MATH 241 (Dr. Rosenberg's sections): Lecture Schedule and Homework, Fall 2010

General Remarks

The schedule below shows what material will be covered and when. The lectures are not a substitute for reading the textbook; rather they are a guide to some of the more important points. So it's usually best to read every section of the book at least twice, once before the class when that material will be covered, and once again afterwards. Try not to get behind in the reading!

MATLAB assignments are are listed at the bottom of this page and are related to the MATLAB lessons. All of these problems must be solved using MATLAB. You are free to use either one of the university computer labs or your own personal computer for this purpose. You might wish to use some of the M-files which you can view and download here on a one-by-one basis or download here and here as zip-archives. (The "NIT" files are a "Numerical Integration Toolbox", useful for the integration section of the course.)

Problems labeled EG refer to Calculus, Sixth Edition, by Robert Ellis and Denny Gulick. These problems are for your own practice and are not to be turned in, but quizzes closely modeled on them will be given in the discussion sections. You may solve these problems with paper-and-pencil or with MATLAB, at your discretion. However you are advised to solve at least some of the Ellis and Gulick problems with paper-and-pencil in order to develop the proficiency you will need for the quizzes and tests.

You may (should!) work on the homework problems, especially the MATLAB problems, in groups. Please make sure that for the MATLAB homework, you only collaborate with other students with the same TA. (No more than three students to a group, please!) In order to receive credit, the name of every member of the group must be included in the submission. In particular, names must be typed in a text cell in the solution printout. A homework group should submit only one copy of an assignment, and all members of a homework group will receive identical grades for that assignment. Each homework submission should adhere to the university honor pledge, and preferably should contain a copy of the pledge in a text cell. MATLAB homework should be prepared in the form of a published M-file or published Mupad notebook (with all the input, output, and graphics included) and turned in to your TA.

The problems listed here are the minimum amount you should do to acquire proficiency in the material of this course. In addition to the assigned problems from Ellis and Gulick, you should solve as many odd-numbered problems as you feel is necessary in order to become comfortable with the techniques introduced in that part of the book.

Grading

Each in-class test is 100 points. Each quiz or MATLAB assignment is 10 points. There are 13 quizzes and we drop the lowest 3. There are 13 MATLAB assignments and we drop the lowest 3. For this reason we will not usually give makeups for quizzes or homeworks missed, except in unusual circumstances. The final exam is 200 points, and is cumulative over the whole course. It is the same for all sections of MATH 241 (including those taught by Drs. Wentworth and Margetis). Also the grading scale on the final is the same for all sections. Calculators are not allowed on the exams (nor are they needed), but you may bring one sheet of notes or formulas to each exam.

