Math 50C - Multivariable Calculus

Homework Assignments

Instructor: David Arnold

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Fall 2007

Assignment #1
- Read the Syllabus
- All students should see me for a login name and password. The price of admission is a ream of paper. Bring one ream of paper to my office and make sure that I check your name in my gradebook as having received your ream of paper. You can then place your ream of paper under the printer in room PS116.
- After logging onto Blackboard at http://bb.redwoods.edu/, update all your personal information. It is very important that your email address is current and correctly entered in Blackboard.
- Once you've updated your personal information, read the welcome message on the Discussion Board and reply to the thread.
Assignment #2
- While seated at the computer, read the activity Vectors and Matrices in Matlab and key in the examples in Matlab as you read. You are welcome to try some of the exercises at the end of the activity, but they are not required.
- In this activity, Plotting in Matlab, do the exercises at the end of the activity: #1, 4, 7, 10, 14, 19. Obtain printouts for each exercise and arrange with Multigraf. You can get a copy of Multigraf for home use at http://math.rice.edu/~dfield/.
Assignment #3
- In Stewart, Early Transcendentals, read Section 10.1, pages 650-656.
- In Stewart, Early Transcendentals, pages 656-659, do exercises #12, 13, 14, and 17 by hand.
- In Stewart, Early Transcendentals, page 658, do all parts of exercise #33. Use Matlab's comet command to verify your results. It is not required to turn in printed plots for these exercises. Rather, you are using the comet command to determine if you have the correct set of parametric equations.
- In Stewart, Early Transcendentals, page 658, do exercises #38 and 39 by hand. Please include labeled sketches with your solutions.
- In Stewart, Early Transcendentals, page 658, use the Geometer's Sketchpad to sketch the locus of point P in exercise #40 as point A moves about the outer circle. Then find parametric equations of the locus by hand. Please include a labeled sketch with your solution. Finally, plot your resulting parametric equations in Matlab and compare it to the result found using the Geometer's Sketchpad. Include printed Sketchpad and Matlab plots with your solution. You must call me into the lab to view your sketch to receive credit for this part.
- In Stewart, Early Transcendentals, page 659, exercise #43, use Matlab to plot the curves presented by the given parametric equations. To find the solution for part (b), you need to animate two plots simultaneously. Matlab doesn't have a function to do this, but the graphing calculator will do an adequate job with this.
  Note: I was a bit frustrated that Matlab didn't have a function for this, so I searched the technical notes at the Mathworks and found this technical note:
  http://www.mathworks.com/support/solutions/files/s37142/comets.m
  I then did the following:
```
t=linspace(0,2*pi,500).';
x1=3*sin(t);y1=2*cos(t);x2=-3+cos(t);y2=1+sin(t);
X=[x1,x2];Y=[y1,y2];
comets(X,Y)
close all, comets(X,Y)
```
  This allowed me to see that there were two points of intersection, but only one "collision point."
Quiz #1

Quiz #1 due Friday 9/7/07.
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Assignment #4
- In Stewart, Early Transcendentals, read Section 10.2, pages 660-666.
- In Stewart, Early Transcendentals, page 667, do exercise #31. Please include a detailed sketch with your work. This can be drawn by hand if you wish, or you may use Sketchpad or Matlab.
- In Stewart, Early Transcendentals, page 667, do exercise #34. Use Matlab to draw a sketch of the Astroid for a = 2, but use the constant a in your calculation.
- In Stewart, Early Transcendentals, page 667, use Matlab to sketch the arcs in exercises #45 and #48. Then use hand calculations to determine the arc length.
- In Stewart, Early Transcendentals, page 667, do exercise #50. Use Matlab to draw the witch over the given region with a = 2. Please ignore the direction about using Simpson's rule; instead, again with a = 2, use Matlab's quad command to find an approximation for your integral.
- In Stewart, Early Transcendentals, page 668, do exercise #62. Use Matlab to draw the curve in the plane over the interval appropriate for the task at hand.
- In Stewart, Early Transcendentals, page 668, do exercise #73, but follow these directions:
  1. First, find the coordinates (x,y) of point P in terms of theta, and verify that your result matches the given result in the text. That is, you may not assume that the equations given are correct. You must prove that the given equations are correct.
  2. Extra points will be given to any student who use the Geometers's Sketcpad to draw the image. You must call me into the lab to view your image to receive this credit.
  3. Perform the appropriate integration to find the length of the involute as theta ranges from 0 to 2π.
Quiz #2

