DAILY ASSIGNMENTS
Week 1 (First day for U of I classes is Tuesday, January 17, 2012)
Jan 17 (Tue) Go to your discussion-recitation section.
Jan 18 (Wed) Read sections 12.1 and 12.2. In 12.1 do #1, 3, 5, 7, 11, 14, 16, 20, 27, 31, 33, 34. Finish the assigned problems by Thursday's discussion. Finish the reading of 12.2 by Friday's lecture.
Jan 20 (Fri) Read sections 12.2 and 12.3. In 12.2 do #4, 6, 11, 17, 19, 23, 24, 37. In 12.3 do #1, 4, 5, 8, 9, 17, 19, 26, 36, 38.
Week 2
Jan 23 (Mon) Read sections 12.4 and 12.5. In 12.3 do #46, 47. In 12.4 do #2, 5, 11, 13, 17, 18, 27, 29, 33, 35, 37, 49.

See the schedule of free tutoring which begins tonight. Note that my own office hours have been modified for this week only.

Jan 25 (Wed) Read section 12.5. Do #1 from section 12.5. Begin working on the problems on the worksheet. Next Tuesday's quiz will cover sections 12.1, 12.2, 12.3 and 12.4. Next Thursday's quiz will cover sections 12.4 (again), 12.5 and 12.6.

My office hours have been changed from today until Thursday (4-5:30pm). See the schedule of free tutoring.

Jan 27 (Fri) Read section 12.6 and explore quadric surfaces at this Interactive Gallery of Quadric Surfaces. Without technology do problems #1, 3, 4, 11, 12, 14, 19, 21–28, 33, 34 from section 12.6. You may also find it useful to further explore surfaces or check your work using Wolfram Alpha. In Wolfram Alpha enter z = 4 − x^2 − y^2. Now enter plot z = 4 − x^2 − y^2. This gives you some information about the surface and then a plot of the surface. To prepare for next week's quizzes, you should know how to solve all of the assigned homework. Select Course Content for MATH 241 at Illinois Compass 2g for solutions to the homework from the textbook. Solutions to the worksheet are also now available.
Week 3 (Deadline to add this course is Monday, January 30, 2012)
Jan 30 (Mon) Read sections 13.1 and 13.2. In 13.1 do #1, 5, 7, 14. In 13.2 do #3, 4, 6, 9, 10, 12, 17, 26, 33, 35.
Jan 31 (Tue) Quiz #1 on sections 12.1, 12.2, 12.3 and 12.4 will be given during today's discussion section.
Feb 1 (Wed) Read section 13.3. In 13.2 do #32. In 13.3 do #1, 5, 11, 14, 15. Prepare for tomorrow's quiz.
Feb 2 (Thu) Quiz #2 on sections 12.4, 12.5 and 12.6 will be given during today's discussion section.
Feb 3 (Fri) Read section 13.4. In section 13.4 do #5, 6, 10, 11, 16, 19, 22. We will begin chapter 14 on Monday.
Week 4
Feb 6 (Mon) Read sections 14.1 and 14.2. In 14.1 do #13, 16, 23, 25, 27, 32, 45, 47, 48. In 14.2 do #5, 6, 7, 8, 9, 13, 17, 25. Thursday's quiz will cover sections 13.1, 13.2, 13.3 and 13.4. There are many parts of sections 13.3 and 13.4 which I have not yet discussed. These parts will not be on the quiz. In particular there will not be any questions about curvature, binormal vectors or osculating planes from 13.3. The questions from 13.4 will be just the straightforward problems on position, velocity, acceleration and speed as found in the assigned homework. I may discuss the skipped parts of 13.3 and 13.4 at a later date.

We discussed functions of several variables, along with domains, graphs of z = f(x,y), level curves, etc.

We then compared the techniques for finding limits of functions from R to R versus limits of functions from R2 to R. If the function is continuous at a given point, we can just plug the value into the function to obtain the limit. Luckily many of the functions we deal with are continuous on their natural domains.

In calculus I, we learned that for a limit to exist, the limit from the left and from the right had to be equivalent. Unfortunately for z = f(x,y), there are many paths to a given point (not just from the left and right). Also l'Hospital's Rule cannot be used for functions from R2 to R. We saw that (x^2 - y^2) / (x^2 + y^2) has different limiting values when (x,y) approaches (0,0) along the x-axis versus approaching (0,0) along the y-axis. We then investigated the limit of (x*y^2) / (x^2 + y^4) as (x,y) approaches (0,0) along y = m*x versus approaching along x = y^2. We'll look mare carefully at examples of limits next time along with the formal definition of limits.

