Mathematics 41, section 1 -- Analysis 3

Old Assignments

• Individual Problem Set 1: From Damiano and Freije, section 1.1: 1/a,c,e, 3/a,b,f, 4/a,b,f, 7 (Hint: take the origin of your coordinate system at the center of the Earth). Due: Friday, 1/16
• Discussion 1
• Individual Problem Set 2: From Damiano and Freije, section 1.3: 1/a,d,e,f, 5, 6 (Hint: consider ||u + v||², expand, and use result of problem 5), 7. Due: Friday, 1/23.
• Lab Day 1
• Individual Problem Set 3: From Damiano and Freije, section 1.4: 1/b,d, 4 (be careful -- use different letters for the parameters on the two lines, since they are independent of each other), 7/a,c,d, 21 (think about the geometry of the dot and cross products). Due: Friday, 1/30.
• Discussion 3
• Lab Day 2
• Lab Day 3
• Exam 1 will be given Friday, February 13. Review sheet and practice exam.
• Solutions for the Sample Exam 1.
• The problems for Lab Day 4 are the three questions in Discussion 15. However, you should be careful -- there are a number of typos. Some of them are just confusing, but 3. below is potentially a big problem:
  1. In question 1 c), the point should be (.2,-.4) as in part 1 b).
  2. In question 2 c), the first sentence should be "Are there qualitative differences between the extreme points of this function and those of the function in question 1?" (not "part a")
  3. The Maple function definition in question 3 is incorrect! There should not be any (x,y) after the f on the left of the assignment operator ( := ). Your function definition should look like the ones in the other two questions. (The factor of sin(x) on the right should be there, though!)
• Individual Problem Set 4: From Damiano and Freije, Section 3.1/1 (i) -- that is, just do the density plot with a 6 x 6 grid of pixels -- also do 2 (a paragraph for each part is sufficient). Section 3.2/8,10,12. Due: Wednesday, February 25.
• Lab Day 5
• Discussion 4
• Exam 2 will be given Friday, March 20. Review sheet and practice exam.
• Answers for Practice Exam 2
• Lab Day 6
• Lab Day 7
• Individual Problem Set 5 -- From Damiano and Freije, Section 4.2/7 a, d, e, h. Due: Friday, March 27
• Individual Problem Set 6 -- From Damiano and Freije, Section 4.3/6, 7 a,c, 8a,14 a. For 14 a, you will probably want to start out by plotting the constraint curve and the contour plot of f to get a rough idea of where to expect the extreme points. You can combine two 2D plots using the display command from the plots package. (This works in a way similar to the display3d command we used in Lab Day 5.) The contour plot can be generated with the contourplot command from the plots package; the constraint curve can be plotted using the ordinary 2D plot command either as two functions y = F(x), or as a parametric curve. Then set up the Lagrange Multiplier equations and solve them for x,y,lambda using the fsolve command in Maple from Lab Day 7. To solve three equations in three variables x,y,l (where l is lambda, the Lagrange Multiplier), you could use a command like this:
  
  fsolve({eqn1,eqn2,eqn3},{x,y,l},{x=a..b,y=c..d,l=e..f});
  

  The range of l values can be -infinity..infinity, but you will need to choose the x and y ranges carefully to make sure you are finding a region containing exactly one solution of the equations! Due: Friday, April 3.

• Discussion 6
• Individual Problem Set 7: From Damiano and Freije, Chapter 5 section 2/ 4 a,c, 6 a, 7 a,c, 8, 10 a,b. Due: Wednesday, April 15. For 10 a,b, you should use the Maple int command. Here is some useful information: Maple can provide numerical approximations to definite integrals of one variable functions. The general format of the command is:
  evalf(int(function,x=a..b));

  (Without the evalf( ) around the int( ), Maple would try to compute an exact value for the integral by the FTC. If the function has no elementary antiderivative, this would fail and the output would just be the integral in mathematical notation.) For example, to approximate the integral of f(x) = cos(x^2) from x = 0 to x = 1, you would use

  evalf(int(cos(x^2),x=0..1));

  (Try it! Your result should be .9045242380. Also try removing the evalf( ) and executing the command again.) Because Fubini's Theorem reduces the problem of computing double integrals to the problem of computing iterated 1-variable integrals, you can also use Maple to do problems like number 10. To compute a double integral of f(x,y) over R = {(x,y) : a <= x <= b, g(x) <= y <= h(x)} you would use a command of the form:

  evalf(int(int(f(x,y),y=g(x)..h(x)),x=a..b));

  For example, if a = 5 and b = 9, then part c of this problem could be done with the command:

  evalf(int(int(1,y=-9*sqrt(1 - x^2/25)..9*sqrt(1 - x^2/25)),x=-5..5));

  since the double integral of the constant function f(x,y) = 1 over a region R computes area(R).

• Discussion 8