Quiz #2 due Friday 9/14/07.
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Assignment #5
- In Stewart, Early Transcendentals, read Section 10.3, pages 669-677.
- Sketch the graphs of r = sin θ, r = sin 2θ, and r = 1 + cos θ on sheets of polar coordinate graph paper. One graph per page, please. See me if you don't have polar graph paper.
- In Stewart, Early Transcendentals, on page 678, sketch the graphs of the polar equations in exercises #33, 42, and 45 without the use of a graphing calculator or Matlab (or any other technology). Just you and your mind and your pencil. Note: It is not expected that you plot a bunch of points on polar graph paper or its equivalent.
- In Stewart, Early Transcendentals, on page 679, use Matlab to draw the "butterfly" in exercise #71. Too fun!
- In Stewart, Early Transcendentals, on page 678, use Matlab to investigate the question posed in Exercise #78. To save paper, use multigraf to arrange your plots. Each plot should have a title indicating value of c and n. Once your convinced what's going on, write and explain it to your reader.
Assignment #6

In Stewart, Early Transcendental, read Section 10.4, pages 679-683.
The polartrc program, and all other M-files needed for this course, are available in the zip file calc3.zip.
In Stewart, Early Transcendental, use polartrc to aid in visualizing the area to be sketched in exercises #27 and 29 on page 683. Obtain a printout of each result. Use integration to find the designated area.
In Stewart, Early Transcendental, page 683, draw the curves in exercises #35 and 41 by hand (you may verify with your calculator or Matlab), then find the remaining requested information by hand calculation only.
In Stewart, Early Transcendental, page 684, find the exact length of the curve in exercise #45. Use hand calculations only.
In Stewart, Early Transcendental, page 684, #49, use Matlab's quad command to find an approximation for the integral required to find the length of the curve. Please attach your M-file with the code for the function representing the integrand and handwrite the command executed at the Matlab prompt on your homework.
In Stewart, Early Transcendental, page 684, do exercise #55.

Assignment #7
- Complete the activity, The Conic Sections in Polar Coordinates. When you've completed the activity, call me to take a look at your Sketchpad drawing.
Assignment #8
- In Stewart, Early Transcendentals, read Section 12.1, pages 793-796.
- In Stewart, Early Transcendentals, read Section 12.2, pages 798-804.
- In Stewart, Early Transcendentals, page 805, exercise #4. For each part of this exercise, recopy the figure in pencil, then use a colored pencil to indicate the vector operation and result.
- In Stewart, Early Transcendentals, page 805, exercise #9, 13. Please draw accurate diagrams on graph paper.
- In Stewart, Early Transcendentals, page 805, do exercises 19, and 26.
- In Stewart, Early Transcendentals, page 806, do exercise #36. Include a sketch with your solution.
- In Stewart, Early Transcendentals, page 806, do exercise #37. Please provide an accurate sketch on graph paper for each part.
- In Stewart, Early Transcendentals, page 806, exercises #39, 40, and 43. Please provide a clearly labeled diagram with each solution.
Assignment #9
- In Stewart, Early Transcendentals, read Section 12.3, pages 807-812.
- InStewart, Early Transcendentals, pages 812-814, do exercises #3, 7, 9, 11, 15, 19, 21, 27, 37, 45, 49, 51, and 56.
Assignment #10
- In Stewart, Early Transcendentals, read Section 12.4, pages 814-820.
- In Stewart, Early Transcendentals, pages 820-821, do exercises #9, 12, 14, 25, 29, 34, 35, 38, 39, and 40.
Quiz #3

Quiz #3 (Due Wednesday 9/26/07)