Feb 8 (Wed) Read section 14.3. In 14.2 do #10, 11, 12, 14, 15, 16, 18. In 14.3 do #5, 16, 17, 21, 23, 27, 39, 40, 43, 45, 51, 57, 72, 94. Prepare for tomorrow's quiz on sections 13.1 –13.4. Wednesday's test will cover sections 12.1 – 12.6, 13.1 – 13.4, 14.1 – 14.4.
Feb 9 (Thu) Quiz #3 on sections 13.1, 13.2, 13.3 and 13.4 will be given during today's discussion section.
Feb 10 (Fri) Read section 14.4. In 14.4 do #1, 4, 6, 12, 15, 17, 19, 20, 21.
Week 5
Feb 13 (Mon) Prepare for Wednesday's test on sections 12.1, 12.2, 12.3, 12.4, 12.5, 12.6, 13.1, 13.2, 13.3, 13.4, 14.1, 14.2, 14.3, 14.4, and the worksheet on lines and planes. No calculators or notes are allowed, and you should bring a student ID. The test will be given during the lecture period. It is expected that you now know how to solve every assigned homework and quiz problem. Detailed solutions to all odd problems and assigned even problems are available at Illinois Compass 2g. Solutions to the quizzes and the worksheet are posted on the course homepage. I also have sent an email to your university email address with a bit more information.

Check the available tutoring hours. No appointment is necessary.

The cover page on your test will include a seating chart. When you pick up your test on Wednesday, one of these seat numbers will be circled and that will be your assigned seat for the first test. Try to arrive early to obtain your assigned seat. Unless you happen to sit at the end of a row, you will be required to stay for the full 50 minute testing period. The TAs will not answer any questions during the test.

Be sure to go to both discussion sections this week.

Feb 15 (Wed) Test 1 (given during lecture)
Feb 16 (Thu) Discussion sections still meet today. The TAs will introduce section 14.5 on the chain rule. For homework do #1–12 in section 14.5.
Feb 17 (Fri) Read sections 14.5 and 14.6. In 14.5 do #1–12, 21, 23, 29, 33, 38, 43.
Week 6
Feb 20 (Mon) From section 14.6 we discussed gradients. You should read about directional derivatives on your own and I will discuss them briefly on Wednesday. For homework do #4, 5, 7, 9, 12, 15, 19, 22, 23, 33, 41, 43, 48, 52 in section 14.6. On Thursday there will be a quiz on sections 14.5 and 14.6.
Feb 22 (Wed) There is no new homework. Prepare for tomorrow's quiz on sections 14.5 and 14.6. In lecture I directional derivatives and went over some homework from 14.5 and 14.6.
Feb 23 (Thu) Quiz #4 on sections 14.5 and 14.6 will be given during today's discussion section.
Feb 24 (Fri) Read section 14.7. For homework do #1, 7, 8, 11, 13, 19, 32, 35, 40, 42, 50, 51 in section 14.7. On Monday I will introduce section 14.8. You will have a quiz Thursday on sections 14.7 and 14.8.
Week 7
Feb 27 (Mon) Read section 14.8. For homework do #5, 6, 8, 11, 28, 30, 38 in section 14.8. You will have a quiz Thursday on sections 14.7 and 14.8. I have cancelled today's office hours, but will be available Tuesday 3-5:30pm in 121/123 Altgeld Hall.

In lecture we looked at the example f(x,y) = x2 y2. We found the critical points to be (a, 0) and (0, b) where a and b are any real numbers. For these values we saw that the discriminant D = 0. For this reason the 2nd derivative test is inconclusive. However we saw in another way that the absolute minimum value of 0 is obtained at each of these critical points.

We discussed the Extreme Value Theorem and where to look for absolute maxima and absolute minima. In particular we found the absolute max/min of f(x, y) = x2 + y2 − 2x + 2y + 5 on the set of (x,y) pairs for which x2 + y2 ≤ 4. For the boundary we used the parameterization x = 2 cos(t), y = 2 sin(t), 0 ≤ t ≤ 2π. The book's solutions show another approach for a similar problem.

