MATH 250, Study Guide for Midterm Test 1

09/30/2015

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)
   4. Normal and tangential components of acceleration. Motion in space (13.4)

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

• Points, vectors, analytic geometry view of the three-dimensional space
• Operations with vectors: addition, subtraction, multiplication by scalar
• i,j,k and component notation for vectors
• Parallel vectors
• The dot product and its uses:
  • length; unit vectors
  • angle; orthogonal vectors
  • components and projections
• The cross product: definition, properties: know with proofs
• The triple scalar product; 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 (one and two sheets)
• Cones
• Elliptic and hyperbolic paraboloids
• Space curve; parametrization (vector function)
• 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
• Arc-length parametrization
• (T,N,B)-frame; curvature, osculating and normal planes
• The normal and tangential components of acceleration

Old material to review:

• Elementary functions and their graphs: 1.2, 6.2, 6.3; Curve sketching techniques: 3.5 -- used for plotting curves and surfaces (Sections 12.6, 13.1-4). You need to know how to plot graphs of elementary functions without point-plotting! Example: y=x^(2/3) is a power function described in Section 1.2; from Section 3.5 and the knowledge of its derivative we know that its derivative becomes infinite (±∞) at zero, thus the origin is a point with a cusp, not a "corner". The function is increasing for (x>0), decreasing for (x<0), and concave down on both of these intervals. The function value at ±1 is 1. Your graph should reflect all this information.
• Conic sections: 10.5, particularly, finding vertices for all types of conic section curves, and asymptotes for hyperbolas -- used for identifying and plotting quadric surfaces (Section 12.6)

Examples of possible test questions:

• Spheres; the distance formula (12.1, problems 19, 22, 40, 41; 12.R, problem 1)
• Prove properties of the dot and/or cross product (12.3, problems 63, 64; 12.4, problems 20-26, 47, 48; 12.R problem 7)
• Use the dot product to find angles; check that vectors are orthogonal by means of the dot product; check that vectors are parallel by seeing whether they are scalar multiples of each other (12.3, problems 19, 33, 55, 56; 12.5, problems 13, 14, 51, 53, 67, 68; 12.R, problem 9)
• Reduce the equation of a quadric surface to one of the standard forms, identify the major axis, find sections (traces) 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 (homework for 12.5; 12.R problems 21-27)
• Differentiate vector function (find the velocity vector), integrate a vector function (find the position vector); Prove one of the properties in Theorem 3; use these properties to compute derivatives (13.2 problems 43-46, 49, 50, 53, 54; homework for 13.4; 13.R problems 2, 11, 17)
• Find the curvature of a space curve, determine the vectors from the (T,N,B)-frame (homework for 13.3)
• Arc length, arc length parametrization (homework for 13.3)
• Find normal and tangential components of acceleration (homework for 13.4)

Good luck on the exam!