Definition of parametric equations:
Suppose that x and y are continuous functions of a third variable t. Then x=f(t) and y=g(t) are called parametric equations for the curve represented by (x,y).

Lines in polar coordinates:
Let and  then the polar equations of the lines x=a and y=b are and  for all values of  

Parametric form of derivatives:
Suppose a smooth curve C is defined by and then the slope of C at (x,y) is given by with

Arc length of parametric curves:
Suppose that and are continuous functions on the interval then the curve and has arc length is given by

Parametric equations are presented to motivate the idea of their derivatives and how they are related to the parametric form. Examples are shown to denote the characteristics of how the derivative formulas affect the vertical tangent and the concept of a singular point.

The arc length of parametric curves is shown with the use of parametric curves. Polar coordinates and their graphical interpretation are shown with the use of examples. Converting rectangular coordinates to polar coordinate is shown to motivate the idea of the arc length in polar coordinates. The polar coordinate graphs are given to discuss the idea of how you represent circles in polar coordinates. Polar coordinates are important to know in a typical Calculus course.

Specific Tutorial Features:

• Several example problems with step by step illustrations of solutions
  
• Graphs showing equations in polar coordinate form and rectangular form.
  
• The different types of representation of the arc length using derivatives are presented in this tutorial.

Series Features:

• Concept map showing inter-connections of new concepts in this tutorial and those previously introduced.
  
• Definition slides introduce terms as they are needed.
  
• Visual representation of concepts
  
• Animated examples—worked out step by step
  
• A concise summary is given at the conclusion of the tutorial.

Parametric Equations
   Examples of parametric equations
   Parametric form of derivatives
   Arc length of parametric curves
Polar coordinates
   Converting Rectangular Coordinates to Polar Coordinates
   Area in polar coordinates
   Arc length in polar coordinates
Graphs in Polar Coordinates