Lagrange Multipliers can be used to minimize or maximize a function f(x,y,z) subject to a constraint g(x,y,z) = c. Our particular example used fewer variables to help us visualize why this technique works. We saw how to find the point closest to the origin on the hyperbola xy = 3. This amounted to minimizing the function f(x,y) = x2 + y2 subject to the constraint xy = 3. Although this problem can be solved without using Lagrange Multipliers, it is helpful to know this technique for problems which cannot be solved easily in other ways.

Next time we will look at more complicated examples of Lagrange Multipliers. I may also introduce section 15.1.

Feb 29 (Wed) Read sections 15.1 and 15.2. In 14.8 do #15, 17. In 15.1 do #11, 12. In 15.2 do #5, 10, 16, 17, 22, 27, 28, 30, 36. Prepare for tomorrow's quiz on sections 14.7 and 14.8.

We sought to minimize f(x,y,z) = 2x2 + y2 + 3z2 under the constraint 2x − 3y = 49 + 4z. We needed to rearrange the constraint equation so that we had only a constant on the right hand side. This gave 2x − 3y − 4z = 49 and informed us to let g(x,y,z) = 2x − 3y − 4z. Using the technique of Lagrange Multipliers we set ∇ f = λ ∇ g to arrive at the equations 4x = 2λ, 2y = −3λ and 6z = −4λ. Together with the constraint equation 2x − 3y − 4z = 49, we solved the system of equations to obtain (x,y,z) = (3, −9, −4). Graphically we should see why there is no maximum, so this point must give us the minimum value f(3, −9, −4) = 147.

When we wish to find the extreme values of f(x,y,z) subject to the two constraint equations g(x,y,z) = C1 and h(x,y,z) = C2, the technique of Lagrange Multipliers generalizes to solving the system of equations obtained from f = λ1 ∇ g + λ2 ∇ h along with the two constraint equations.

I introduced volumes over a rectangular region as a limit of double Riemann sums. I defined double integrals using this limit and we used it to determine certain volumes. If f is continuous over a rectangular region, then Fubini's theorem tells us we will obtain the same result regardless of whether we integrate with respect to x first then y or with respect to y first then x. For the homework problems you should always solve the double integrals both ways if possible. It is only after attempting this for each problem that you will learn to anticipate which approach will be easiest.

Mar 1 (Thu) Quiz #5 on sections 14.7 and 14.8 will be given during today's discussion section.
Mar 2 (Fri) Read section 15.3. In 15.3 do #5, 10, 13, 19, 21, 25, 32, 33, 39, 41, 43, 45, 47, 49, 55. Quiz #6 on sections 15.1, 15.2 and 15.3 will be given during Tuesday's discussion section. Quiz #7 will be a take-home quiz on sections 15.4, 15.5 and possibly 15.6. It will be due in lecture next Friday.
Week 8 (Deadline to drop this course without a grade of W is Friday, March 9, 2012)
Mar 5 (Mon) Read section 15.4. Do #9, 11, 17, 19, 22, 27, 29, 30, 31, 32 in 15.4.
Mar 6 (Tue) Quiz #6 on sections 15.1, 15.2 and 15.3 will be given during today's discussion section.
Mar 7 (Wed) Read section 15.5. In 15.5 do #1, 3, 5, 7, 10, 11, 12, 13, 14. More problems from this section will be assigned Friday along with some problems from 15.6. The take-home quiz will now be distributed on Friday and due on Monday.
Mar 8 (Thu)
Mar 9 (Fri) Read section 15.6. In 15.6 do #3, 5, 11, 18, 19, 27, 31, 37, 39. Turn in quiz #7 at the beginning of Monday's lecture.
Week 9
Mar 12 (Mon) Prepare for Wednesday's test on sections 14.5, 14.6, 14.7, 14.8, 15.1, 15.2, 15.3, 15.4, 15.5 and 15.6. No calculators or notes are allowed, and you should bring a student ID. The test will be given during the lecture period. It is expected that you now know how to solve every assigned homework and quiz problem. Detailed solutions to all odd problems and assigned even problems are available at Illinois Compass 2g. Solutions to the quizzes are posted on the course homepage. For sections 15.5 and 15.6, I will limit the applications to those on the assigned homework. In particular this test will not have questions on moments of inertia or probability.

Check the available tutoring hours. No appointment is necessary.

The cover page on your test will include a seating chart. When you pick up your test on Wednesday, one of these seat numbers will be circled and that will be your assigned seat for this test. Try to arrive early to obtain your assigned seat. Unless you happen to sit at the end of a row, you will be required to stay for the full 50 minute testing period. The TAs will not answer any questions during the test.

