Brief Study Guide for Chapters 10 and
11 |
Preview For Chapter 10 |
Differential
Equations |
In Calculus 1, solving equations involving a derivative was
limited to antidifferentiating an expression involving only one variable, usually x.
The expression might look like y' = x2. The
solution would be y = x3/3 + C. In Chapter 10 we solve a
differential equation, an equation involving x, y, and y', all sort of mixed
in together. |
Many techniques for solving differential equations are
presented in our course with that name, Differential Equations (MAT212). We present
but one method in Calculus 2 named "Separation of Variables". It is
presented in section 10.4 and is used in solving many growth and decay problems. |
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Calculus
2 Lesson 10.4 Separation of Variables |
The following topics are the most
important. Typical exercises from Calculus, 5th edition, by James Stewart, are
assigned at the end of each objective. |
1. |
Solve growth and decay
problems by the method of separation of variables. P656#2,3,5,8,11,17,18 |
Maple Assignment: none |
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Preview For
Chapter 11 |
Parametric Equations and
Polar Coordinates |
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. |
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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?
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Calculus
2 Lesson 11.1
Parametric Equations |
The following topics are the most
important. Typical exercises from Calculus, 5th edition, by James Stewart, are
assigned at the end of each objective. |
1. |
Eliminate the parameter in parametric
equations to find the rectangular equation of the parameterized curve. P692#1,3,7,11,18 |
2. |
Graph the curve defined by parametric
equations. P692#1,3,7,11,18 |
3. |
Give parameterizations for popular curves
like the straight line, parabola and circle. P692#18,31,33 |
4. |
Answer the Brachistochrone
problem. |
Graphing
Calculator Hints show how to graph a parametric equation using the
TI-86. |
Maple Assignment: See
Maple Assignment |
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Calculus
2 Lesson 11.2
Calculus with Parametric Curves |
The following topics are the most
important. Typical exercises from Calculus, 5th edition, by James Stewart, are
assigned at the end of each objective. |
1. |
Find the first and second derivatives
(dy/dx and d2y/dx2) for parametric equations. P702#11,13 |
2. |
Find the slopes of the tangent line to a
parametric curve. P702#3,7 |
3. |
Give the equation of the tangent line in
rectangular form and parametric form. P702#3,7 |
4. |
Find the points on a parametric curve
where the tangent line is horizontal or vertical. P703#17 |
5. |
Find the area "under a parametric
curve". P703#33,35 |
6. |
Find the length of a parametric curve.
P703#37,41 |
7. |
Find the surface area of revolution of parametric curves
revolved about the x or y axes. P703#57,59 |
Maple Assignment:
See Maple Assignment |
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Calculus
2 Lesson 11.3
Polar Coordinates |
The following topics are the most
important. Typical exercises from Calculus, 5th edition, by James Stewart, are
assigned at the end of each objective. |
1. |
Graph curves expressed in polar
coordinates. P714#29-41 odd,69,71,73 |
2. |
Change equations back and forth between
rectangular and polar coordinates. P714#15-25 odd |
3. |
Find the slope of the tangent line for a
polar curve. P714#55 |
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 |
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Calculus
2 Lesson 11.4
Areas and Lengths |
The following topics are the most
important. Typical exercises from Calculus, 5th edition, by James Stewart, are
assigned at the end of each objective. |
1. |
Find the area bounded by curves in polar coordinates.
P719#3,7,17,25,29 |
2. |
Find the arc length of curves in polar coordinates.
P720#45 |
3. |
Find the surface area of revolution of polar curves revolved
about the x or y axes. P720#55 |
Maple Assignment:
See Maple Assignment |
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