MATH 250, Study Guide for the Final Exam

12/4/2015

Final Exam Date and Time: Monday December 14, 12:45pm - 2:45pm, in CR 5117.

Exam coverage

1. Analytic Geometry in Three-dimensional Space (Chapter 12)
   1. Three-dimensional Coordinate Systems (12.1)
   2. Vectors in Two and Three Dimensions (12.2)
   3. The Dot Product (12.3)
   4. The Cross and Triple Products (12.4)
   5. Lines and Planes (12.5)
   6. Cylinders and Quadric Surfaces (12.6)
2. Vector Functions and Geometry of Curves (Chapter 13)
   1. Vector Functions and Space Curves (13.1)
   2. Derivatives and Integrals of Vector Functions (13.2)
   3. Arc Length and Geometry of Curves (13.3)
3. Derivative for functions of several variables (Chapter 14)
   1. Functions of several variables, domains, ranges, graphs (14.1)
   2. Limits and Continuity (14.2)
   3. Partial Derivatives (14.3)
   4. Tangent Planes and Linear Approximations. Differentials (14.4)
   5. The Chain Rule (14.5)
   6. Directional Derivatives and the Gradient Vector (14.6)
   7. Maximum and Minimum Values (14.7)
   8. Method of Lagrange Multipliers (14.8)
4. Multiple Integrals (Chapter 15)
   1. Double Integrals over Rectangles (15.1)
   2. Iterated Integrals (15.2)
   3. Double Integrals over General Regions (15.3)
   4. Double Integrals in Polar Coordinates (15.4)
   5. Triple Integrals (15.7)
5. Vector Calculus (Chapter 16)
   1. Vector Fields (16.1)
   2. Line Integrals (16.2)
   3. The Fundamental Theorem of Line Integrals (16.3)
   4. Green's Formula, the Divergence Theorem in 2D (16.4)

Key concepts (in addition to those listed for Midterm I and Midterm II)

• Computing double integrals using polar coordinates
• Using double and triple integrals to compute volumes
• Changes of order of integration in double and triple integrals
• Vector fields, potential (gradient, conservative) fields
• Two kinds of line integrals: "arc length" and "work done by a vector field"
• The fundamental theorem of calculus for line integrals. Path independence. Closed loop property. The "equality of mixed partials" (Q_x=P_y) condition
• Method for finding a potential under the "equality of mixed partials" condition
• Using Green's formula to compute line integrals and areas.

Review problems grouped into types:

• Spheres, lines or planes: equations, intersections with coordinate (or other) planes; parallel, skew, or intersecting lines(12.1, problems 21, 41; 12.R problems 1, 21, 22 (hint: shortest distance is obtained by going perpendicular to the line); 23-26)
• Dot, cross, or triple products. Examples: 12.4, problems 37, 47, 48, 50, 53; 12.R, problems 5, 6, 9, 10.
• Reduce the equation of a quadric surface to one of the standard forms, find sections by coordinate or other suitable planes, sketch the surface; particularly review hyperbolas (finding vertices, asymptotes) and hyperboloids/hyperbolic paraboloids. Examples: 12.6: 31, 33-36; 12.R: 33, 35-38.
• Parametric curves: curve sketching; review helixes, and curves obtained as intersection of surfaces Examples: 13.1, problems 10, 15, 40-42; 13.R problems 1, 3, 6.
• Parametric curves: calculus; derivatives, integrals, arc length, curvature, vectors from the moving trihedral ((T,N,B)-frame) Examples: 13.3 problems 4, 17, 32, 48; 13.R problems 5, 10, 11.
• Domains and contour maps: Examples: Problem 1 in Midterm 2; 14.1 problems 45, 49, 50; 14.R problems 1, 2).
• Continuity and Limits: Path limits to show that a limit does not exist; polar coordinates, or facts about elementary functions to show that a limit does exist. Region of continuity: every point in the domain if a function is given by a formula, plus all points where the limit is equal the value of the function. Examples: 14.2 problems 8, 13, 16, 37, 38; 14.R problems 9, 10).
• Partial derivatives, the chain rule, linear approximations, differentials. Conditions for differentiability. Examples: 14.5 35, 41, problems 14.R problems 32-35, 40.
• Gradients; directional derivatives, tangent planes to surfaces. Examples: 14.R problems 29, 43, 44, 47, 48.
• Find all critical points of a function and determine their type (apply the second derivative test). Examples: 14.R problems 52-54.
• Solve a constrained optimization problem using Lagrange's method (one or two constraints) Examples: 14.R problems 59, 62.
• Compute integrals: using elementary geometry (volumes of prisms, cylinders and spheres); approximately using Riemann sums, exactly using iterated integrals. Examples: Homework for 15.1-4.
• Compute the volume of a 3D region using double or triple integral; use polar coordinates if appropriate Examples: Homework for 15.3,4,7.
• Compute a double integral, change order of integration, sketch a 2D region of integration (Homework for 15.3; 15.R problems 11, 13, 17, 19)
• Compute a line integral using parametric representation of a curve (Homework for 16.2; 16.R problem 5)
• Determine if a vector field is potential, find a potential function, use it to compute a line integral (Homework for 16.3; 16.R problems 11, 13)
• Use Green's formula to compute a line integral or an area (Homework for 16.4)

A certain emphasis will be put on topics in the second half of the course, particularly double integrals and line integrals. Problems on optimization (local extrema/saddle points, Lagrange multipliers) are usually on the exam as well.

You should review your old midterms and quizzes to find out which topics require additional work.

Good luck on the exam!