Be sure to go to both discussion sections this week.

Mar 14 (Wed) Test 2 (given during lecture)
Mar 15 (Thu) Discussion sections still meet today. The TAs will introduce section 15.7 on cylindrical coordinates. For homework do #1, 3, 5, 9, 15, 16, 17, 21, 22 in section 15.7.
Mar 16 (Fri) Read section 15.8. Finish the homework assigned yesterday from section 15.7. In section 15.8 do #1, 3, 7, 11, 12, 18, 23, 25, 27, 28, 34. In lecture we discussed integration in cylindrical and spherical coordinates. On Monday after break I'll do a couple of problems from these sections but then introduce section 15.9 on change of variables. Students generally view this as a difficult section. After that we will finish the course with chapter 16. Chapter 16 is considered by most students to be far more difficult than the earlier chapters.
Week 10 (Spring Break!)
Week 11
Mar 26 (Mon)
Mar 28 (Wed) Read section 15.9 very carefully. In section 15.9 do #1, 2, 3, 7, 8, 12, 13, 15. Prepare for tomorrow's quiz on sections 15.7 and 15.8.

We discussed change of variables in the following contexts.

• We solved a simple integral using u-substitution as seen in Calculus I.
• We solved #28 in 15.7 by switching from rectangular to cylindrical coordinates.
• We solved #39 in 15.8 by switching from rectangular to spherical coordinates.
• We found the area of the ellipse x2/a2 + y2/b2 = 1 by substituting u = x/a and v = y/b.
• We saw that the change in variables u = 3x − 2y and v = x + y resulted in dxdy = (1/5)dudv so that a double integral with respect to x and y can be converted to a double integral in terms of u and v.
• I quickly defined the Jacobian and showed how it is used to transform dxdy into dudv in order to change the variables of integration.
Mar 29 (Thu) Quiz #8 on sections 15.7 and 15.8 will be given during today's discussion section.
Mar 30 (Fri) Reread section 15.9 carefully. Read 16.1. In section 15.9 do #19, 20, 21, 22.

I discussed the Jacobian in greater detail today and used it in the following problems.

• We saw that x = r cos(θ) and y = r sin(θ) leads to dy dx = r dr dθ as expected when converting from rectangular to polar coordinates.
• We saw that x = ρ sin(φ) cos(θ), y = ρ sin(φ) sin(θ) and z = ρ cos(φ) leads to dz dy dx = ρ2 sin(φ) dρ dθ dφ as expected when converting from rectangular to spherical coordinates.
• To evaluate the double integral of sqrt( (x − y) / (x + y + 1) ) over the square with vertices (0, 0), (1, −1), (2, 0), and (1, 1), we used the change of variables u = x − y and v = x + y. I believe that I mistakenly used the partials of u and v with respect to x and y instead of the partials of x and y with respect to u and v to compute the Jacobian. This resulted in a Jacobian of 2 instead of the correct Jacobian of 1/2. I'll post a corrected solution here this weekend.
• We began the set-up of #15 in section 15.9 noting that the region is not transformed into a nice polygon.
I'll post the solutions to section 15.9 on Illinois Compass 2g this evening.
Week 12
Apr 2 (Mon) Read section 16.1. In section 16.1 do #1, 2, 6, 11, 15, 21, 23. Quiz #9 on sections 15.9 and 16.1 will be given on Thursday.

I spent time in lecture discussing how to decide what change of variables to make. We mostly concentrated on simplifying the region of integration but also looked at simplifying the integrand.

For example, problem 15 from section 15.9 included boundaries given by the hyperbolas xy = 1 and xy = 3. By letting u = xy we now have u ranging from 1 to 3 and the region of integration is simplified. We also let v = y and did a careful analysis of how to solve this problem.

We also looked at problems 19, 20 and 21 from section 15.9 to see how we arrive at a reasonable substitution.

I introduced vector fields and mentioned their applications to wind patterns, fluid flow, force fields, electric/magnetic fields. We then looked at graphs for the following vector fields.

• F = 2ij = ⟨2, −1⟩
• F = xi = ⟨x, 0⟩
• F = xi + yj = ⟨x, y⟩
• F = −yi + xj = ⟨−y, x⟩
I noted how we've seen vector fields before in the form of gradient vector fields.
Apr 4 (Wed) Read section 16.2. In section 16.2 do #3, 5, 8, 11, 13, 19, 21, 33, 39, 40, 41. Prepare for tomorrow's quiz on sections 15.9 and 16.1.

