Multivariable Calculus, MATH 241

Prof. Gareth Roberts


Tuesday, May 14, 8:00 - 10:30 am, Smith Labs 154

The final exam is cumulative, that is, it covers all of the material from the first day of class onwards. Approximately 15 - 20% will cover material since the third midterm exam. This is Sections 13.3 - 13.5. It is recommended that you go over homework problems (both written and WebAssign), worksheets, the midterm exams, computer projects, and your class notes. Many of the problems and questions we discuss in class are excellent examples of test questions. The solutions to your WebAssign problems can be seen by clicking "View Key" near the top of each assignment. You can also click on "Practice Another Version" to redo most homework problems.

Important Note: You will be allowed a scientific calculator for the exam which does NOT have graphing capabilities. Please bring your own "certified" calculator to the exam. 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 should help you review for the exam.

Click here for some sample final exam questions. Click here for the solutions to these problems.
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 other review questions, see the previous exam review sheets: Exam 1, Exam 2, Exam 3.

The exam will be designed to take two hours (twice the length of a midterm) although you will have the full 2.5 hours to complete the exam.

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.), conic sections, 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 special curves (e.g., circles, line segments, parabolas), 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 2d and 3d vector field, divergence, Green's Theorem.

Chapter 13 Review, pp. 975-976
Problems:   10, 11, 13, 14, 15, 16, 17, 18, 35

The answers to the evens are:
10.   (a) (3π - 9)/2, (b) -3π/4.
14.   curl F = 0 and the domain of F is simply connected. The value of the line integral is 2 using the Fundamental Theorem of Line Integrals.
16.   3.
18.   curl F = <-e-y cos(z), - e-z cos(x), -e-x cos(y)>, div F = -e-x sin(y) - e-y sin(z) - e-z sin(x) .