Math 234 — About the Exams

• Final
• Midterms
• Quizzes

The Final

Extra Office hours: Wednesday and Friday 10am–3pm.

Exam time&location. December 21, 5:05 ^pm—7:05 ^pm. The exam will be in Psychology 105 and Psychology 113. Which classroom you should go to depends on your TA.

• If your TA is Christopher Breeden, Christian Geske, Yuan Liu, or Yi Li, then go to Psychology 105
• If your TA is Mahadharshini Devanesan or Thomas Morrell, then go to Psychology 113

Topics covered. The final exam is cumulative. Exam problems will cover the following topics:

• Parametric curves: know how to parametrize a straight line, a line segment connecting two points, a circle. Find the unit tangent and normal to given parametrized plane curves.
• Chain rule for $g(t) = f(x(t), y(t))$, or $g(u,v)=f(x(u,v), y(u,v))$; second derivatives too. For practice, see problem 2 on the 2013 final.
• Clairaut’s theorem, and how to find a function $f(x,y)$ if you are given its partial derivatives $f_x(x,y)=P(x,y)$ and $f_y(x,y)=Q(x,y)$. Note the even though there may not be an explicit question of this form on the exam, knowing how to find $f$ can be very useful when doing line integrals.
• Maxima and minima of functions of two or three variables (the theorems: continuous function on a closed and bounded set always has a max and a min; if a max or min is an interior point, then it’s a critical point)
• Second derivative test (say in words why it works? Know how to analyze the second order terms by completing a square)
• Lagrange multipliers (don’t forget the so-called “exceptional case”)
• Double & Triple integrals (finding the integration bounds from a drawing or description of the domain; drawing the domain given the integration bounds; changing the order of integration)
• Know that $\int_\cC1\,ds$ is the length of the curve $\cC$, $\iint_\cR1\,dA$ is the area of the two dimensional region $\cR$, and $\iiint_\cR 1dV$ is the volume of $\cR$ if $\cR$ is a three dimensional region.
• Polar, Cylindrical, and Spherical Coordinates (recognize regions that are “blocks” in either cylindrical or spherical coordinates; understand the drawings that shows $dA=r\, dr\,d\theta$, $dV=r\,dz\,dr\,d\theta$, and $dV = \rho^2\sin\phi\,d\rho\,d\theta\,d\phi$)
• Line integrals (work and flux integrals; differential form version) and Green's theorem: the two forms (“work” and “flux” integrals). The fundamental theorem (about the line integral when the vector field is the gradient of a function.)
  See a short summary of line integrals and the Extra problems about line integrals (with answers)

Midterms

Midterm 1— Wednesday, September 30

Topics: See this page for the list of topics covered on the first midterm.

Location: The midterm will be in our usual classroom, B102 Van Vleck Hall and in B130 Van Vleck Hall.

Solutions to the exam: Here is a copy of the exam, with solutions.

Why the problem 4 score? The final exam will have one extra problem which will be very similar to problem 4 from the first midterm. Students’ scores on that problem will not count toward the final, but instead will be added to their score on problem 4, midterm 1 (the maximal score on midterm 1 therefore becomes 115 points instead of 100).

Midterm 2— Wednesday, October 28

Topics: See this page for the list of topics covered on the second midterm.

Location: Same as for the first midterm

The exam, with solutions. Here is a copy.

Midterm 3— Wednesday, December 2

Topics: See this page for the list of topics covered on the third midterm.

Location: Same as for the first two midterms, namely:

The midterm will be in our usual classroom, B102 Van Vleck Hall and in B130 Van Vleck Hall.

• If your TA is Christian Geske, Christopher Breeden, or Thomas Morrell, then go to B102 (the usual class room).
• If your TA is Yu Li, Yuan Liu, or Mahadharshini Devanesan, then go to B130 Van Vleck.

The exam, with solutions. Here is a copy.

Quizzes

• Sept 10. Topics: The homework problems from chapter 1.
• Sept 24. Topics: The quiz will cover the homework problems since the previous quiz. I.e. the problems from chapter 2, the extra vector function problems , and mostly problems 3 and 5 from chapter 3.
• Oct 8. Topics: The quiz will cover:
  • level sets and domain of functions (Chapter 3, problem 7, page 47).
  • partial derivatives (Chapter 4, §3, problems 2, 6, page 52).
  • linear approximation (as done in lecture, and like problems 1 and 8 from Chapter 4, §7, pages 61/62).
• Oct 22. Topics: The quiz will cover:
  • Tangent planes or lines. These have come up in the following different settings:
    • graph of $z=f(x,y)$
    • level set of $z=f(x,y)$
    • level set of $F(x,y,z)$
    In each case, is the tangent a line or a plane? What are the equations for the tangent line or plane in these cases?
  • The chain rule: know how to compute the first and second order partial derivatives of $g(u,v) = f(x(u,v), y(u,v))$ if you are given the two functions $x(u,v)$ and $y(u,v)$. (like problem 5, page 76, and problem 15, page 82)
  • Mixed partials and Clairaut’s theorem: know how to tell if there is a function $z=f(x,y)$ with two given partial derivatives $f_x=P(x,y)$ and $f_y=Q(x,y)$, and know how to compute $f$ when it exists. (e.g. problems 12, 13, 14 on page 81/82).
• Nov 5. Topics:
  • The two main theorems: You should know, and know how to use the two main theorems, namely §2.1 (page 84), and §4.2 (page 86). These can be used to predict critical points, and to find (local) maxima and minima (as in §4.4 and §4.5 on page 88). Typical problems are §6, problem 1a–1r.
  • The second derivative test: compute the second order Taylor expansion of a function at a critical point, and use the expansion to decide if the critical point is a local max/min, or perhaps something else. Typical problems: Chapter 5,§10, problem 5: there you are asked to apply the second derivative test to the critical points that you found in §6, problem 1. The online version of the text has answers to these problems.
• Nov 19. Topics:
  • Lagrange multipliers. Keep in mind that there are at least two parts of a Lagrange multiplier problem:
    1. setting up the equations you must solve
    2. solving them (and then deciding which has the lowest/highest function value, if needed)
    There is no recipe for solving the equations that is guaranteed to work always, but it is a good idea to systematically eliminate one variable after another (solve one equation for $x$, substitute the result in the other equations; then get rid of the next variable, until you are done.) Even if you get stuck in the solving part, you can still get partial credit for step 1, the formulation of the equations, provided (!) this is neatly written up.
  • Double integrals: Expect a problem like one of those in problems 5 or 6, from Ch.6 §3. You should be able to
    1. draw the region $D$
    2. set up the integral
    3. compute it (and choose the other order of integration if you run into difficult integrals)
    On timed tests the integrals (i.e. the antiderivatives) you will be asked to find will always be fairly easy, with a simple substitution being the most complicated trick required. So if you run into an integral that isn’t easy, then you should choose the other order of integration (or possibly you made a mistake).