In lecture we discussed smooth curves, parametrizations and line integrals. We saw that line integrals can be used to compute area, average value, mass, work, etc. We spent some time on deriving how to compute the work done by a force field in moving an object along a curve. For our examples we calculated the following.

• The average temperature along the border of a circular plate of radius 3 centered at the origin when the temperature is given by T(x,y) = 100(x2 + 2y2).
• The work done by a force field F = −yi + xj = ⟨−y, x⟩ in moving an object along the curve C given parametrically by x = t, y = t2, 0 ≤ t ≤ 1.
We solved this last problem in multiple ways to show the different approaches discussed in section 16.2.
Apr 5 (Thu) Quiz #9 on sections 15.9 and 16.1 will be given during today's discussion section.
Apr 6 (Fri) Read section 16.3. In section 16.3 do #3, 4, 6, 7, 9, 12, 14, 15, 18.

The plan for the rest of the semester is to cover sections 16.4 – 16.9 in order.

Week 13
Apr 9 (Mon) Read section 16.4 very carefully. I won't lecture on this section until Wednesday. For additional homework in 16.3 do #19, 20, 21, 22. Thursday's quiz will cover sections 16.2 and 16.3.

I began class by answering questions from earlier sections. We looked at the following equivalent conditions.

Let F = ⟨P, Q⟩ have continuous first partial derivatives in an open connected region R and let C be a piecewise smooth curve in R. The following conditions are equivalent.

• F is conservative. That is, F = ∇ f for some function f.
• The integral along C of F • dr is independent of path.
• The integral along C of F • dr = 0 for every closed curve C in R.
This means that for conservative vector fields, it sometimes is worthwhile to integrate along a different path.

For our main example we let C1 be the path from (0,0) to (2,0) along the top half of a circle of radius 1 centered at (1, 0). Given F = ⟨y3 + 1, 3xy2 + 1⟩, we found the integral along C1 of F • dr in three ways.

1. We parametrize C1 using r(t) = ⟨1 − cos(t), sin(t)⟩ for 0 ≤ t ≤ π. Then dr / dt = ⟨sin(t), cos(t)⟩ and F = ⟨sin3(t) + 1, 3(1 − cos(t))sin2(t) + 1⟩. Now the integral along C1 of F • dr = the integral from 0 to π of = F • (dr / dt) dt = the integral from 0 to π of ( sin4(t) + sin(t) + 3(1 − cos(t))sin2(t)cos(t) + cos(t) ) dt. This is very messy but the integral can be evaluated to give an answer of 2.
2. Using the techniques in section 16.3 we know that F is a conservative vector field (see theorem 6). We find a potential function f = xy3 + x + y. Now the integral along C1 of F • dr = the integral along C1 of ∇ f • dr = f(2,0) − f(0,0) = 2.
3. Since F is conservative, we know that the value of the line integral is independent of path. Thus we replace the semicircular path with the simpler straight line path C2 from (0,0) to (2,0). The parametrization we use for C2 is r(t) = ⟨t, 0⟩ for 0 ≤ t ≤ 2. Then dr / dt = ⟨1, 0⟩ and F = ⟨1, 1⟩. Now the integral along C1 of F • dr = the integral along C2 of F • dr = the integral from 0 to 2 of F • (dr / dt) dt = the integral from 0 to 2 of 1 dt = 2.
Apr 11 (Wed) Read section 16.4. In section 16.4 do #3, 4, 5, 7, 8, 11, 13, 17, 18, 19. Thursday's quiz will cover sections 16.2 and 16.3. There will be two quizzes next week.

We looked at the statement and proof of Green's Theorem along with a number of examples.

Apr 12 (Thu) Quiz #10 on sections 16.2 and 16.3 will be given during today's discussion section.
Apr 13 (Fri) Read section 16.5. In section 16.5 do #2, 4, 8, 10, 11, 12, 14, 15, 16, 18, 19, 20, 21, 22, 24. There will be two quizzes next week. The quiz on Tuesday covers sections 16.4 and 16.5.

We looked at an example with Green's Theorem when the curve was not positively oriented.

I proved the theorem given as problem #27 in section 16.3. Afterwards we looked at the definitions, physical interpretations and some theorems for curl F and div F.

