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.

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

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.

 

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?

 

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

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

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

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
 

Maple worksheets to be submitted for a grade for Chapter 10&11: 

See Maple Assignment