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MA102: Single-Variable Calculus II

Unit 4: Parametric Equations and Polar Coordinates   So far we have worked in Cartesian (rectangular) coordinates where there has been one dependent variable, say x, and one dependent variable y=f(x).  At times this has been inconvenient.  Think about the equation describing the graph of a circle, x2 + y2 = r2: here, y cannot be given as an explicit function of x.  For situations like this one there are other ways of describing graphs which make calculations much simpler.

You studied parametric equations and polar coordinates in subunits 4.2 and 4.5 of Precalculus II (MA003), so for an alternate approach, you may review those subunits.  Make sure to come back to this unit, because MA003 does not cover as much material.

Unit 4 Time Advisory
This unit should take you 12.5 hours to complete.

☐    Subunit 4.1: 3.5 hours

☐    Subunit 4.2: 4 hours

☐    Reading: 0.75 hours

☐    Lecture: 0.75 hours

☐    Assessment: 2.5 hours

☐    Subunit 4.3: 2.5 hours

☐    Subunit 4.4: 1.25 hours

☐    Subunit 4.5: 1.25 hours

Unit4 Learning Outcomes
Upon successful completion of this unit, the student will be able to: - Graph parametric equations. - Find derivatives of parametric equations. - Convert between Cartesian and polar coordinates. - Graph equations in polar coordinates. - Compute derivatives of equations in polar coordinates. - Compute areas under curves described by polar coordinates. - Compute arc length for curves given in polar coordinates.

4.1 Parametric Equations and Their Derivatives   Parametric equations treat x and y each as functions of a third variable, typically t.  It is helpful to think of t as time, and the equations as instructing how a curve is to be drawn, giving the pen’s coordinates for each point in time.  This easily extends to curves in three-dimensional space by adding an equation for z as a function of t.

  • Reading: University of Michigan’s Scholarly Monograph Series: Wilfred Kaplan’s and Donald J. Lewis’s Calculus and Linear Algebra Vol. 1: “3-9 Parametric Equations” Link: University of Michigan’s Scholarly Monograph Series: Wilfred Kaplan’s and Donald J. Lewis’s Calculus and Linear Algebra Vol.1: “3-9 Parametric Equations” (HTML)

    Instructions: Please click on the link above, and read the assigned section.  Use the “previous” and “next” links at the bottom of each webpage to navigate through the reading.  This reading discusses parametric equations for curves and how to differentiate them.  Note the similarity to related rates.  Ignore the reference in the first paragraph to vector equations; that is Calculus III material.

    Studying this reading should take approximately 30 minutes.

    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Lecture: Khan Academy's “Parametric Equations I” and “Parametric Equations II”; MIT: David Jerison’s “Lecture 31: Parametric Equations, Arclength, Surface Area” and “Lecture 32: Polar Coordinates; Area in Polar Coordinates” Khan Academy's “Parametric Equations I” (YouTube) and “Parametric Equations II” (YouTube); MIT: David Jerison’s “Lecture 31: Parametric Equations, Arclength, Surface Area” (YouTube)

    Also Available in:
    iTunes U

    and “Lecture 32: Polar Coordinates; Area in Polar Coordinates” (YouTube)

    Also Available in:
    iTunes U

    Instructions: Please click on the links above to watch Salman Khan’s “Parametric Equation I” and “Parametric Equations II.”  Then, watch Professor Jerison’s Lecture 31 from time 40:35 minutes to the end and Lecture 32 from the beginning up to time 22:50 minutes.  Note that lecture notes are available in PDF; the link is on the same page as the lecture.

    Salman Khan’s first video gives a very intuitive example of the concept of parametric curves—two-dimensional motion with a time parameter.  The second video shows how to eliminate the parameter and gives a second example.  In the lectures from MIT, Professor Jerison will work through a more advanced example and discuss how to calculate arc length for curves expressed by parametric equations.

    Viewing these lectures and pausing to take notes should take approximately 1 hour.

    Terms of Use: The Khan videos above are released under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.  They are attributed to Khan Academy.

    The MIT videos above are released under a Creative Commons Attribution-Share-Alike License 3.0.  They are attributed to MIT.

