Preview For Chapter 11 |
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| 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. |
| 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 11.1
Parametric Equations |
| The following topics are the most
important. Typical exercises from Calculus, 6th edition, 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. P662#1,3,7,11,17 |
| 2. |
Graph the curve defined by parametric
equations. P662#1,3,7,11,17 |
| 3. |
Give parameterizations for popular curves
like the straight line, parabola and circle. P662#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 11.2
Calculus with Parametric Curves |
| The following topics are the most
important. Typical exercises from Calculus, 6th edition, 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 first and second derivatives
(dy/dx and d2y/dx2) for parametric equations. P672#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. P672#3,7 |
| 4. |
Find the points on a parametric curve
where the tangent line is horizontal or vertical. P672#17 |
| 5. |
|
Find the area "under a parametric
curve". P673#33,35 |
| 6. |
Find the length of a parametric curve. P673#37,41 |
| 7. |
Find the surface area of revolution of parametric curves
revolved about the x or y axes. P763#57,59 |
| Maple Assignment: See
Maple Assignment |
Calculus
2 Lesson 11.3
Polar Coordinates |
| The following topics are the most
important. Typical exercises from Calculus, 6th edition, 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. P684#29-41 odd,71,73,75 |
| 2. |
Change equations back and
forth between rectangular and polar coordinates. P684#15-25 odd |
| 3. |
Find the slope of the
tangent line for a polar curve. P684#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 11.4
Areas and Lengths |
| The following topics are the most
important. Typical exercises from Calculus, 6th edition, 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.
P689#3,7,17,25,29 |
| 2. |
Find the arc length of curves in polar coordinates.
P690#45 |
| 3. |
Find the surface area of revolution of polar curves revolved
about the x or y axes. P690#55 |
| Maple Assignment: See
Maple Assignment |
Calculus
2 Lesson 10.4 Models
for Population growth |
| The following topics are the most
important. Typical exercises from Calculus, 6th edition, 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. P634#1,3,5,7,13,14,15,17,18 |
| Maple Assignment: none |
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