconclusive.

Absolute extrema on closed interval

On [a,b], the largest and smallest values of a continuous f occur at critical points inside (a,b) or at the endpoints a and b.

Integration by parts

The rule ∫u v′ dx = uv − ∫u′ v dx. Choose u first in the ILATE order: inverse trig, logarithmic, algebraic, trigonometric, exponential.

Area between two curves

For curves y=f(x) above y=g(x) between x=a and x=b, the area is ∫ₐᵇ (f(x) − g(x)) dx. Use absolute values where the curves cross.

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 times that factor.

Order and degree of ODE

Order is the highest derivative appearing. Degree is the power of that highest derivative once the equation is polynomial in all derivatives.

Scalar triple product

The value a·(b×c), equal to the determinant of the three vectors' components. It gives the parallelepiped volume and is zero when vectors are coplanar.

Plane equation with normal vector

For normal n=(l,m,n) through point a, the plane is n·(r−a)=0, giving lx+my+nz=d with d=n·a.

Corner point theorem

For a linear program with a bounded feasible region, the optimal value of the objective is attained at a vertex of the feasible polygon.

Feasible region in linear programming

The set of points satisfying all constraints and non-negativity conditions simultaneously. It is a convex polygon, possibly unbounded.

Bayes' theorem

P(Aᵢ|B) = P(Aᵢ)P(B|Aᵢ) divided by the sum of P(Aⱼ)P(B|Aⱼ) over all partition events Aⱼ.

Binomial distribution mean and variance

For n independent trials with success probability p, the mean is np and the variance is np(1−p), with standard deviation the square root of that.

Perpendicular lines slope condition

Two non-vertical lines are perpendicular exactly when the product of their slopes is −1, that is m₁m₂ = −1.

Eccentricity of an ellipse

For x²/a² + y²/b² = 1 with a>b, eccentricity e = √(1 − b²/a²). The foci lie at (±ae, 0) and 0<e<1.

Quadratic discriminant

For ax²+bx+c=0, D=b²−4ac. Positive D gives two distinct real roots, zero gives equal real roots, negative gives complex roots.

Standard limit sin x over x

As x approaches 0, (sin x)/x approaches 1 when x is measured in radians. This underlies the derivatives of sine and cosine.

Composition of functions

(g∘f)(x) = g(f(x)), defined only when the range of f lies inside the domain of g. Composition is associative but generally not commutative.

Inverse of a composition

For invertible f and g, the inverse of g∘f is f⁻¹∘g⁻¹. The order of the functions reverses, mirroring matrix inverse products.

Determinant of adjugate

For an n×n matrix A, |adj A| = |A|^(n−1). Also A(adj A) = (adj A)A = |A|I, which underlies the inverse formula.

Chain rule

The derivative of f(g(x)) is f′(g(x))·g′(x). For parametric curves, dy/dx equals (dy/dt) divided by (dx/dt).

Related rates

If y=f(x) and x changes at rate dx/dt, then dy/dt = f′(x)·dx/dt. Differentiate the relation with respect to time.

Fundamental theorem of calculus

If F is an antiderivative of continuous f on [a,b], then ∫ₐᵇ f(x)dx = F(b) − F(a). Swapping the limits negates the value.

Homogeneous differential equation

An equation dy/dx = F(y/x). Substitute y = vx, giving dy/dx = v + x dv/dx, which then separates into v and x terms.

Angle between line and plane

If b is the line direction and n the plane normal, then sinθ = |b·n| divided by the product of their magnitudes.