Parametric Equations and Vectors

Study Notes

Parametric Equations

Parametric equations represent a curve where the x and y coordinates are defined in terms of a third variable, often denoted as 't', called a parameter.
The parameter 't' often represents time, but can be any variable.
To find the rectangular equation (or Cartesian equation), solve for 't' in one equation and substitute into the other.
Parametric equations are useful for describing curves that are not easily described by a single equation, like cycloids or spirals.
The derivatives of x and y with respect to 't' can be used to determine the slope of the tangent line to the curve.
The arc length of a curve defined parametrically is given by [ \int_{t_1}^{t_2} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt ]
Parametric equations can often be used for modeling motion.

Vectors

A vector is a quantity with both magnitude and direction.
Vectors can be represented graphically by arrows.
Vectors can be added and subtracted using the parallelogram law or tip-to-tail method.
Scalar multiplication of a vector multiplies both its magnitude and direction.
Components of a vector can be determined using the coordinates on a coordinate system.
Vectors can be used to represent displacements, forces, velocities, and other physical quantities.
Magnitude/length of a vector: [ ||\bold{a}|| = \sqrt{a_x^2 + a_y^2 + a_z^2} ]
Unit vector: a vector with a magnitude of 1.

Vector-Valued Functions

A vector-valued function is a function that maps a scalar input to a vector output. This creates a curve in space.
They are often written as (\bold{r}(t) = <x(t), y(t), z(t)>) where 't' is a parameter.
Derivative of a vector-valued function represents the tangent vector to the curve.
The derivative at a point represents the instantaneous velocity.
Integrals of vector-valued function represent the accumulation of position of the vector from a specific time.

Polar Functions

Polar coordinates represent a point in the plane using a distance from the origin (r) and an angle from the positive x-axis.
The relationship between polar coordinates and Cartesian coordinates is:
x = r cos(θ)
y = r sin(θ)
Polar equations are frequently used to define specific curves, such as circles, roses, limaçons, and spirals.
Conversion between polar and rectangular equations is useful for graphing polar curves.
Solving polar equations might involve trigonometry or algebraic manipulation.
Polar graphs are inherently different than graphs in the rectangular plane.

Integration Techniques

Substitution: Replacing part of the integral with a new variable.
Integration by parts: Using the product rule in reverse. The formula is (\int u dv = uv-\int vdu). Choose (u) and (dv) based on the integral to reduce complexity.
Partial Fraction Decomposition: Decomposing a rational function into simpler fractions before integrating. This technique is helpful for integrating rational functions that have repeating factors or denominators.
Trigonometric Integrals: Using trigonometric identities to simplify integrals involving trigonometric functions.
Trigonometric Substitution: Substituting trigonometric functions for algebraic expressions.
Table of integrals: Using a table of common integrals to look for patterns.
Recognizing basic integral patterns: Memorizing basic or common integrals for immediate substitution and streamlining the solution process.
Recognizing and applying specific properties of definite integrals, like the order, limits, or intervals of integration, is essential.

Description

This quiz explores the concepts of parametric equations and vectors. Discover how parametric equations define curves through a parameter and learn about the representation and manipulation of vectors. Understand applications in motion modeling and geometry.