Week 14
Apr 16 (Mon) Read section 16.6. In section 16.6 do #3, 5, 6, 20, 23, 24, 34, 35, 38, 42, 43. Prepare for tomorrow's quiz on sections 16.4 and 16.5.
Apr 17 (Tue) Quiz #11 on sections 16.4 and 16.5 will be given during today's discussion section.
Apr 18 (Wed) Read section 16.7 carefully. In section 16.7 do #6, 11, 12, 14, 16, 19, 21, 24, 25, 29, 37, 38. There will be a take-home quiz on sections 16.6 – 16.7 and parts of sections 16.8 – 16.9. It will be distributed on Friday and due at the beginning of Monday's lecture. Check back here later today for a list of homework problems from 16.7.
Apr 19 (Thu) Quiz #12 will be a take-home quiz on parts of sections 16.6 – 16.9. It will be due at the beginning of Monday's lecture.
Apr 20 (Fri) Since we've been rushing through recent material fairly quickly, I've decided to hold off on the last two important sections until after the test. These important sections cover Stokes' Theorem (16.8) and the Divergence Theorem (16.9). There has also been a change in plans for the last quiz since I want it to cover these important sections. Quiz #12 will now be a take-home quiz given on Monday (April 30) and due Wednesday (May 2).

The test on Wednesday will be on the following sections. I've included the section heading and the homework for each of these sections.

• Section 15.7 (Triple Integrals in Cylindrical Coordinates): #1, 3, 5, 9, 15, 16, 17, 21, 22.
• Section 15.8 (Triple Integrals in Spherical Coordinates): #1, 3, 7, 11, 12, 18, 23, 25, 27, 28, 34.
• Section 15.9 (Change of Variables in Multiple Integrals): #1, 2, 3, 7, 8, 12, 13, 15, 19, 20, 21, 22.
• Section 16.1 (Vector Fields): #1, 2, 6, 11, 15, 21, 23.
• Section 16.2 (Line Integrals): #3, 5, 8, 11, 13, 19, 21, 33, 39, 40, 41.
• Section 16.3 (The Fundamental Theorem for Line Integrals): #3, 4, 6, 7, 9, 12, 14, 15, 18, 19, 20, 21, 22.
• Section 16.4 (Green's Theorem): #3, 4, 5, 7, 8, 11, 13, 17, 18, 19.
• Section 16.5 (Curl and Divergence): #2, 4, 8, 10, 11, 12, 14, 15, 16, 18, 19, 20, 21, 22, 24.
• Section 16.6 (Parametric Surfaces and Their Areas): #3, 5, 6, 20, 23, 24, 34, 35, 38, 42, 43.
• Section 16.7 (Surface Integrals): #6, 11, 12, 14, 16, 19, 21, 24, 25, 29, 37, 38.
Week 15
Apr 23 (Mon) Test help session
Apr 24 (Tue) Discussion section
Apr 25 (Wed) Test 3 (given during lecture)
Apr 26 (Thu) Discussion section: TAs will introduce Stokes' Theorem from section 16.8.
Apr 27 (Fri) Read section 16.8. In 16.8 do #2, 3, 4, 7, 8, 10, 13.
Week 16 (Last day for U of I classes is Wednesday, May 2, 2012)
Apr 30 (Mon) Read section 16.9 and look at the summary in section 16.10. In 16.9 do #2, 5, 7, 9, 10, 12, 13, 19, 20. Turn in quiz #12 at the beginning of Monday's lecture.

My office hours this week are Tuesday/Thursday 4-5:30pm. The tutoring room will be definitely be open Monday, Tuesday and Wednesday. If there are additional hours I will post them here.

May 1 (Tue) Last discussion section meeting
May 2 (Wed) Quiz #12 is due at the beginning of lecture.

The cumulative final exam for section DL1 will be held Wednesday, May 9, 8:00 AM – 11:00 AM in 314 Altgeld Hall. Here are the sections covered.

• Test 1 material (12.1, 12.2, 12.3, 12.4, 12.5, 12.6, 13.1, 13.2, 13.3, 13.4, 14.1, 14.2, 14.3, 14.4, lines/planes worksheet)
• Test 2 material (14.5, 14.6, 14.7, 14.8, 15.1, 15.2, 15.3, 15.4, 15.5, 15.6)
• Test 3 material (15.7, 15.8, 15.9, 16.1, 16.2, 16.3, 16.4, 16.5, 16.6, 16.7)