  • Assessment: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Parameterizations of Plane Curves” and “Calculus with Parameterized Curves” Link: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Parameterization of Plane Curves” (PDF) and “Calculus with Parameterized Curves” (PDF)

    Also Available in:

    HTML (“Parameterization of Plane Curves”)

    HTML (“Calculus with Parameterized Curves”)

    Instructions: Please click on the first link above, and work through each of the nine examples on the page.  As in any assessment, solve the problem on your own first.  Solutions are given beneath each example.  Do the same with the second link; work through each of the eight examples on the page on your own before checking them with the given solutions.

    Completing these assessments should take approximately 2 hours.

    Terms of Use: The material above has been reposted by the kind permission of Elizabeth Wood, and can be viewed in its original form here (HTML) and here (HTML).  Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.

4.2 Polar Coordinates   Note: Polar Coordinates are both a different coordinate system to describe two-dimensional space and, when related back to the “x-y plane,” a different parameterization for curves in that system.  Instead of representing location by horizontal and vertical distance from the origin, we represent it by straight-line distance from the origin and angle from the positive x-axis (measured counter-clockwise).

  • Reading: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 7: Trigonometric Functions: “Section 7.7: Polar Coordinates” Link: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 7: Trigonometric Functions: “Section 7.7: Polar Coordinates” (PDF)

    Instructions: Please click on the link above, and read Section 7.7 in its entirety (pages 406 through 411).  Polar coordinates use two parameters, angle and radius, to describe the graphs of curves.

    Studying this reading should take approximately 45 minutes.

    Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0 (HTML).  It is attributed to H. Jerome Kiesler and the original version can be found here (PDF).

  • Lecture: University of Houston: Selwyn Hollis’s “Video Calculus: Polar Coordinates and Graphs” Link: University of Houston: Selwyn Hollis’s “Video Calculus: Polar Coordinates and Graphs” (QuickTime)

    Instructions: This lecture will cover subunits 4.2 and 4.3.  Please click on the link, scroll down to Video 41: “Polar Coordinates and Graphs,” and watch the entire lecture.  In this video, you will learn how to graph a number of well-known figures in polar coordinates, such as cardioids, roses, and limaçons.  You will also learn more about derivatives and tangent lines in polar coordinates.

    Viewing this lecture and pausing to take notes should take approximately 45 minutes.

    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Dan Reich’s “Plotting Points in Polar Coordinates” Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Dan Reich’s “Plotting Points in Polar Coordinates” (HTML)

    Instructions: Click on the link above.  Then, click on the “Index” button.  Scroll down to “5. Geometry, Curves, and Polar Coordinates,” and click button 181 (Sketching Polar Curves).  Do five of the problems in the module as they are presented to you.  If at any time a problem set seems too easy for you, feel free to move on.

    Completing this assessment should take approximately 30 minutes.

    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Assessment: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Polar Coordinates” Link: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Polar Coordinates” (PDF)

    Also Available in:

    HTML

    Instructions: Please click on the link above, and work through each of the eighteen examples on the page.  As in any assessment, solve the problem on your own first.  Solutions are given beneath each example.

    Completing this assessment should take approximately 2 hours.

    Terms of Use: The linked material above has been reposted by the kind permission of Elizabeth Wood, and can be viewed in its original form here (HTML).  Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.

4.3 Derivatives and Curve Sketching in Polar Coordinates   - Reading: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 7: Trigonometric Functions: “Section 7.8: Slopes and Curve Sketching in Polar Coordinates” Link: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 7: Trigonometric Functions: “Section 7.8: Slopes and Curve Sketching Polar Coordinates” (PDF)

 Instructions: Please click on the link above, and read Section 7.8
in its entirety (pages 412 through 419).  Here, you will learn tips
and tricks for graphing equations in polar coordinates and discover
how to take derivatives of such functions.  

 Studying this reading should take approximately 45 minutes.  

 Terms of Use: The article above is released under a [Creative
Commons Attribution-NonCommercial-ShareAlike License
3.0](http://creativecommons.org/licenses/by-nc-sa/3.0/) (HTML).  It
is attributed to H. Jerome Kiesler and the original version can be
found [here](http://www.math.wisc.edu/~keisler/calc.html) (PDF).
  • Activity: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Dan Reich’s “Sketching Curves in Polar Coordinates” Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Dan Reich’s “Sketching Curves in Polar Coordinates” (HTML)

    Instructions: Click on the link above.  Then, click on the “Index” button.  Scroll down to “5. Geometry, Curves, and Polar Coordinates,” and click button 183 (Sketching Polar Curves).  Do each of the problems (six in total).  This is an exploratory graphing assessment which is more like an applet.  If at any time a problem set seems too easy for you, feel free to move on.

