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# Unit1.pdf

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School
Department
Mathematics
Course
Mathematics 1225A/B
Professor
Vicki Olds
Semester
Winter

Description
Math1225ABUnit1ExponentialandLogarithmicFunctionstextreferenceSections51and52mixedtogethercustomtextpgs319cVOlds2012Unit111ExponentialandLogarithmicFunctionsWestartthecoursewithareviewofexponentialandlogarithmicfunctionsandtheirpropertiesTheninthenextunitwewillinthecontextofthesekindsoffunctionsreviewwhatyoulearnedinIntroductoryCalculusthevariousdierentiationrulesaswellasusingcalculusforgraphingandalsotheconceptsofimplicitandlogarithmicdierentiationExponentialFunctionsxDenition11Thefunctionfxbwithb0andb1iscalledtheexponentialfunctionwithbasebThatiseverypositivenumberotherthan1isthebaseofanexponentialfunctionandthevariableinthefunctionistheexponentiethepowertowhichthebaseisraisedRecallthatthedomainofafunctionfxisthesetofallxvaluesforwhichthatfunctionisxdenedForanyb0withb1thefunctionfxbisdenedeverywhereieforallrealvaluesofxThereforethedomainofanyexponentialfunctionisthesetofallrealnumbersietheintervalSimilarlyrecallthattherangeofafunctionfxisthesetofallvalueswhichthefunctionhasanywhereinitsdomainThatisforyfxtherangeisthesetofyvalueswhichcanoccurForanyexponentialfunctionwhenapositivebaseisraisedtoanypowernomatterwhetherpositiveornegativeintegerorotherwisetheresultisalwaysapositivenumberThatiswhenb0andxb1thevalueofbisalwayspositivenomatterwhatthevalueofxisAndanypositivevaluecanbeobtainedsotherangeofanyexponentialfunctionisthesetofallpositiverealnumbersietheinterval0GraphsofExponentialFunctionsxForanyb1thegraphofyblookslikethegraphinFigure1belowNoticethatthisgraphisincreasingthroughoutitsdomainisconcaveupwardeverywhereandpassesthroughthepoint01xxAlsonoticethattheliney0isahorizontalasymptotewithlimb0whilelimbxxFigure1Anexponentialfunctionwithbaseb1
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