Class Schedule

MATLAB Assignments

1. (due Tuesday 9/7)
   1. Graph the function f(x) = x⁵ - 10x⁴ + 36x² - 10x - 15 in a way that displays all the interesting features of the graph. Compute all the zeros and critical points, using both solve and fzero. How can you be sure you are not missing any zeros or critical points?
   2. Compute the integral of (1 - x³)^1/2 from 0 to 1, both numerically and symbolically. The symbolic integral comes out in terms of a function you may not know much about --- what is it? Apply double to the symbolic integral to get a numerical value. Compare with the result of using quadl (using format long to look at many decimal places).
2. (due Thursday 9/16)
   1. Find the equation of the plane P through the points (1,2,4), (1,4,5), and (-1, 2, 6).
   2. Next, find the (parametric vector) equation of the line L through (0, 1, 1) perpendicular to the plane P.
   3. Show the line L and the plane P on the same set of axes.
   4. Finally, determine where the line L and the plane P intersect. This is the point on P closest to (0, 1, 1).
3. (due Thursday 9/23) Read the hints from Additional Problems 1 and 2 in the lesson on curves. Then for each of the following graphs, plot the curve, find the arclength, and plot the curvature as a function of the parameter t. Finally, explain what the curvature calculation tells you about the geometry.
   1. the circular arc y = sqrt(1 - x²) between x = -1 and x = 1.
   2. the eight-petal rose r = cos 4θ.
4. (due Thursday 9/30) For each of the following functions of two variables, draw a contour plot and a 3D graph.
   1. f(x, y) = sin(x)sin(y), -2π ≤ x, y ≤ 2π.
   2. g(x, y) = x²y² - 5x³ + 4y², -3 ≤ x, y ≤ 3.
5. (due Thursday 10/14) This assignment keeps the notation of the last assignment.
   1. For each of the two functions in Assignment 4, superimpose a plot of the gradient vectors on the contour plot.
   2. What do you notice about the relationship between the vectors and the contour lines?
   3. Find the tangent plane to the 3D graph z = g(x, y) at the point (1, 1, 0), and superimpose a plot of the tangent plane on the 3D graph.
6. (due Thursday 10/21) Let f(x, y) = x⁴ + 3xy² - 2x² + 7y. Find and classify the critical points of f using each of the following methods:
   1. by looking at a contour plot of f. This can only give an approximate answer.
   2. by plotting f_x = 0 and f_y = 0 on the same set of axes, and looking for the intersection points. (Then looking at contours should help you classify them.)
   3. by solving the equations f_x = 0 and f_y = 0 using solve. (If you say syms x y, you should find five solutions, but only one of them should be real-valued. If you say syms x y real, that will disallow complex values and avoid this problem.)
7. (due Tuesday 10/26) This assignment keeps the notation of the last assignment.
   1. Find the constrained critical points of f on the circle (x + 1)² + (y - 1)² = 4, using the method of Lagrange multipliers, and find the maximum and minimum values of f on this circle.
   2. Redo the same calculation by letting x = -1 + 2 cos(θ), y = 1+ 2 sin(θ) to parameterize the circle, then substituting into the formula for f to get a function of the single variable θ, 0 ≤ θ ≤ 2π. This reduces you to an optimization problem in a single variable. Plotting f as a function of θ should show you where the critical points are, and then you can locate them more exactly with fminbnd.
8. (due Tuesday 11/2) Set up the integral ∫∫_R x² dA, where R is the region in the plane bounded by the parabolas y = 4 - x² and y = x² + x + 1.
   1. First draw a picture of R.
   2. Write the integral as an iterated integral and evaluate it (exactly) using int twice.
   3. Convert your answer to (b) to a decimal.
   4. Redo the calculation using dblquad applied to a function of x and y that gives x.^2 multiplied by something that is 1 when (x, y) lies in R and 0 if it doesn't. Recall that dblquad requires x (but not y) to be vectorized. Type help dblquad for an example.
   5. Redo the calculation once more using numint2 from the lesson on double integrals.
9. (due Tuesday 11/9) Set up the integral ∫∫∫_D dV for the volume of the region D in 3-space bounded below by the elliptic paraboloid z = x² + 3y² and above by the elliptic paraboloid z = 4 - x² - 3y².
   1. First draw a picture of D and of its projection R into the x-y plane. You might want to try the M-files viewSolid or newviewSolid.
   2. Write the integral as an iterated integral and evaluate it using MATLAB. Try if possible to get an exact answer; otherwise, get a numerical one. You have a choice of many methods.
10. (due Tuesday 11/18)
    1. Redo the problem from the Assignment 9 using modified cylindrical coordinates: x = r cos θ, y = (r sin θ)/√3.
    2. Find the area of the region in the first quadrant bounded by the parabolas y = x² and y = 2 x² and by the circle x² + y² = 4.
11. (due Tuesday 11/23)
    1. Parametrize the portion of the hyperboloid of one sheet x² + y² - z² = 9 between z = -1 and z = 1, and use your parametrization to plot it with MATLAB.
    2. Compute the surface area of the piece of surface you plotted in (a).
    3. Find the volume enclosed by the surface and the planes z = -1 and z = 1, using the divergence theorem to relate this volume to the flux of the vector field F = xi + yj through the surface.
12. (due Thursday 12/2)
    1. Define a vector field F by
       [ y*z - 6*x*y*sin(x^2*y), x*z - 3*x^2*sin(x^2*y), exp(z) + x*y]
       (in MATLAB notation). (Note that once you've defined x, y, and z to be symbolic variables, you can copy this and paste it into your MATLAB command window.)
    2. Show that F is conservative, by computing the curl.
    3. Integrate the vector field along the helical path
       r(t) = [cos(t), sin(t), t]
       from t = 0 to t = π. If MATLAB won't do the integral symbolically, do it numerically.
    4. Now integrate F^.dr along the straight-line path
       r(t) = [1 - 2t, 0, πt]
       from t = 0 to t = 1 (which has the same endpoints) and show that you get the same answer, at least up to round-off error.
    5. Find a potential function for the vector field, and use it to compute the line integrals in (c) and (d) exactly.
13. (due Thursday 12/9)
    1. Define a vector field F in the plane by
       [ x^2 + y, x^2 - 3*y]
       (in MATLAB notation). (Note that once you've defined x and y to be symbolic variables, you can copy this and paste it into your MATLAB command window.)
    2. Plot the vector field in the region where x and y go from -2 to 2, using quiver.
    3. Superimpose a plot of the circle x² + y² = 4 on top of your quiver plot. From looking at your picture, if you traverse the circle counterclockwise, would you expect the line integral to be positive or negative?
    4. Compute the line integral of F^.dr around the circle (going counterclockwise), by direct computation (using int).
    5. Now compute the same integral using Green's Theorem, and show that it agrees with what you got before.