    Completing this assessment should take approximately 45 minutes.

    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Assessment: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Graphing in Polar Coordinates” Link: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Graphing in Polar Coordinates” (PDF)

    Also Available in:

    HTML

    Instructions: Please click on the link above, and work through each of the eight examples on the page.  As in any assessment, solve the problem on your own first.  Solutions are given beneath each example.

    Completing this assessment should take approximately 1 hour.

    Terms of Use: The material above has been reposted by the kind permission of Elizabeth Wood, and can be viewed in its original form here (HTML).  Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.

4.4 Areas with Polar Coordinates   - Reading: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 7: Trigonometric Functions: “Section 7.9: Area in Polar Coordinates” Link: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 7: Trigonometric Functions: “Section 7.9 Area in Polar Coordinates” (PDF)

 Instructions: Please click on the link above, and read Section 7.9
in its entirety (pages 420 through 424).  

 Studying this reading should take approximately 30 minutes.  

 Terms of Use: The article above is released under a [Creative
Commons Attribution-NonCommercial-ShareAlike License
3.0](http://creativecommons.org/licenses/by-nc-sa/3.0/) (HTML).  It
is attributed to H. Jerome Kiesler and the original version can be
found [here](http://www.math.wisc.edu/~keisler/calc.html) (PDF).
  • Lecture: YouTube: MIT: David Jerison’s “Lecture 33: Exam 4 Review” Link: YouTube: MIT: David Jerison’s “Lecture 33: Exam 4 Review” (YouTube)

    Also Available in:
    iTunes U

    Instructions: Please watch Lecture 33 from the beginning to time 34:58.  Note that lecture notes in PDF are available for this video; the link is on the same page as the lecture.  In this lecture, Professor Jerison will touch on computing area under curves described by polar coordinates and also do several more examples of curve sketching in polar coordinates.

    Viewing this lecture and pausing to take notes should take approximately 45 minutes.

    Terms of Use: The article above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML).  It is attributed to MIT and the original version can be found here (Flash or MP4).

  • Lecture: University of Houston: Selwyn Hollis’s “Video Calculus: Areas and Length Using Polar Coordinates” Link: University of Houston: Selwyn Hollis’s “Video Calculus: Areas and Lengths Using Polar Coordinates” (QuickTime)

    Instructions: This lecture will cover subunits 4.4 and 4.5.  Please click on the link, scroll down to Video 42: “Areas and Lengths Using Polar Coordinates,” and watch the entire video.  This video discusses area (slides 1-5) and arc length (slides 7-9) in polar coordinates.

    Viewing this video lecture should take approximately 30 minutes.

    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

4.5 Arc Length with Polar Coordinates   - Reading: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 7: Trigonometric Functions: “Section 7.10: Length of a Curve in Polar Coordinates” Link: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 7: Trigonometric Functions: “Section 7.10: Length of a Curve in Polar Coordinates” (PDF)

 Instructions: Please click on the link above, and read Section 7.10
in its entirety (pages 425 through 427).  

 Studying this reading should take approximately 15-20 minutes.  

 Terms of Use: The article above is released under a [Creative
Commons Attribution-NonCommercial-ShareAlike License
3.0](http://creativecommons.org/licenses/by-nc-sa/3.0/) (HTML).  It
is attributed to H. Jerome Kiesler and the original version can be
found [here](http://www.math.wisc.edu/~keisler/calc.html) (PDF).
  • Assessment: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Integration in Polar Coordinates” Link: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Integration in Polar Coordinates” (PDF)

    Also Available in:

    HTML

    Instructions: Please click on the link above, and work through each of the nine examples on the page.  As in any assessment, solve the problem on your own first.  Solutions are given beneath each example.  The examples cover both Arc Length and Area in Polar Coordinates.

    Completing this assessment should take approximately 1 hour.

    Terms of Use: The linked material above has been reposted by the kind permission of Elizabeth Wood, and can be viewed in its original form here (HTML).  Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.