Parametric Curves

Astroid Curve and Tangents

Parametric equations allow us to describe curves using a parameter, typically denoted as t. The astroid is a classic example of a parametric curve, often defined by equations such as x = a cos3 t and y = a sin3 t. The tangent lines to such curves can be classified as horizontal or vertical based on the derivatives.

• Horizontal Tangent: Occurs when dy/dt = 0 and dx/dt ≠ 0.

• Vertical Tangent: Occurs when dx/dt = 0 and dy/dt ≠ 0.

• Example: For x = 4 cos t, y = 16 sin t, horizontal tangents are found by setting dy/dt = 0, vertical tangents by dx/dt = 0.

Formulas:

• and are computed from the parametric equations.

• The slope of the tangent is .

Arc Length of Parametric Curves

The arc length of a parametric curve defined by x = f(t) and y = g(t) for a ≤ t ≤ b is given by:

• Example: For x = cos3 t, y = sin3 t, the total length is four times the length in the first quadrant.

Polar Coordinates

Introduction to Polar Coordinates

The polar coordinate system represents points in the plane using a distance r from the origin (pole) and an angle θ from the polar axis. This system is an alternative to the rectangular (Cartesian) coordinate system.

• Conversion Formulas:

  • From polar to rectangular: ,

  • From rectangular to polar: ,

• Graphing: In polar coordinates, curves are described by equations of the form r = f(θ).

Basic Curves in Polar Coordinates

• Circle:

• Line through the pole:

• Spiral:

Graphing Polar Equations

There are three main methods to graph polar equations:

1. Table of Values: Compute r for various θ and plot the points.

2. Convert to Rectangular: Use conversion formulas to rewrite the equation in terms of x and y.

3. Use Cartesian Graph as Guide: Sketch the Cartesian graph and use it to inform the polar graph.

Calculus in Polar Coordinates

Slope of Tangent Line in Polar Coordinates

Given a polar equation r = f(θ), the slope of the tangent line at a point is found using parametric representations:

• The slope is

• Where and

Area in Polar Coordinates

The area enclosed by a curve r = f(θ) between θ = α and θ = β is:

• Example: Area of a circle r = 2 \cos \theta is calculated by integrating over the appropriate interval.

Area Between Two Polar Curves

If a region is bounded by two curves r = f(θ) and r = g(θ), the area is:

• Example: Area between r = 1 + \cos \theta and r = \cos \theta.

Conic Sections

Introduction to Conic Sections

Conic sections are curves obtained by intersecting a plane with a double cone. The main types are:

• Parabola

• Ellipse

• Hyperbola

The Parabola

A parabola is the set of points equidistant from a fixed point (focus) and a fixed line (directrix).

• Standard Equation: or

• Focus: Located at or depending on orientation.

• Directrix: Line or .

• Example: For , focus at , directrix .

The Ellipse

An ellipse is the set of points where the sum of distances from two fixed points (foci) is constant.

• Standard Equation:

• Vertices: or

• Foci: or where

• Major Axis: Length

• Minor Axis: Length

• Eccentricity: ,

• Example: has vertices at , foci at .

The Hyperbola

A hyperbola is the set of points where the absolute difference of distances from two fixed points (foci) is constant.

• Standard Equation: or

• Vertices: or

• Foci: or where

• Asymptotes:

• Eccentricity: ,

Summary Table: Conic Sections

Additional info: Some details and examples were inferred for completeness and clarity, especially where original notes were fragmented or brief.