MATH 250, Study Guide for Midterm Test 1
6/11/2012
Test coverage
- Analytic Geometry in Three-dimensional Space (Chapter 12)
- Three-dimensional Coordinate Systems (12.1)
- Vectors in Two and Three Dimensions (12.2)
- The Dot Product (12.3)
- The Cross and Triple Products (12.4)
- Lines and Planes (12.5)
- Cylinders and Quadric Surfaces (12.6)
- Vector Functions and Geometry of Curves (Chapter 13)
- Vector Functions and Space Curves (13.1)
- Derivatives and Integrals of Vector Functions (13.2)
- 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
- Applications: velocity vectors, forces in static equilibrium, work
- 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
Basic types of questions (the list is not all-inclusive, but if you know
all these you should be able to figure out the rest)
- Apply basic concepts of analytic geometry: points, vectors, lengths, triangles
- Find an equation of a sphere; work with an equation to find radius and center (complete the squares)
- Operations on vectors: geometric and algebraic representations (picture problems and algebra problems)
- Given a vector, find a unit vector in the same direction (normalize to unit length)
- Find angles, components, projections
- Compute cross-products, use the determinant formula
- Find volumes using triple scalar product
- Equations of lines and planes satisfying various conditions
- Points of intersection of lines with planes/surfaces
- Work with a quadric surface equation to bring to the standard form and sketch the surface
- Parametrize a curve given as intersection of surfaces
- Use surfaces to help sketch a space curve
- Compute a derivative of a vector function, use rules of differentialtion, find a tangent vector to a curve
- Compute an integral of a vector function
- Compute the length of a curve
- Parameterize a curve by its arc length
- Find the curvature of a space curve, determine the vectors from
the (T,N,B)-frame, find an osculating plane
Homework problems are the main resource, they cover all topics and should provide enough practice
if you do them all.
Practice tests (for 60 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
(12.R is Chapter 12, Review Problems.)
Solutions to most even problems are available on the main page (see under homework solutions).
Good luck on the exam!