Mathematics Review Sheet

In 2013, after a near decade hiatus from teaching calculus to first–year students, I returned from an instructional schedule that was almost completely consumed by teaching a growing student body topics from Fourier analysis, differential equations and linear algebra to one that emphasized multi–variate calculus. This document contains a list of topics with problems that I would expect someone who has studied single–variable calculus with series and sequence to have either seen or be able to digest. The narrative around the associated problems, whose answers are nearly always given, is meant to give some derivations that might be missing in ones recall and quickly summarize my own thoughts on the topics. Please report any errors to [email protected]. 1. ALGEBRA 1.1. Quadratic forms of a single variable. The study of quadratic forms is a deep mathematical topic and speaks to the fact that nonlinearities significantly impede mathematical analysis. With that in mind we review properties of the simplest of all quadratic forms. 1. Prove that the equation f (x) = ax2 +bx + c has roots given by x± = −b ± p b2 −4ac 2a . 2. Plot f (x) = ax2 +bx + c assuming that a,b,c ∈R. 1.2. Partial fraction decomposition. Polynomials functions are not closed under the operation of division.2 While differential calculus applied to such rational functions is straightforward, integral calculus does not feel the same. This is because rational functions must be reduced to other algebraically equivalent forms so that the results of integral calculus interface nicely. The fundamental theorem of algebra says that for a reciprocated polynomial of degree N we have f (x) = 1 N ∑ n=0 an x n = a −1 N N ∏ n=1 x −xn (2) where xn is the nth−root of the polynomial [ f (x)]−1. Partial fractions tells us that there exist αn such that a−1 N N ∏ n=1 x −xn = N ∑ n=1 αn x −xn (3) which is useful for the purposes of integration since each term in this sum is integrable through the use of logarithms. To find the coefficients αn we must find a common denominator to relate both sides of the previous equality. This is the procedural essence of constructing partial fraction decompositions. It is important to note here that some of the roots xn may be complex numbers, which means that the coefficients αn may be complex. In this case one may decompose the polynomial up to linear terms multiplying quadratic polynomials. While this makes the integration more difficult, it is often desirable when one wants to work in only the real number system. The following problems provide some practice on the procedure of partial fractions. 1. Using partial fractions, show that 1 x2 +2x −3 = 1 4 ( 1 x −1 − 1 x +3 ) . 2. Using partial fractions and synthetic division, show that x3 +16 x3 −4x2 +8x = 1+2 ( 1 x + x x2 −4x +8 ) . Last compiled on 18/08/2014 at 00:43. For historical versions see this site. 1A quadratic form is a scalar valued function of a vector valued argument which has the matrix representation, f (x) = x Ax+bt x+ c. (1) A single–variable quadratic polynomial is a special case of this formula. If x = [x y] then we have a function whose roots are the conic section. If x = [x y z] then we have a function whose roots are quadric surfaces. 2A set has closure under an operation if performance of that operation on members of the set always produces a member of the same set; in this case we also say that the set is closed under the operation. For example, the real numbers are closed under subtraction, but the natural numbers (non-negative integers) are not: 3 and 8 are both natural numbers, but 3−8 is −5, which is not. 1