MATH 250, Study Guide for the Final Exam

12/11/2013

Final Exam Date and Time: Monday, Dec 16, 2013, 12:45-2:45pm, in CR 5123

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. Vector Calculus (Chapter 16)
   1. Vector Fields (16.1)
   2. Line Integrals (16.2)
   3. The Fundamental Theorem of Line Integrals (16.3)
   4. The Green's Formula (16.4)

Key concepts (list in addition to Midterms I and II)

• Computing double integrals over regions of Type I and II, changing the order of integration
• Using double integrals to compute volume
• Computing double integrals using polar coordinates
• Vector fields, gradient (potential, conservative) fields
• Line integrals with respect to arc length
• Line integrals of vector field. Work done by a vector field
• The fundamental theorem for line integrals. Path independence. Closed loop property
• Conditions under which a vector field satisfies the path independence property
• Method to find the potential function for a gradient vector field
• Green's formula. Computing line integrals over closed loops using double integrals
• Computing areas using line integrals

Here's a somewhat abbreviated list of possible types of final exam questions:

• Play around with equations of spheres (12.1, problems 21, 41; 12.R, problem 1)
• Use properties of the dot or cross product (12.4, problems 47, 48, 12.R, problems 5-7)
• Reduce the equation of a quadric surface to one of the standard forms, find sections by coordinate or other suitable planes, sketch the surface (12.6; 12.R: 29-37 (odd))
• Parametrize a curve given as intersection of surfaces. Use surfaces to help sketch a space curve (13.1, problems 40-44; 13.R problems 1, 3, 6)
• Use triple scalar product to find volumes and to verify if the vectors are coplanar (12.4 problems 35-38; 12.4 problem 10)
• Lines and planes: equations, parallel, intersecting, skew, distance formulas (12.5; 12.R problems 21-27)
• Find the curvature of a space curve, determine the vectors from the (T,N,B)-frame (13.3 problems 17, 48; 13.R problems 1, 10, 11)
• Find the equation of a tangent plane and a normal line to a graph or a level surface (14.4,6; 14.R problems 29, 31)
• Compute a limit using the 'squeeze principle' or show that the limit does not exist; check a function for continuity (14.2 problems 13, 16, 37, 38)
• Applied problems on chain rule: examples from geometry and physics/other sciences (14.5: Example 2, problems 35-44)
• Use differentials to estimate increments of functions (14.4; 14.R problems 33, 34)
• Find all points in the domain of a function where a directional derivative or the gradient satisfy certain conditions. Find the maximum rate of increase/decrease of a function at a given point (14.6 problems 25, 28, 29; 14.R problems 47, 48)
• Find all critical points of a function and determine their type (apply second derivative test) (14.7; 14.R problems 53, 54)
• Solve a constraint optimization problem using Lagrange's method (one or two constraints) (14.8; 14.R problems 59, 62)
• Use the geometric definition of integral and symmetry to evaluate double and triple integrals without computation (15.1 problems 11-14, 15.7 problems 37, 38; know formulas for the areas of circles, trapezoids, volumes of cylinders, spheres, and cones)
• Find the volume of a 3D region using double integral; use polar coordinates if appropriate (15.3,4; 15.R problems 29, 31, 32)
• Compute a double integral, change order of integration, sketch a 2D region of integration (15.3; 15.R problem 11, 13, 17, 19)
• Compute a line integral using parametric representation of a curve (16.2; 16.R problem 5)
• Determine if a vector field is conservative, find a potential function, use it to compute a line integral (16.3; 16.R problems 11, 13)
• Use Green's theorem to compute a line integral or area of a plane region (use one of the formulas for the area) (16.4; 16.R problem 17)

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 guaranteed to be on the exam.

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

Review problems per chapter

Chapter 12, Review, page 859: 5-7, 9-11, 18-23; 29-37 (odd); 12.4: 43, 47, 48

Chapter 13, Review, page 898: 1, 10, 11; 13.1: 40-44; 13.3:17, 48

Chapter 14, Review, page 992: 47, 48, 53, 54, 59, 62; 14.6: 25, 28, 29

Chapter 15, Review, page 1074: 11, 13, 17, 19, 21, 29, 31, 33; 15.1 problems 11-14

Chapter 16, Review, page 1161: 5, 11, 13, 15, 17; 16.4: 19.

Good luck on the exam!