Assignment #11
- In Stewart, Early Transcendentals, read Section 12.5, pages 822-829.
- In Stewart, Early Transcendentals, pages 829-831, do exercises #10, 18, 29, 31, 35, 42, 53, 58, 63, and 65.
Assignment #12
- In this activity, Lines and Planes in Matlab do all of the exercises at the end of the activity.
Assignment #13
- In Stewart, Early Transcendentals, read Section 12.6, pages 832-837.
- In Stewart, Early Transcendentals, page 837, sketch the cylinders in Exercises #3, 4, and 5.
- In Stewart, Early Transcendentals, page 838, disregard the directions for Exercises #11, 13, 15, 17, and 19. Instead, follow the directions on your handout.
  - First, sketch the trace of the surface in each of the coordinate planes. That is, let x = 0, y = 0, and z = 0, then sketch the resulting trace in the appropriate coordinate plane.
  - Sketch parallel cross sections in a direction appropriate for the problem.
  - Darken all lines visible to the eye.
- In Stewart, Early Transcendentals, page 838, consider Exercise #43, but follow these special directions.
  - Find a parametrization for the curve y = x² in terms of t. That is, find x = x(t), y = y(t), so that the resulting parametrization provides the same curve as the original Cartesian equation y = x².
  - Now, rotate the parametrized curve around the y-axis. Let θ be the angle of rotation.
  - Parametrize the resulting surface in terms of t and θ. That is, find a point (x,y,z) on the surface in terms of t and θ, with x = x(t,θ), y = y(t,θ), and z = z(t,θ).
  - Use Matlab to draw the resulting surface. Create a mesh on t and θ, then use your parametrization from the preceding part to define x, y, and z. Then, sketch the surface.
Assignment #14
- In this activity, Cylindrical Coordinates and Quadric Surfaces do all of the exercises at the end of the activity.
Assignment #15
- In Stewart, Early Transcendentals, read Section 13.1, pages 849-855. No grade will be given for this assignment, but you should give this section a good read.
Asignment #16
- In Stewart, Early Transcendentals, read Section 13.2, pages 857-860.
- In Stewart, Early Transcendentals, pages 861, do exercises #3, 5. Include hand drawn drawings of the curve, position, and tangent vectors for the given t-values.
- In Stewart, Early Transcendentals, page 861, do exercises #15, 16, and 17.
- In Stewart, Early Transcendentals, page 861, do exercises #25. Use Matlab to draw the curve and the tangent line.
- In Stewart, Early Transcendentals, page 861-862, do exercise #29c, 31, 39, 47, and 48.
Quiz #4

Quiz #4 due Friday 10/5/07.
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Assignment #17
- In Stewart, Early Transcendentals, read Section 13.3, page 862-868.
- In Stewart, Early Transcendentals, page 868, do exercises #1 and 5.
- In Stewart, Early Transcendentals, use Matlab's quad command to do exercise #7. Ignore the direction about Simpson's Rule.
- In Stewart, Early Transcendentals, page 868-869, do exercises #9, 11, 12, 15, 19, 25, 36, 37, and 39.
- In Stewart, Early Transcendentals, page 869, use Matlab to complete exercise #43.
Assignment #18
- While sitting at the computer, work through the activity The Osculating Circle.
- Once you complete the activity, do the homework exercise described at the end of the activity. When you have completed the sketch in the Geometer's Sketchpad, find me. I will come in and view your results, then assign a grade.
Assignment #19
- In Stewart, Early Transcendentals, read Section 14.1, pages 887-897.
- In Stewart, Early Transcendentals, on page 898, without the aid of a computer, determine the domain and range of the function given in Exercise #8. Next, draw the function on the domain D={(x,y): -3<x,y<3}. There might be difficulties with complex numbers on D, so draw your surface with the command mesh(x,y,real(z)). Obtain a printout of the result, then explain on the printout how the picture helps you "see" the domain and range of the given function.
- In Stewart, Early Transcendentals, page 899, do exercises #32, 35, and 36.
- In Stewart, Early Transcendentals, page 899, sketch by hand several level curves of the functions in exercises 41 and 43.
- In Stewart, Early Transcendentals, page 900, use Matlabs's contour command to sketch level isothermals for the temperature function given in Exercise #47. I'd like labels for the contours as well, so sketch the contours with [c,h]=contour(x,y,z), then label them with clabel(c,h).
- In Stewart, Early Transcendentals, page 55, use Matlab's meshc command to sketch both the surface and contours for exercise #55.
- In Stewart, Early Transcendentals, page 900, use Matlab to sketch level surfaces f(x,y,z) = c, for c = 15, 30, and 45, using the function given in exercise #59.
Assignment #20
- In Stewart, Early Transcendentals, read Section 14.2, pages 902--908.
- In Stewart, Early Transcendentals, perform each of the following tasks for exercises #7, 13, and 16.
  1. Sketch the surface represented by the given equation. Can you see any paths on the surface that lead to a different limit as (x,y)->(0,0)?
  2. Sketch a contour plot with [c,h]=contour(x,y,z) and label it with clabel(c,h,'manual'). Are there any paths you can travel along that lead to a different limit as (x,y)->(0,0)?
  3. Use the evidence found in the first two parts to analyze the function along two different paths leading to different limits.
- In Stewart, Early Transcendentals, page 909, use Matlab to draw the surfaces in Exercises #25 and 26. Explain the phenomenon observed in terms of any disontinuities.
- In Stewart, Early Transcendentals, page 909, do exercise #40. Use Matlab to sketch the surface represented by the given function.
Quiz #5

