Vocabulary Flashcards – Calculus, Algebra, Probability & Geometry

Key vocabulary terms and definitions covering derivatives, algebraic equations, sequences and series, probability, conic sections, differential equations, vectors and coordinate geometry from the lecture notes.

Derivative

Instantaneous rate of change of a function with respect to an independent variable.

Rate of Change

Change of one quantity per unit variation of another; measured by the derivative.

Increasing Function

Function whose derivative is positive throughout an interval.

Decreasing Function

Function whose derivative is negative throughout an interval.

Tangent (to a curve)

Straight line that touches a curve at a point without crossing it; slope equals the derivative at that point.

Normal (to a curve)

Line perpendicular to the tangent of a curve at the point of contact.

Absolute Maximum

Greatest value a function attains over its entire domain.

Relative (Local) Maximum

Greatest value a function attains within some neighborhood of a point.

Absolute Minimum

Smallest value a function attains over its entire domain.

Relative (Local) Minimum

Smallest value a function attains within some neighborhood of a point.

Point of Inflection

Point where a curve changes concavity; second derivative changes sign.

Monotonic Function

Function that is either entirely non-increasing or non-decreasing on its domain.

Linear Approximation

Estimating a function near a point using its tangent line; first-degree Taylor polynomial.

Newton’s Method

Iterative procedure using tangents to approximate roots of equations.

Linear Equation

Polynomial equation of degree one; general form ax + b = 0.

Quadratic Equation

Polynomial equation of degree two; general form ax² + bx + c = 0.

Cubic Equation

Polynomial equation of degree three; general form ax³ + bx² + cx + d = 0.

Factorization Method

Solving a quadratic by expressing it as product of two linear factors.

Sridharacharya’s Formula

Quadratic-formula solution x = [-b ± √(b²-4ac)] /(2a).

Arithmetic Progression (AP)

Sequence in which each term differs from the previous by a constant difference d.

Common Difference

Fixed amount added (or subtracted) to get successive terms of an AP.

nth Term of an AP

Tₙ = a + (n-1)d, where a is first term.

Sum of n Terms of an AP

Sₙ = n/2 [2a + (n-1)d] or n/2 (a + l).

Arithmetic Mean (A.M.)

Value b such that a, b, c are in AP; b = (a + c)/2.

Geometric Progression (G.P.)

Sequence in which each term is obtained by multiplying previous term by constant ratio r.

Common Ratio

Fixed factor by which successive terms of a GP are multiplied.

nth Term of a GP

Tₙ = arⁿ⁻¹, with first term a.

Sum of n Terms of a GP

Sₙ = a(rⁿ–1)/(r–1) for r≠1 (or a(1–rⁿ)/(1–r) if |r|<1).

Sum to Infinity (GP)

S∞ = a /(1–r) for |r|<1.

Geometric Mean (G.M.)

Mean between a and b given by √(ab).

Harmonic Progression (H.P.)

Sequence whose reciprocals form an arithmetic progression.

Harmonic Mean (H.M.)

For numbers a, b; H = 2ab /(a + b).

Relation A > G > H

For two positive numbers, Arithmetic mean exceeds Geometric mean, which exceeds Harmonic mean.

Binomial Expression

Algebraic expression containing exactly two unlike terms, e.g., x + y.

Binomial Theorem

Expansion (x + y)ⁿ = Σ₀ⁿ nCᵣ xⁿ⁻ʳ yʳ.

Binomial Coefficient

nCᵣ = n! /(r!(n–r)!), constant multiplier in binomial expansion.

Conditional Probability

Probability of event A given event B has occurred; P(A|B) = P(A∩B)/P(B).

Joint Probability

Probability of two events occurring together; P(A∩B).

Marginal Probability

Unconditional probability of a single event; P(A).

Bayes’ Theorem

P(Eᵢ|A) = P(Eᵢ)P(A|Eᵢ) / Σ P(Ek)P(A|Ek).

Prior Probability

Initial probability P(Eᵢ) before new evidence.

Posterior Probability

Updated probability P(Eᵢ|A) after considering evidence A.

Bernoulli Trial

Single experiment with exactly two outcomes: success (prob. p) or failure (prob. q).

Binomial Distribution

Discrete distribution giving probability of x successes in n Bernoulli trials; P(x) = nCₓ pˣ qⁿ⁻ˣ.

Mean of Binomial

μ = np.

Variance of Binomial

σ² = npq.

Derivative (dy/dx)

Limit of Δy/Δx as Δx → 0; primary tool of differential calculus.

Continuity

Function with no breaks; limit equals value at every point in domain.

Differentiability

Property of a function having a finite derivative at each point in its domain.

Exact Differential Equation

Equation Mdx + Ndy = 0 with ∂M/∂y = ∂N/∂x; integrable via potential function.

Integrating Factor

Function μ(x, y) multiplied to make a differential equation exact.

Order of a Differential Equation

Highest derivative present in the equation.

Degree of a Differential Equation

Power of the highest derivative after rationalization.

General Solution

Family of functions containing arbitrary constants satisfying a differential equation.

Particular Solution

Specific solution obtained by assigning definite values to constants in the general solution.

Circle (Conic)

Set of points equidistant from a fixed centre; equation x² + y² = r² (centre at origin).

Ellipse

Locus of points whose distances from two foci sum to a constant; eccentricity e<1.

Parabola

Set of points equidistant from a focus and a directrix; eccentricity e = 1.

Hyperbola

Locus of points where the difference of distances from two foci is constant; eccentricity e>1.

Eccentricity (Conic)

Ratio of focal distance to directrix distance; classifies conics.

Latus Rectum

Chord of a conic through a focus perpendicular to the major axis.

Major Axis

Longest chord of an ellipse or hyperbola passing through both foci.

Minor Axis

Shortest chord of an ellipse through its centre perpendicular to the major axis.

Focus (Conic)

Fixed point used in conic’s definition via distance ratio.

Directrix

Fixed line used in conic’s focus–directrix definition.

Direction Cosines

Cosines of angles between a vector and the coordinate axes; denoted l, m, n.

Direction Ratios

Any set of numbers proportional to the direction cosines; denote vector’s direction.

Plane Equation (Normal Form)

x cosα + y sinα + z cosγ = d; distance d from origin, α, β, γ direction angles.

Distance Formula (2-D)

Distance between (x₁,y₁) and (x₂,y₂) is √[(x₂−x₁)² + (y₂−y₁)²].

Distance Formula (3-D)

Distance between (x₁,y₁,z₁) and (x₂,y₂,z₂) is √[(Δx)²+(Δy)²+(Δz)²].

Sphere Equation

(x−a)² + (y−b)² + (z−c)² = r², centre (a,b,c), radius r.