PEPCVF - Parametric Equations, Polar Coordinates, and Vector Functions Module Overview

Parametric Equations, Polar Coordinates, and Vector Functions Module Overview

Introduction

A curve in the plane consists of a collection of points and the defining parametric equations. What is different here from your previous experience is that two distinct curves can have the same graph because the curve is traced out in different ways on the graph. Vector-valued functions incorporate parametric equations as component functions of the vector function. Likewise with polar curves, parametric equations serve as the vehicle for finding tangents to polar curves. Finding areas defined by polar coordinates requires a new geometric perspective, pie-shaped wedges, rather than the rectangular strips used with rectangular coordinates.

Essential Questions

What are some advantages of using parametric equations for graphing?
How do you find the length of a curve given in parametric form?
How do you find the area enclosed by a polar curve?
What considerations are needed when analyzing planar curves given in polar form?
How can technology help when investigating graphs of parametric and polar curves?
How do you compute velocity and acceleration of vector-valued functions?
What advantages exist when using parametric, polar, and vector functions to analyze curves?

Key Terms

The following key terms will help you understand the content in this module.

Acceleration vector - The second derivative of a vector function r(t) given by $LaTeX: \textbf{a}\left(t\right)=\textbf{v'}\left(t\right)=\textbf{r''}\left(t\right)=\textbf{x''}\left(t\right)i+y''\left(t\right)\textbf{j}$ $\textbf{a}\left(t\right)=\textbf{v'}\left(t\right)=\textbf{r''}\left(t\right)=\textbf{x''}\left(t\right)i+y''\left(t\right)\textbf{j}$

Component form of a vector - An ordered set that represents a vector and is denoted by $LaTeX: \textbf{v}=<v_1,\:v_2>$ $\textbf{v}=<v_1,\:v_2>$

Derivative of a vector function - Defined as $LaTeX: \vec{r}\,'(t)=\frac{dr}{dt}=\lim_{h \to 0}\frac{r(t+h)-r(t)}{h}$ $\vec{r}\,'(t)=\frac{dr}{dt}=\lim_{h \to 0}\frac{r(t+h)-r(t)}{h}$ , if this limit exists.'

Dot product - The scalar product of two vectors u = < u₁, u₂ >, and v = < v₁, v₂ > and given by u ⋅ v = u₁v₁+u₂v₂.

Eliminating the parameter - Finding a rectangular equation that represents the graph of a set of parametric equation.

Length (Magnitude) of a vector - The length of a vector v = < v₁, v₂ >, given by $LaTeX: \left|\textbf{v}\right|=\sqrt[]{v^2_1+v^2_2}$ $\left|\textbf{v}\right|=\sqrt[]{v^2_1+v^2_2}$

Parameter - An arbitrary constant whose value affects the specific nature but not the formal properties of a mathematical expression.

Parametric equations - A set of equations expressing a number of quantities as explicit functions of the same set of independent variables (parameters) and equivalent to some direct functional relationship between these quantities.

Plane curve - The parametric equations x = f(t) and y = g(t), where f and g are continuous functions of t, taken together with the graph of the parametric equations.

Polar axis - A horizontal ray beginning at the pole and extending to the right.

Polar coordinates - A pair of coordinates (r, ϴ) that locates a point in a plane by its distance r from the origin/pole and its angle of inclination theta ϴ.

Polar graph - The set of points whose polar coordinates r and ϴ satisfy the equation r = f(ϴ).

Position vector - A vector represented by a directed line segment whose initial point is the origin.

Smooth curve - A curve with parametric representation x = f(t) and y = g(t) on an interval I where f and g have continuous first derivatives on I and are not simultaneously 0, except possibly at the endpoints of I.

Speed - The magnitude of the velocity vector and given by $LaTeX: \left|\textbf{v'}\left(t\right)\right|=\left|\textbf{r'}\left(t\right)\right|=\sqrt[]{\left[x'\left(t\right)\right]^2+\left[y'\left(t\right)\right]^2}$ $\left|\textbf{v'}\left(t\right)\right|=\left|\textbf{r'}\left(t\right)\right|=\sqrt[]{\left[x'\left(t\right)\right]^2+\left[y'\left(t\right)\right]^2}$ .

Tangent vector - The derivative vector r'(t) to the curve defined by r at the point P, provided that r'(t) exists and r'(t) ≠ 0.

Unit vector - Any vector of length 1.

Unit tangent vector - A vector defined by $LaTeX: \vec{T}\,(t)=\frac{\vec{r}\,'(t)}{|\vec{r}\,'(t)|}$ $\vec{T}\,(t)=\frac{\vec{r}\,'(t)}{|\vec{r}\,'(t)|}$

Vector - A term used to describe a quantity that has both magnitude and direction.

Vector-valued function - A function whose domain is a set of real numbers and whose range is a set of vectors.

Velocity vector - The tangent vector (first derivative of the position vector) that points in the direction of the tangent line and defined by v(t) = r'(t) = x'(t) i > + y'(t)j = $LaTeX: \frac{d\vec{t}\,}{dt}=\frac{\vec{r}\,(t+h)-\vec{r}\,(t)}{h}$ $\frac{d\vec{t}\,}{dt}=\frac{\vec{r}\,(t+h)-\vec{r}\,(t)}{h}$

Zero vector - The only vector with length 0 and no specific direction, which is denoted by 0 = < 0, 0 >

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