Quiz #5 due Friday 10/19/07.
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Assignment #21
- In Stewart, Early Transcendentals, read Section 14.3, pages 909--919.
- In Stewart, Early Transcendentals, page 920, do exercises #5, 6, 8, 18, 24, and 30.
  1. Note: You can check your answers to differentiation problems using Matlab. This is particularly important on these exercises because you are not provided with the even solutions in your text or student solution manual.
  2. For example, let check exercise #30.
```
>> format compact
>> syms x y z
>> u=x^(y/z)
u =
x^(y/z)
```
    Now, the partial derivative with respect to x:
```
>> ux=diff(u,x)
ux =
x^(y/z)*y/z/x
>> pretty(ux)
 
                                    (y/z)
                                   x      y
                                   --------
                                     z x
```
    Similarly, the derivative with respect to z:
```
>> uz=diff(u,z)
uz =
-x^(y/z)*y/z^2*log(x)
>> pretty(uz)
 
                                  (y/z)
                                 x      y log(x)
                               - ---------------
                                        2
                                       z
```
- In Stewart, Early Transcendentals, page 921, do exercises #39, 41, 43, 55, 67, and 69.
- In Stewart, Early Transcendentals, page 922, do exercise #77.
- In Stewart, Early Transcendentals, page 922, use Matlab to plot each request in exercise #84.
Assignment #22
- In Stewart, Early Transcendentals, read Section 14.4, pages 923--929.
- In Stewart, Early Transcendentals, on page 930, do exercise #7. Use Matlab to sketch the surface and the tangent plane to the surface at the given point. However, don't bother with the zoom-in instruction given in the text.
- In Stewart, Early Transcendentals, on pages 930-931, do exercises #19, 23, 27, 29, 31, 33, and 37.
Quiz #6

Quiz #6 due Monday 10/29/07.
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Assignment #23
- In Stewart, Early Transcendentals, read Section 14.5, pages 931--937.
- In Stewart, Early Transcendentals, on page 938-939, do exercises #5, 9, 13, 15, 23, 39, 33, 39, 43, 45, 47, and 51.
Assignment #24
- In Stewart, Early Transcendentals, read Section 14.6, pages 940-950.
- In Stewart, Early Transcendentals, on pages 951-952, do exercises #12, 15, 31, and 37.
- In Stewart, Early Transcendentals, on pages 952, use Matlab to complete exercise #45.
- In Stewart, Early Transcendentals, on pages 952-953, do exercises #47, 52, and 53.
Assignment #25
- Do all the exercises at the end of the activity The Gradient.
Assignment #26
- In Stewart, Early Transcendentals, read Section 14.7, pages 953-961.
- In Stewart, Early Transcendentals, pages 961-962, perform each of the following tasks for exercises #5, 7, 11, and 19.
  1. Use Matlab to draw the surface over a domain. Adjust your domain so that the extrema are visible on the surface.
  2. Using the same domain as in part (1), use Matlab to create a contour plot with labeled contours. Place a grid on your plot, then use the "Text" and "Arrow" tools off the Figure Toolbar to annotate your plot with the coordinates of the extrema and the type of extrema. Note that there are several ways to use the contour plot.
    - [c,h]=contour(x,y,z), followed by clabel(c,h) gives the default labeled contour plot. However, some find that this places far too many labels on the contours and prefer [c,h]=contour(x,y,z), followed by clabel(c,h,'manual') which allows the user to use the mouse to place labels judiciously.
    - [c,h]=contour(x,y,z,n) will place n contours on your plot, where n is a positive integer. For example, to get 20 contours, you would use [c,h]=contour(x,y,z,20), to get 30 contours, you would use [c,h]=contour(x,y,z,30), etc.
    - You can also force contours at certain levels. For examples to draw the level curves f(x,y)=c, for c=1, 2, 3, 4, and 5, you would exercute [c,h]=contour(x,y,z,[1,2,3,4,5]) or, alternatively, [c,h]=contour(x,y,z,1:5). This allows very selective choice of contours.
- Find and analyze extrema using the second derivative test shown in class. Arrange your work in a table as follows:
  
