Multivariable Calculus, MATH 241
Final Exam
Monday, May 11, 3:00 - 5:30 pm (Section 01)
Wednesday, May 13, 11:30 - 2:00 pm (Section 02)
Swords 328
The final exam is cumulative, that is, it covers all of the material from the first
day of class onwards. Approximately 20 - 25% will cover material since the third midterm exam. This is
Sections 13.1 - 13.4. It is recommended that you go over homework
problems, class notes, the midterm exams, and the three labs. Many of the problems and questions we discuss in
class are excellent examples of test questions. Some sample final exam questions are available
here. The solutions to these questions are available
here.
In addition, some review problems from the Chapter 13 Review Exercises are listed below.
The odd answers are in the back of the book while the evens are listed here.
The Concept Check problems at the end of each chapter (before the Exercises) are also a good source of questions.
For problems from the earlier chapters see the previous Exam Review sheets.
The final exam will be approximately twice the length of a midterm exam.
Note: You will be allowed a scientific calculator for the exam which does NOT have graphing capabilities.
Please bring your own calculator. In addition, 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. Creating this
reference paper will be an excellent opportunity to review topics for the exam.
Exam Review Session: Sunday, May 10, 6:30 - 8:00 pm in O'Neil 112. Please come prepared
with specific questions.
List of Key Topics By Chapter
- Chapter 9: Three-dimensional coordinates, distance formula, vectors in the plane or in
space, length of a vector, unit vectors, vector addition, vector between two points, dot product and its geometric properties,
cross product and its geometric properties, equation of a plane, parametric equations of
a line, equations for special surfaces (bowl, saddle, cone, sphere, etc.),
cross-sections, cylindrical and spherical coordinates.
- Chapter 10: Parametrized curves (lines, circles, helix, etc.) in the plane or in space,
finding the velocity, speed, or acceleration of a parameterized curve r(t), arc length of a curve,
reparametrizing a curve with respect to arc length, curvature, the unit tangent vector T(t),
the TNB-frame, the tangential and normal components of acceleration.
- Chapter 11: Functions of two or three variables, domain and range, graphs of functions,
contour diagrams, level curves, level surfaces,
limits of a function of two variables, continuity, first- and second-order partial derivatives
(limit definition, computation of, and their qualitative meaning), tangent plane,
chain rule (tree diagrams), directional derivative, gradient vector and its qualitative properties,
finding and classifying critical points (local mins, maxs, saddles),
second derivative test, global extrema, constrained optimization, Lagrange multipliers.
- Chapter 12: Integration (Riemann sums and geometric understanding of),
double and triple integrals, changing the order of integration
in a double integral (horizontal versus vertical cross sections), changing coordinates to
evaluate an integral (polar, cylindrical, or spherical coordinates),
finding the area of a region in the plane or the volume of a solid in space.
- Chapter 13: Vector fields, line integrals (physical interpretation as work and calculation of),
parametrizing circles and line segments, conservative vector fields
(properties of, how to check if a given vector field is conservative, path independence), the Fundamental Theorem of Line Integrals,
finding a potential function for a conservative vector field, the curl of a planar vector field, Green's Theorem.
Chapter 13 Review, p. 975-976
Problems: 1a, 4, 6, 7, 9, 11, 13, 14, 15, 16, 17
The answers to the evens are:
4. 0 (what type of vector field is F?)
6. e - 9/70
14. Don't worry about showing that F is conservative. You can assume it is. The value
of the line integral is 2 using the Fundamental Theorem of Line Integrals.
16. 3.