Curves, Parametric Equations, and Coordinate Systems in Calculus III

Curvas no Plano (Curves in the Plane)

Gráfico de uma Função (Graph of a Function)

In Calculus, the graph of a function y = f(x) represents the set of all points (x, f(x)) in the plane. This is a fundamental concept for visualizing mathematical relationships and understanding the behavior of functions.

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

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

Cônicas (Conic Sections)

Conic sections are curves obtained by intersecting a plane with a cone. These include the parabola, hyperbola, circle, and ellipse. Each type of conic section has unique geometric and algebraic properties.

• Types of Conics:

  • Parabola

  • Hyperbola

  • Circle

  • Ellipse

• Application: Conic sections appear in physics, engineering, and astronomy (e.g., planetary orbits).

Curvas Parametrizadas (Parametric Curves)

Equações Paramétricas (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. This allows for the representation of more general curves than those described by y = f(x) alone.

• Definition: A parametric curve is given by: $x = x(t), \quad y = y(t), \quad t \in I$

• Interpretation: The parameter t can often be interpreted as time, and the vector $\gamma(t) = (x(t), y(t))$ gives the position of a particle at time t.

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

Sistemas de Coordenadas (Coordinate Systems)

Coordenadas Cartesianas (Cartesian Coordinates)

Cartesian coordinates describe the position of a point P in the plane using two perpendicular axes (x, y). This is the most common coordinate system in mathematics.

• Definition: The coordinates of a point P are (x, y), where x is the horizontal distance and y is the vertical distance from the origin.

Coordenadas Polares (Polar Coordinates)

Polar coordinates describe the position of a point P using the distance from the origin (r) and the angle (θ) from the positive x-axis.

• Definition: The coordinates of a point P are (r, θ), where: $x = r \cos \theta$ $y = r \sin \theta$

• Conversion between systems:

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

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

• Example: The point (1, 1) in Cartesian coordinates corresponds to $r = \sqrt{2}$, $\theta = \frac{\pi}{4}$ in polar coordinates.

Summary Table: Cartesian vs. Polar Coordinates

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

Additional info: These concepts are foundational for studying multivariable calculus, including limits, derivatives, and integrals in higher dimensions.