  (a,b) f(a,b) f_xx(a,b) f_xx(a,b)f_yy(a,b)-f²_xy(a,b) Classification
  
  (1,2) 8 -4 (2)(2)-(1)²=3>0 local maximum
- To save paper, you might arrange so that your contour printout is printed near the top of the page, which would allow you to use the bottom half of the page for your calculations. Page Setup off the File menu in the Figure Window will allow you to set up the page with the plot at the top of the page.
Assignment #27
- On page 962, exercises #27, 32, and 33, perform each of the following tasks.
  1. Draw a contour map of the given function, then superimpose the graph of the given boundary on your contour map. Mark interior critical points and points on the boundary where you expect you might find extrema.
  2. Use the method shown in class to find the absolute extrema.
Assignment #28
- In Stewart, Early Transcendentals, read Section 15.1 and 15.2, pages 981-994 (Here comes page 1000!).
- In Stewart, Early Transcendentals, read Section 15.3, pages 995-1001. (Yahoo! We just passed page 1000!).
- In Stewart, Early Transcendentals, page 1002-1003, do exercises 9, 10, 17, 19, 25, 27, 28, and 45. You are encouraged to employ fdomain5 and sketch the region of interest in each exercise, be it the indicated volume of a domain of integration.
Quiz #7

(a,b)	f(a,b)	f_xx(a,b)	f_xx(a,b)f_yy(a,b)-f²_xy(a,b)	Classification
(1,2)	8	-4	(2)(2)-(1)²=3>0	local maximum

Quiz #7 due Monday 11/19/07.
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Assignment #29
- In Stewart, Early Transcendentals, read Section 15.7, pages 1023-1030.
- In Stewart, Early Transcendentals, pages 1030-1031, do exercises #9, 11, 13, and 19.
- In Stewart, Early Transcendentals, pages 1031, do exercise #31. Use fdomain5 to verify each of your setups.
Assignment #30
- In Stewart, Early Transcendentals, read Section 15.8, pages 1033-1037.
- In Stewart, Early Transcendentals, page 1038, do exercises #14, 23, 24, and 32. In each case, use Matlab to sketch the indicated volume (or at least the surfaces bounding the indicated volume).
Quiz #8

Quiz #8 due Monday 11/26/07.
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Assignment #31
- In Stewart, Early Transcendentals, read Section 16.1, pages 1055-1060.
- In Stewart, Early Transcendentals, page 1060, sketch the vector field in Exercises #5 and #6 by hand.
- In Stewart, Early Transcendentals, page 1061, use Matlab to answer Exercise #19. Obtain a printout.
- In Stewart, Early Transcendentals, page 1061, use Matlab to complete Exercise #27.
Assignment #32
- In Stewart, Early Transcendentals, read Section 16.2, pages 1062-1071.
- In Stewart, Early Transcendentals, page 1071, do Exercise #11, two ways:
  1. First, parametrize the given line segement in terms of time t, then evaluate the integral.
  2. Second, parametrize the given line segment in terms of arc length, then evaluate the integral.
- In Stewart, Early Transcendentals, page 1072, do Exercises #15 and 19.
- In Stewart, Early Transcendentals, page 1072, use Matlab to sketch the vector field of exercise #23 and superimpose the given curve C. Use your graph to predict whether the given integral will be positive, negative, or zero, then evaluate the integral and compare the result.
- In Stewart, Early Transcendentals, page 1073, do exercise #40.
- In Stewart, Early Transcendentals, page 1073, do the following for exercise #43.
  1. For part (a), use Matlab to sketch a constant force field of your choice, superimpose the graph of the unit circle, the explain why the field does zero work on the particle as it moves around the unit circle (once, uniformly)?
  2. For part (a), let the vector field be defined by F(x,y)=c₁i + c₂j, then evaluate the appropriate integral to show that the work done by the field on the particle as it moves around the unit circle is zero.
  3. For part (b), use Matlab to sketch the given force field for k = 1/2, then superimpose the unit circle. What can you predict from your sketch about the work done by the field on the particle as it moves around the unit circle (once, uniformly)?
  4. Verify your prediction in part (c) by evaluating the appropriate integral.
Assignment #33
- In Stewart, Early Transcendentals, read Section 16.3, pages 1074-1081.
- In Stewart, Early Transcendentals, page 1081, perform each of the following tasks for exercises #3, 5, and 7.
  1. Sketch the vector field on the domain {(x,y): -3 < x,y < 3}.
  2. Examine your sketch. How can you use the sketch of the vector field to determine whether or not the field is conservative?
  3. Check your result of the previous question by answering the questions posed in the text.
- In Stewart, Early Transcendentals, page 1082, do exercises #13 and 21.
- In Stewart, Early Transcendentals, pages 1082-1083, do exercises #33 and 34(a).
Assignment #34
- In Stewart, Early Transcendentals, read Section 16.4, pages 1083-1088.
- In Stewart, Early Transcendentals, pages 1089-1090, do exercises #3, 9, 11, 15, and 18.
Assignment #35
- In Stewart, Early Transcendentals, read Section 16.5.
- In Stewart, Early Transcendentals, pages 1096-97, do exercises #1, 6, 13, 16, 21, 22 31, and 36.