July 08, 2026 - Calculus 2 - Taylor Series and Polar Coordinates

A set of vocabulary flashcards covering Taylor polynomials, power series convergence, and polar coordinate conversion and functions based on the lecture transcript.

Order (Derivatives)

The degree representing the highest derivative involved, such as first or second order.

Maclaurin Series

A Taylor series that is specifically centered at zero ($a = 0$).

Taylor Polynomial ($P_k$)

The $k$-th partial sum of the Taylor series that stops at the derivative term specified by $k$.

Center of Convergence ($a$)

The value around which a Taylor series is expanded, such as $x = 1$ for the natural log expansion.

Radius of Convergence

The distance from the center ($a$) within which a power series is guaranteed to converge.

Ratio Test

The mathematical test used to determine the radius of convergence of a Taylor series.

Cartesian Coordinate System

A system that describes a point in space by its horizontal distance ($x$) and vertical distance ($y$).

Polar Coordinates ($r, \theta$)

A system describing every point in space using its distance from the origin ($r$) and the angle ($\theta$).

Radius ($r$) in Polar Coordinates

The distance from the origin to the point $(x, y)$, calculated as $r = \sqrt{x^2 + y^2}$.

Angle ($\theta$) in Polar Coordinates

The value representing the rotation from the pole, calculated using the relationship $\tan(\theta) = \frac{y}{x}$.

Coordinate Conversion for $x$

The relationship used to find the horizontal distance from polar values, defined as $x = r \cos(\theta)$.

Coordinate Conversion for $y$

The relationship used to find the vertical distance from polar values, defined as $y = r \sin(\theta)$.

Spiral

A polar function where the radius ($r$) is a linear function of the angle ($\theta$).

Cardioid

A polar curve represented by the form $1 \pm \sin(\theta)$ or $1 \pm \cos(\theta)$.

Limacon

A French term for a polar curve that often contains a loop in the middle.

Rose Curves

Polar functions where the number of petals depends on the coefficient of the angle within the trigonometric function.

Area of a Sector

The formula $\frac{1}{2} r^2 \theta$ used to find the area of a slice of a circle.

Polar Integration

The method of finding area under a polar curve by integrating $\int_{\alpha}^{\beta} \frac{1}{2} [f(\theta)]^2 \,d\theta$.

Harmonic Series

A series that resembles the Taylor expansion of $\ln(x)$ at certain points and is known to diverge.

Alternating Series Test

A convergence test used for series where the sign of each term alternates between positive and negative.