Multivariable Calculus MA 241-02

Final Exam

Tuesday, May 10, 2:30 pm

The final exam is cumulative, that is, it covers all the material from the first day of class onwards. Approximately 25-35% will cover material since the second exam. This is Sections 16.4, 16.5, 17.1 -- 17.4, and Chapter 18. It is recommended that you go over the homework problems as well as your class notes. Many of the problems and questions we discuss in class are excellent examples of test questions. I have also listed some sample problems from the Chapter 16, 17 and 18 Reviews and Check Your Understanding Sections below. For problems from the earlier chapters see the previous Exam Review sheets.

Note that these problems are a SAMPLE selection. Certain topics are covered in the reviews better than others. The odd answers are in the back of the book while the evens are listed here. The exam will be designed to take 2 hours although you will have the full 3 hour exam period.

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. Creating this reference paper will be an excellent opportunity to review topics for the exam.

EXAM REVIEW SESSION: Sunday, May 8, 2:00 - 4:00 pm in SWORDS 359. Please come prepared with specific questions.

List of Topics By Chapter

• Chapter 12: 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 13: Vectors in the plane and in space, unit vectors, dot product and its geometric properties, cross product and its geometric properties, equation of a plane.
• Chapter 14: Partial derivatives (limit definition and algebraic computation of), linearization and the tangent plane, gradient vector and its qualitative properties, directional derivative, chain rule, second-order partials, first and second order Taylor approximations
• Chapter 15: Finding and classifying critical points (local mins, maxs, saddles), second derivative test, global extrema, constrained optimization, Lagrange multipliers.
• Chapter 16: 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 (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 17: Parameterizing curves (lines, circles, ellipses, etc.) 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 a curve, vector fields, the flow of a vector field
• Chapter 18: Line integrals (physical interpretation as work and calculation of), path-independent or conservative vector fields (how to check if a given vector field is conservative), Fundamental Theorem of Line Integrals, finding a potential function for a path-independent vector field, properties of a path-independent vector field, the curl of a planar vector field, Green's Theorem.

Chapter 16 Review, pp. 777 - 779
Problems:   1, 3, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 27, 33, 35, 41

The answers to the evens are:
6.   0 <= r <= 3, 0 <= z <= 2, 0 <= theta <= 2Pi
8.   0 <= rho <= 5, 0 <= phi <= Pi, 0 <= theta <= Pi/2
10.   85/12
12.   10(e-2)
16.   7Pi/3
18.   Pi(3 - 2 Ln 2)
24.   positive

Chapter 16 Check Your Understanding, pp. 780 - 781
Problems:   1, 3, 5, 7, 11, 15, 19, 21, 23, 25, 27, 28

The answers to the evens are:
28.   True

Chapter 17 Review, pp. 819 - 822
Problems:   1, 3, 5, 7, 9, 10, 11, 13, 15, 17, 22, 25

The answers to the evens are:
10.   x = 3 - 2t, y = -2 + t, z = t
22.   (a) First show x^2 + y^2 = R^2. Radius = R, ccw motion, period = 2Pi/omega.  (b) velocity is perpendicular to position r, speed = R omega.  (c) acceleration a = -omega^2 r, ||a|| = omega^2 R

Chapter 17 Check Your Understanding, pp. 822 - 823
Problems:   1, 3, 5, 7, 9, 15, 17, 19, 21, 23, 25, 30, 31

The answers to the evens are:
30.   False

Chapter 18 Review, pp. 857 - 860
Problems:   1, 3, 5, 7, 9, 10, 11, 12, 13, 17, 19, 21, 23

The answers to the evens are:
10.   (ii) and (iv)
12.   Path-independent since F = grad f where f = y^2/2

Chapter 18 Check Your Understanding, pp. 860 - 861
Problems:   1, 5, 6, 7, 9, 13, 15, 19, 23, 27, 29, 31, 39

The answers to the evens are:
6.   False