Vector, Polar, and Parametric Functions

Flashcards designed to help students understand and memorize key concepts related to vector, polar, and parametric functions, as well as integration techniques and infinite series.

Parametric Functions

Functions that express quantities in terms of another variable, often denoted by time (t).

Velocity Vector

A vector that describes the rate of change of the position vector, denoted as 〈x(t), y(t)〉.

Acceleration Vector

A vector that represents the rate of change of the velocity vector.

Speed

The magnitude of the velocity vector, representing how fast an object is moving.

Arc Length

The length of a curve defined parametrically over a given interval [a, b].

Polar Area

The area bounded by a polar curve, calculated using integrals with respect to the angle θ.

Infinite Series

A sum of infinitely many terms, which can converge or diverge based on certain conditions.

Converge or Diverge?

A method to determine if an infinite series approaches a finite limit (converges) or grows indefinitely (diverges).

P-Series

A specific type of infinite series of the form ∑ 1/n^p, which converges if p > 1 and diverges if p ≤ 1.

Integration by Parts

A technique used to integrate products of functions, based on the rule ∫udv = uv - ∫vdu.

U-Substitution

A method for simplifying integrals by substituting a part of the integrand with a new variable.

Improper Integral

An integral that has one or more infinite limits or an integrand that approaches infinity.