Curves in the Plane, Conics, and Coordinate Systems – Calculus III Study Notes

Curves in the Plane

Graphs of Functions

In calculus, a curve in the plane can often be represented as the graph of a function y = f(x). The graph is the set of all points (x, f(x)) for each x in the domain of f.

• Definition: The graph of a function y = f(x) is the set of points (x, f(x)) in the plane.

• Example: The parabola y = x^2 is the set of all points (x, x^2).

Conics

Conic sections are curves obtained by the intersection of a plane with a double-napped cone. These curves include the parabola, ellipse, circle, and hyperbola.

• Definition: A conic is any curve formed by the intersection of a plane and a cone.

• Types of Conics:

  • Circle: Plane perpendicular to the cone's axis.

  • Ellipse: Plane at an angle, not parallel to the side of the cone.

  • Parabola: Plane parallel to the side of the cone.

  • Hyperbola: Plane intersects both nappes of the cone.

Parametric Curves

Parametric Equations

A parametric curve in the plane is defined by a pair of functions x = x(t) and y = y(t), where t varies over an interval I of real numbers. The parameter t often represents time.

• General Form: $x = x(t)$ $y = y(t)$ $t \in I$

• Interpretation: t can represent time, and \gamma(t) = (x(t), y(t)) gives the position of a particle at time t.

• Example: The unit circle can be parametrized as $x = \cos t$ $y = \sin t$ $t \in [0, 2\pi]$

Coordinate Systems

Cartesian and Polar Coordinates

Points in the plane can be described using either Cartesian coordinates (x, y) or polar coordinates (r, \theta).

• Cartesian Coordinates: (x, y) specify the horizontal and vertical distances from the origin.

• Polar Coordinates: (r, \theta) specify the distance r from the origin and the angle \theta from the positive x-axis.

Conversion Between Cartesian and Polar Coordinates

• From Polar to Cartesian: $x = r \cos \theta$ $y = r \sin \theta$

• From Cartesian to Polar: $r = \sqrt{x^2 + y^2}$ $\cos \theta = \frac{x}{r}$ $\sin \theta = \frac{y}{r}$ $\theta = \arctan\left(\frac{y}{x}\right)$

Summary Table: Cartesian vs. Polar Coordinates

Cartesian (x, y)	Polar (r, \theta)
$x = r \cos \theta$ $y = r \sin \theta$	$r = \sqrt{x^2 + y^2}$ $\theta = \arctan\left(\frac{y}{x}\right)$

Additional info:

• These topics are foundational for Calculus III, especially for understanding curves, surfaces, and coordinate transformations.

• Conic sections are important in geometry, physics, and engineering applications.

• Parametric equations and polar coordinates are essential for describing more general curves and for integration in multiple variables.