Preview For Chapter 10
To date, we have portrayed functions numerically, graphically and algebraically but always in terms of the rectangular coordinate system.  So algebraically, we have written  y = f(x).  In this chapter we introduce parametric equations and polar equations.
 Parametric Equations allow us to write both the x and the y from rectangular coordinates in terms of a parameter, usually t.  If we think of t as time, then we can interpret x(t) and y(t) as a point (x,y) which is moving as time passes.  Thus instead of fixed curves in a rectangular coordinate system, we now can see a point moving along a curve in a rectangular coordinate system.

Polar Coordinates allow us to simplify the consideration of functions and their curves which have pronounced symmetry about the origin.

We consider the "dozen" aspects of the calculus of both of these new topics including slopes of tangents, equations of tangents, areas under curves, arc length, surface area and other applications.
 
A student completing this chapter successfully will be able to:
bulletGraph functions defined parametrically.
bulletUse derivatives and integrals of parametric functions to find areas, arc length and surface area.
bulletGraph functions defined using polar coordinates.
bulletUse derivatives and integrals of polar functions to find areas, arc length and surface area.
bulletSolve boundary value problems for growth and decay models.
 

What Can Parametric Equations Do For You?

Suppose you have been assigned the responsibility of designing the water slide for an amusement park.  To maximize the thrill level and make the lines move faster, your slide is to take the person from the top of the slide to the pool in the shortest amount of time.  The slide begins at the top of the tower 150 feet high and ends in the pool 200 feet away from the base of the tower. What path will take the thrill seeker from the top to the bottom in the shortest time?  The path is the inverted cycloid which is explored in the Maple worksheet from this chapter named Brachist.mws.

 

 

Calculus 2     Lesson 10.1     Parametric Equations
The following topics are the most important. Typical exercises from Calculus, 7th edition-Early Transcendentals, by James Stewart, are assigned at the end of each objective.  

In smaller print following the objective is a guide for the reading assignment in the form of page numbers in the text and notes regarding the reading about the objective
1. Eliminate the parameter in parametric equations to find the rectangular equation of the parameterized curve. P641#1,3,7,11,15
2. Graph the curve defined by parametric equations. P641#1,3,7,11,15
3. Give parameterizations for popular curves like the straight line, parabola and circle. P641#14,31,33
4. Answer the Brachistochrone problem.
Maple Assignment:  See Maple Assignment
 Graphing Calculator Hints show how to graph a parametric equation using the TI-86.

 

Calculus 2     Lesson 10.2     Calculus with Parametric Curves
The following topics are the most important. Typical exercises from Calculus, 7th edition-Early Transcendentals by James Stewart, are assigned at the end of each objective.   Calculus, 6th edition,

In smaller print following the objective is a guide for the reading assignment in the form of page numbers in the text and notes regarding the reading about the objective
1. Find the first and second derivatives (dy/dx and d2y/dx2) for parametric equations. P651#11,13
2. Find the slopes of the tangent line to a parametric curve. P672#3,7
3. Give the equation of the tangent line in rectangular form and parametric form. P651#3,7
4. Find the points on a parametric curve where the tangent line is horizontal or vertical. P651#17
5. Find the area "under a parametric curve". P651#33,35
6. Find the length of a parametric curve. P651#37,41
7. Find the surface area of revolution of parametric curves revolved about the x or y axes. P652#57,59
Maple Assignment:  See Maple Assignment

 

Calculus 2     Lesson 10.3     Polar Coordinates 
The following topics are the most important. Typical exercises from Calculus, 7th edition-Early Transcendentals, by James Stewart, are assigned at the end of each objective.  

In smaller print following the objective is a guide for the reading assignment in the form of page numbers in the text and notes regarding the reading about the objective
1. Graph curves expressed in polar coordinates. P663#29-41 odd,71,73,75
2. Change equations back and forth between rectangular and polar coordinates. P663#15-25 odd
3. Find the slope of the tangent line for a polar curve. P664#57
4. Explain dr/(d theta) geometrically. Compare with dy/dx.    (dr/(d theta) pos => ?, dr/(d theta ) neg => ?) .   Derivative of r w/r theta Interpretation
 Graphing Calculator Hints show how to graph a polar equation using the TI-86.
Maple Assignment:  none

 

Calculus 2     Lesson 10.4     Areas and Lengths 
The following topics are the most important. Typical exercises from Calculus, 7th edition-Early Transcendentals, by James Stewart, are assigned at the end of each objective.  

In smaller print following the objective is a guide for the reading assignment in the form of page numbers in the text and notes regarding the reading about the objective
1. Find the area bounded by curves in polar coordinates. P668#3,7,17,25,29
2. Find the arc length of curves in polar coordinates. P669#45
3. Find the surface area of revolution of polar curves revolved about the x or y axes. P670#55
Maple Assignment:  See Maple Assignment
 


Calculus 2     Lesson 9.4    Models for Population growth
The following topics are the most important. Typical exercises from Calculus, 7th edition-Early Transcendentals, by James Stewart, are assigned at the end of each objective.  

In smaller print following the objective is a guide for the reading assignment in the form of page numbers in the text and notes regarding the reading about the objective
1. Solve growth problems using various models. P613#1,3,5,7,9,13,14,15,17,18
Maple Assignment:  none