Math 444HH
Cumulative Exam
Topics List
1) POLYNOMIALS
A. The technique of synthetic division
B. The remainder theorem
C. The factor theorem
D. Rational Root Theorem
E. Descartes Rule of Signs
F. Upper and lower bounds for roots
G. Graphing polynomial functions
H. The Fundamental Theorem of Algebra
I. Solving Inequalities
(i) polynomial inequalities
(ii) rational inequalities
(iii) absolute value inequalities (graphically)
2) COMPLEX NUMBERS AND TRIGONOMETRY
A. Trigonometric inequalities (e.g. cos2ϑ + cosϑ +1> 0, -2π≤ ϑ ≤ 2π )
B. Argand diagrams and conversion formulas
C. DeMoivre’s theorem and applications to powers and roots
D. Applications of sets of complex numbers to the structure of a GROUP
E. Applications of Demoivre’s theorem to rotations in the plane
3) ANALYTIC GEOMETRY INCLUDING CONIC SECTIONS
A. Formula for the distance from a point to a line.
B. Relationship between slope and the angle of inclination of a line.
C. Formula for the angle between two intersecting lines.
D. Ellipse
E. Parabola
F. Hyperbola
G. Rotated conics of the type ax2 +bxy+cy2+dx+ey+f = 0, b≠0
H. The discriminant
I. TI ready form for a conic section.
4) FUNCTIONS
A. Challenging domain/range problems including the
trigonometric and inverse trigonometric functions, and restricting the domain of
functions so that the inverse will be a function.
B. Problems involving restrictions on domain/range when composing functions,
including the int(x), sgn(x), and piece-wise defined functions
C. Odd and even functions-- properties
D. Sets of functions which do/don’t form a GROUP under the binary operations
of composition and addition
E. Mathematical modeling, such as the setting up of min/max problems which can be
solved using the TI.
5) GRAPHING TECHNIQUES, RATIONAL FUNCTIONS AND LIMITS
A. Transformations and graphing functions
B. Rational functions
(I) x and y intercepts
(II) Vertical, horizontal, and slant asymptotes
(III) Point and infinite discontinuities
(IV) rational functions which cross their horizontal asymptote
C. Algebraic techniques for evaluating limits
D. Formal definition of continuity
(I) Types of discontinuity, also classifying as removable or non-removable
(II) Redefine a function to remove the discontinuity
lim
E.
6)
θ →0
sin(θ )
θ
=1
and limit problems which require using that result.
INTRODUCTION TO THE DERIVATIVE
A. The derivative as the slope of the tangent line to a curve
B. Contrast between the average rate of change and the instantaneous rate of change
for a given function over a given interval including the alternate definitions for
derivative.
C. Relationship between the graph of a function and the graph of it’s derivative;
using the TI to graph derivatives of known functions whose derivatives are unknown
D. Apply power rule for f(x) = xn.
E. Applications of the results for the derivative of Sin(x) and Cos(x).
7) SEQUENCES AND SERIES
A. Definition
B. Arithmetic and Geometric (formulas for the nth term and sum of the first n
terms)
D. Recursively defined sequences other than the Fibonacci sequence, including
the recursive forms for arithmetic and geometric sequences
E. Infinite series-- Precise definition for convergence of an infinite series in terms
of its sequence of partial sums --- geometric, harmonic.
F. The comparison test using geometric and harmonic series.
G. Functions which are expressed as infinite series--interval of convergence, polynomial
approximator, error function
H. BASIC program