Curves in the Plane, Conic Sections, and Coordinate Systems

Curves in the Plane

Graphs of Functions

The graph of a function y = f(x) represents the set of all points (x, f(x)) in the plane, where x belongs to the domain of the function. This is a fundamental concept in calculus, as it allows us to visualize the behavior of functions and analyze their properties.

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

• Example: For f(x) = x2, the graph is a parabola opening upwards.

Conic Sections

Definition and Types

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. The type of curve depends on the angle and position of the intersection:

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

• Ellipse: Intersection at an angle, but not parallel to the base or side.

• Circle: Special case of an ellipse where the plane is perpendicular to the axis of the cone.

• Hyperbola: Intersection cuts through both nappes of the cone.

These curves are fundamental in calculus and analytic geometry, with applications in physics, engineering, and astronomy.

Parametric Curves

Parametric Equations

A parametric curve in the plane is defined by a pair of functions x(t) and y(t), where t varies over an interval I. The set of points (x(t), y(t)) traces out the curve as t changes.

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

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

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

Coordinate Systems

Cartesian and Polar Coordinates

Points in the plane can be represented using different coordinate systems:

• Cartesian coordinates: Each point is given by (x, y), where x and y are distances along perpendicular axes.

• Polar coordinates: Each point is given by (r, θ), where r is the distance from the origin and θ is the angle from the positive x-axis.

Conversion between Cartesian and Polar Coordinates

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

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

Example:

• Given a point (3, 4) in Cartesian coordinates: $r = \sqrt{3^2 + 4^2} = 5$ $\theta = \arctan\left(\frac{4}{3}\right)$

Summary Table: Coordinate Conversion

Cartesian (x, y)	Polar (r, θ)
x = r cos θ	r = √(x² + y²)
y = r sin θ	θ = arctan(y/x)

Applications: Polar coordinates are especially useful for describing curves with circular or rotational symmetry, such as circles, spirals, and some conic sections.

Additional info: These topics are foundational for calculus involving functions of several variables, vector calculus, and advanced integration techniques.