1. Algebra1.1. Quadratic forms of a single variable1.2. Partial fraction decomposition1.3. Exponential and trigonometric functions2. Limits2.1. Continuity2.2. Squeeze theorem3. Differential Calculus3.1. The product rule3.2. The chain rule3.3. L'Hôpital's rule4. Integral Calculus4.1. Fun integrations/anti–derivatives4.2. Integration by parts4.3. Integration by substitution4.4. The fundamental theorem of calculus with chain–rule5. Power–Series5.1. Taylor/Maclaurin series for common functions5.2. Complex exponential functions5.3. Hyperbolic trigonometric functionsMATHEMATICS REVIEW SHEETSCOTT A. STRONGBender: Hm. I forgot you could tempt me with things I want. Well, I suppose I’ve always wondered what it would be like to be more annoying. [Futurama – The Devil’sHand are Idle Play Things (2003)]ABSTRACT. In 2013, after a near decade hiatus from teaching calculus to first–year students, I returned from an instructional schedule that wasalmost completely consumed by teaching a growing student body topics from Fourier analysis, differential equations and linear algebra to onethat emphasized multi–variate calculus. This document contains a list of topics with problems that I would expect someone who has stud-ied 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 ownthoughts on the topics.Please report any errors [email protected]. ALGEBRA1.1. Quadratic forms of a single variable. The study of quadratic forms1is a deep mathematical topic and speaks to the fact thatnonlinearities significantly impede mathematical analysis. With that in mind we review properties of the simplest of all quadraticforms.1. Prove that the equation f (x) =ax2+bx +c has roots given by x±=−b ±pb2−4ac2a.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.2While differential cal-culus applied to such rational functions is straightforward, integral calculus does not feel the same. This is because rational functionsmust be reduced to other algebraically equivalent forms so that the results of integral calculus interface nicely. The fundamentaltheorem of algebra says that for a reciprocated polynomial of degree N we havef (x) =1NXn=0anxn=a−1NNYn=1x −xn(2)where xnis the nth−root of the polynomial [f (x)]−1. Partial fractions tells us that there exist αnsuch thata−1NNYn=1x −xn=NXn=1αnx −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 thecoefficients αnwe must find a common denominator to relate both sides of the previous equality. This is the procedural essence ofconstructing partial fraction decompositions. It is important to note here that some of the roots xnmay be complex numbers, whichmeans that the coefficients αnmay be complex. In this case one may decompose the polynomial up to linear terms multiplyingquadratic polynomials. While this makes the integration more difficult, it is often desirable when one wants to work in only the realnumber system. The following problems provide some practice on the procedure of partial fractions.1. Using partial fractions, show that1x2+2x −3=14µ1x −1−1x +3¶.2. Using partial fractions and synthetic division, show thatx3+16x3−4x2+8x=1 +2µ1x+xx2−4x +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) =xTAx +btx +c.(1)A single–variable quadratic polynomial is a special case of this formula. If xT=£x y¤then we have a function whose roots are the conic section. If xT=£x y z¤thenwe 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 saythat 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: 3and 8 are both natural numbers, but 3 −8 is −5, which is not.12 SCOTT A. STRONG3. Using partial fractions and synthetic division, show thatx9−2x6+2x5−7x4+13x3−11x2+12x −4x7−3x6+5x5−7x4+7x3−5x2+3x −1= x2+3x + 4 +1(x −1)+1(x −1)3+x +1x2+1+1(x2+1)2.1.3. Exponential and trigonometric functions. If look ahead to section 5.2 of this document then you will find that the name ofthis subsection should really just be exponential functions. You will also revisit the following questions which are easily answeredusing the concept of power–series representations.31. Graph ex,sin(x),cos(x). Make sure to include both x and y intercepts.2. Explain why ea+b=eaeb.3. Prove that sin2(x) +cos2(x) =1 and using this show that sec2(x) −tan2(x) =1.4. Prove that sin(α ±β) =sinαcosβ ±cosαsin β.5. Prove that cos(α ±β) =cosαcosβ ∓sinαsin β.6. Show that sin(2x) =2sin(x)cos(x).7. Show that cos(2x) =cos2(x) −sin2(x) =2cos2(x) −1.2. LIMITSWhile both differential and integral calculus are useful/usable to/in applied science and engineering, they exist as a consequenceof the mathematical notion of a limit. For those that study mathematics the exact language of a limit is discussed. Through thislanguage we justify the existence of a limit. For most this is a topic best left alone in favor of developing trust in the concept ofcontinuity. Before we mention this property we note the following properties of limits,limx→p(f (x) +g (x)) = limx→pf (x) + limx→pg (x),(4)limx→p(f (x) −g (x)) = limx→pf (x) − limx→pg (x),(5)limx→p(f (x) ·g (x)) = limx→pf (x) · limx→pg (x),(6)limx→p(f (x)/g (x)) = limx→pf (x)/ limx→pg (x), provided that limx→pg (x) 6=0,(7)where f , g are functions such that all stated limits exist.2.1. Continuity. For a function, f , to be continuous at a point, x =x0∈R, we must havelimx→x0f (x) exists ,(8)limx→x0f (x) = f (x0).(9)Roughly speaking, a continuous function takes in “small” changes and produces “small” changes. Otherwise, a function is said to bea discontinuous function. When one says that a function is continuous, it is implied at all points in its domain. It is
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