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MATH137 Fall 2013

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MATH 137
Sean Speziale

Return to University Notes indexMATH 137Calculus ISection 003InstructorName Sean SpezaileEmail sspezailuwaterloocaOffice M3 2128Advising Hours Monday Wednesday at 12001330 MC 4023Office hours starts week 3 details TBA wip9913PreliminariesIntegers Z21012pRational numbers QpZqZqIrrational numbers cannot be expressed as ratio nonterminating and nonrepeating decimal expansions eln22Real number denotedcontains and irrationals any number with decimal expansion as opposed toRZQcompleximaginary numbersComplex number denotedcontains real and imaginary partsCIntervalsSetsinclusive closed interval fromto abableft exclusive left open interval fromtonot including ababaright exclusive right open interval fromtonot including ababb is an element ofand xabxabxRaxbInfinity can be used as an interval endpoint but is always excluded as it is a limit not a number 4Inequalitiesif and only if first implies second and second implies firstSolve 613x10613x10Isolate x53x953x953x31Solve 2x12x1When x012xx0 then x21When x012xx0 then x211If x0 and x then x221If x0 and x then x02Therefore x012Absolute ValueDistance between number and 0 on the real number linexif x0xxif x0 is the distance betweenandon the real number linexaxaxaif x0xaxif x0Solve by considering both cases Deal with each case separatelySolve 2x34Open the inequality 42x34fggfg1712x7x22Simplify xy1y1x1xif x0y1xif x011913FunctionsA function is a rule that assigns a single output to an inputx independent variableffx or y dependent variableA function maps values in the set known as a domain into values in the set known as a rangeDomain set of allowable values for the independent variableCommon exclusions from domainsDividing by 0Log of nonpositive numberEven root of negative numberA function has an inverse if and only if every unique value of the independent variable in the domain has aunique value of the dependent variableie onetooneRange set of possible values for the dependent variableFinding the range can be significantly more difficult than finding the domain which is simply a matter oflooking at the functionfxx1Restriction x1D1R01fx21xRestriction x0D1111R01A true function must have a unique output for each input Graphically the vertical line test passes for allpoints vertical line drawn on graph will only intersect it at most once if it is a graph of a true functionNonfunctions can be described using functions by combining multiple functions into one22Unit circle yxy2222This can be written as two functionsand yxyyxyHorizontal line test a function is onetoone if any horizontal line drawn on its graph intersects it at mostonceFunction examplesnPolynomialn is a positive integer domain real numbersxRationaldomain real numbers such that fxgxgx0TrigonometricInverse trigonometric
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