Preview For Chapter 10 and 9
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  Chapter 10 we introduce parametric equations and polar equations.  We consider initial value problems and boundary value problems in Chapter 9 as the last topic.
 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
P636 Look at the method of eliminating t in both Example 1 on this page and Example 2 on the next page. 
  P637 Look at the video in the ebook.
2. Graph the curve defined by parametric equations. P641#1,3,7,11,15
P636 Look at the respective graphs for Examples 1, 2, 3, 4 and 6.  Note that  Examples 5 and 6 use  the graphing calculator to do the respective graphs. Example 7 shows the power of the graphing utility (graphing calculator or Maple).
  P637 Look at the video in the ebook.
3. Give parameterizations for popular curves like the straight line, parabola and circle. P641#14,31,33
P637 The outline on how to do this parameterization is found in Example 4 and in general just below Example 5.
  P638 Look at the video in the ebook.
4. Answer the Brachistochrone problem.
P640 Figure 14 and the paragraphs in the middle of the page describe the Brachistochrone problem as finding the curve along which a particle will slide in the shortest time (under the influence of gravity) from a point A to a lower point B not directly under A.
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
P645 Box 1 gives the formula for the first derivative.   The formula for the second derivative is given just below that box.  Examples 1 and 2 are helpful.
  P646 Look at the video in the ebook.
2. Find the slopes of the tangent line to a parametric curve. P672#3,7
P645 Box 1 gives the formula for the first derivative which is the slope.  Example 1 is  helpful.
3. Give the equation of the tangent line in rectangular form and parametric form. P651#3,7
P646 Example 1 is  helpful.
4. Find the points on a parametric curve where the tangent line is horizontal or vertical. P651#17
P646 Example 1 is  helpful.
5. Find the area "under a parametric curve". P651#33,35
P647 The formula is given in the middle of this page.   Example 3 is  helpful.
  P647 Look at the video in the ebook.
6. Find the length of a parametric curve. P651#37,41
P649 The formula for the arc length is given in box 5 as a theorem.  The derivation of the formula is given on pages 648 and 649.  Look at examples 4 and 5 to see how to use the formula. 
  P649 Look at the video in the ebook.
7. Find the surface area of revolution of parametric curves revolved about the x or y axes. P652#57,59
P650 The formula for the surface area is given in box 6   Look at example 6 to see how to use the formula. 
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
P656 Look at examples 4, 5, 6, 7,  and 8 to see how to graph these functions.  Look at examples 10 and 11 to see the use of graphical devices in graphing polar functions.
  P656 Look at the video in the ebook. Example 4
  P658 Look at the video in the ebook. Example 7
  P662 Look at the video in the ebook. Example 11
2. Change equations back and forth between rectangular and polar coordinates. P663#15-25 odd
P655 The formulas to change back and forth are given in boxes 1 and 2. Look at example 6 to see how to use these formulas in converting equations.    Examples 1, 2, and 3 show how to convert points using these formulas.
3. Find the slope of the tangent line for a polar curve. P664#57
P659 The general formula is found in box three.  The formula at the pole is given at the bottom of the page as a special case.  Look at example 9 to see how to use these formulas. 
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
P686 The formula is given in boxes 3 and 4. Look at examples 1 and 2.
  P666 Look at the videos in the ebook. Examples 1 and 2
2. Find the arc length of curves in polar coordinates. P669#45
P668 The formula is given in box 5. Look at example 4.
  P668 Look at the video in the ebook.
3. Find the surface area of revolution of polar curves revolved about the x or y axes. P670#55
P670 The formula for the surface area is given in the problem. Use the formula in box 5 as the formula of the arc length to derivate the formula given in the problem.
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
  P606 The general solution of the differential equation where the rate of change of a function is proportional to the function is given in box 2.
  P606 The differential equation where the rate of change of a function is proportional to the function, but harvesting also occurs is discussed at the bottom of the page.
  P607 The logistic model used in growth problems is discussed on this page and the following ones.  Look at Examples 1 and 2. 
  P607 Look at the video in the ebook
  P610 The comparison of natural growth and logistic models is discussed at the top of this page.  Look at example 3.
  P610 Look at the video in the ebook
  P612 The other models are briefly explained here.
Maple Assignment:  none