Preview For Chapter 11 |
|
| 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 |
|
P657 |
Look at the method of eliminating t in both Example 1
on this page and Example 2 on the next page. |
| 2. |
Graph the curve defined by parametric
equations. P662#1,3,7,11,17 |
|
P657 |
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). |
| 3. |
Give parameterizations for popular curves
like the straight line, parabola and circle. P662#14,31,33 |
|
P659 |
The outline on how to do this parameterization is found
in Example 4 and in general just below Example 5. |
| 4. |
Answer the Brachistochrone
problem. |
|
P661 |
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 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 |
|
P666 |
Box 2 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. |
| 2. |
Find the slopes of the tangent line to a
parametric curve. P672#3,7 |
|
P666 |
Box 2 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. P672#3,7 |
|
P667 |
Example 1 is helpful. |
| 4. |
Find the points on a parametric curve
where the tangent line is horizontal or vertical. P672#17 |
|
P667 |
Example 1 is helpful. |
| 5. |
|
Find the area "under a parametric
curve". P673#33,35 |
|
P668 |
The formula is given at the bottom of this page.
Example 3 is helpful. |
| 6. |
Find the length of a parametric curve. P673#37,41 |
|
P670 |
The formula for the arc length is given in box
6 as a theorem. The derivation of the formula is given on pages 669 and
670. Look at
examples 4 and 5 to see how to use the formula. |
| 7. |
Find the surface area of revolution of parametric curves
revolved about the x or y axes. P763#57,59 |
|
P671 |
The formula for the surface area is given in
box 7.
Look at example 6 to see how to use the formula. |
| 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 |
|
P677 |
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. |
| 2. |
Change equations back and
forth between rectangular and polar coordinates. P684#15-25 odd |
|
P676 |
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. P684#57 |
|
P680 |
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 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 |
|
P686 |
The formula is given in boxes 3 and 4. Look at examples
1 and 2. |
| 2. |
Find the arc length of curves in polar coordinates.
P690#45 |
|
P688 |
The formula is given in box 5. Look at example 4. |
| 3. |
Find the surface area of revolution of polar curves revolved
about the x or y axes. P690#55 |
|
P690 |
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 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 |
| |
P627 |
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. |
| |
P628 |
The differential equation where the rate of
change of a function is proportional to the function, but harvesting also
occurs is discussed at the top of the page. |
| |
P628 |
The
logistic model used in growth problems is discussed on this page and the
following ones. Look at Examples 1 and 2. |
| |
P632 |
The
comparison of natural growth and logistic models is discussed at the
top of this page. Look at example 3. |
| |
P633 |
The other
models are briefly explained here. |
| Maple Assignment: none |
|