MATH 250, Study Guide for Midterm Test 1

10/01/2012

Test 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)

Key concepts (review from your lecture notes or the textbook)

• Points, vectors, analytic geometry view of the three-dimensional space
• Tools of three-dimensional analytic geometry: operations with vectors. Geometric and algebraic views
• Basic vector operations: addition, subtraction, multiplication by scalar
• The standard unit vectors i, j, k; component and (i,j,k)-notation for vectors
• Length (magnitude), the dot product, angles, orthogonal vectors, components and projections
• Determinants, orientation, cross products, triple scalar products and volumes
• Lines and planes
• Vector, parametric, and symmetric equations for lines
• Planes, distance from a point to a plane
• Types of quadric surfaces: know the names, the meanings of parameters, and be able to sketch, using the method of sections
• Cylinders
• Ellipsoids
• Elliptic hyperboloids (in one and two sheets)
• Cones
• Elliptic and hyperbolic paraboloids
• Vector functions and parametric curves
• Continuity, limits, derivatives and integrals of vector functions
• Rules of differentiation for vector functions: dot and cross products, the chain rule
• Formula for the arc length
• Geometry of curves. The moving trihedral ((T,N,B)-frame)
• Arc length parameterization
• Tangent vectors, the unit tangent vector T
• Curvature, curvature radius, curvature vector
• Principal normal N, binormal B, osculating plane, osculating circle

Here's a list of possible types of final exam questions (the list is not all-inclusive)

• Use the distance formula to answer questions about geometry (12.1, problems 7-10)
• Play around with equations of spheres (12.1, problems 21, 22, 41; 12.R, problem 1)
• Vector operations in geometric and algebraic representations (12.2, problems 7, 8, 26, 29; 12.3, problems 11, 12, 23)
• Use properties of the dot or cross product (12.3, problems 63, 64; 12.4, problems 47, 48, 12.R, problems 5-7)
• Use the dot product to find angles, between vectors, planes, intersecting curves (12.3, problems 19, 33, 55, 56; 12.5, problems 51, 53; 12.R, problem 9, 13.R, problem 9)
• 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 problems 13-19 (odd), 29, 31; 12.R: 29-37 (odd))
• Parametrize a curve given as intersection of surfaces. Use surfaces to help sketch a space curve (13.1, problems 27, 28, 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)
• Compute a derivative/integral of a vector function, find a (unit) tangent vector; use rules of limits/derivatives (13.2 problems 17, 27, 37, 39, 53, 54; 13.R problems 2, 5, 11)
• 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)
• Use the arc length formula to compute the length of a curve (13.3 problems 1-6; 13.R problem 8)

Practice tests (for 75 minutes):

Test 1: 12.1: 20, 12.R: 10, 12.5: 68, 12.6: 33, 45, 13.1: 41, 13.2: 27, 49, 13.3: 4, 20

Test 2: 12.1: 9(a), 12.3: 64, 12.4: 43, 12.R: 11, 12.6: 16, 48, 13.1: 14, 13.2: 25, 13.3: 32, 50

Good luck on the exam!