Multivariable Calculus (Both sections)
Final Exam Review Sheet
Friday, Dec. 7, 8:30 - 11:00 am, O'NEIL 123 (Section 01 -- 2pm class)
Wednesday, Dec. 12, 8:30 - 11:00 am, HAB 238 (Section 02 -- 9am class)
The final exam is cumulative, that is, it covers all the material from the first
day of class onwards. Approximately 50-60% will cover material since the second exam. This is
sections 15.1, 15.2, 15.3, 15.5, 15.6, 16.1, 16.2, 17.1, 18.1, 18.2, 18.3, and 18.4.
You should go over homework problems and class notes. You should also review basic
trigonometry (again!) Some sample problems from the remaining chapters are listed below.
The exam will be designed for 100 minutes (twice the length of the
mid semester exams) so you should have plenty
of time to complete it in the alloted 2.5 hour time slot.
Note: You will be allowed one "cheat sheet" 8.5 x 11 piece of paper, front and back, full of whatever formulas,
graphs, etc. you wish. No calculators will be allowed for the final, however.
EXAM REVIEW SESSION: Wednesday 3:00 - 5:00 pm in SWORDS 359. Please come prepared
with specific questions.
List of Topics By Chapter
- Chapter 11: Visualizing functions of two and three variables, cross-sections, level curves,
level surfaces, contour plot, distance formula, linear functions, limits of functions of two variables, equations
for special surfaces (bowl, saddle, cone, sphere, etc.)
- Chapter 12: Vectors in the plane and in space, unit vectors, dot product and its properties, cross product
and its properties, equation of a plane.
- Chapter 13: Partial derivatives (limit definition and algebraic computation of),
linearization and tangent plane, gradient vector and its properties, directional derivative,
chain rule, second-order partials, Taylor approximations, differentiability.
- Chapter 14: Finding and classifying critical points (local mins, maxs, saddles),
second derivative test, global extrema,
constrained optimization, Lagrange multipliers.
- Chapter 15: Integration, double and triple integrals, changing the order of integration
in a double integral (horizontal versus vertical cross sections), changing coordinates to
evaluate an integral (in the plane -- polar coordinates, in space -- cylindrical or spherical
coordinates), finding the area of a region in the plane or the volume of a solid in space.
- Chapter 16: Parameterizing curves in the plane or in space, how to alter the direction
the curve is traversed, finding the velocity, speed and acceleration of a parameterized curve, arclength of
- Chapter 18: Vector fields (see section 17.1), line integrals (physical interpretation
and calculation of), path-independent or conservative vector fields (how to check if you have such a thing),
Fundamental Theorem of Line Integrals,
finding a potential function for a path-independent vector field, curl, properties
of a path-independent vector field, Green's Theorem.
Three items certain to be on the exam:
- Finding the tangent plane to a surface at a given point.
- Using Lagrange multipliers to solve a constrained optimization problem.
- Finding a potential function for a conservative vector field.
Some sample problems from Ch. 15-18 are below. Note that this is only a partial list.
In many cases, the homework problems more accurately reflect the material covered
in class. See previous exam review sheets
for review problems from earlier chapters.
Chapter 15 review, pp. 269-272
Problems: 3, 5, 7, 12, 14, 18, 19, 20
Chapter 16 review, pp. 319-324
Problems: 5, 7, 9, 11, 14, 16, 19
Chapter 18 review, pp. 381-384
Problems: 1, 3, 4, 6-9, 11, 14, 18 (use Green's Theorem)