CK12 Calculus - Single Variable
10. Answers and Differentiationand Integrationof Logarithmicand ExponentialFunctions LearningObj ...
We wantto applythis definitionto get the derivativeto our logarithmicfunction Usingthe defi- nitionof the derivativeand the rule ...
Thuswe conclude and in the special case where To generalize,if is a differentiablefunctionof and if then the abovetwo equations, ...
Example2: Find. Solution: Example3: Find Solution: Herewe use the ChainRule: Example4: Find the derivativeof Solution: Herewe us ...
Find the derivativeof Solution: We use the QuotientRule and the naturallogarithmrule: IntegralsInvolvingNaturalLogarithmicFuncti ...
The formalway of solvingsuchintegralsis to use -substitutionby letting equalthe denominator. Here, let and Substituting, Remark: ...
Derivativesof ExponentialFunctions We havediscussedabovethat the exponentialfunctionis simplythe inversefunctionof the logarithm ...
Example9: Find the derivativeof. Solution: Applyingthe rule for differentiatingan exponentialfunction, Example10: Find the deriv ...
IntegralsInvolvingExponentialFunctions Associatedwith the exponentialderivativesin the box aboveare the two correspondingintegra ...
Find of Find of Find of Find of Find of Evaluate Evaluate Evaluate Evaluate Evaluate Evaluate Answers ...
7. 8. 9. 10. 11. 12. ExponentialGrowthand Decay LearningObjectives A studentwill be able to: Applythe laws of exponentialand lo ...
givesus y So the solutionto the equation has the form The box belowsummarizesthe detailsof this function. The Law of Exponential ...
The equationthat is usedfor radioactivedecayis We wantto find an expressionfor the half-life of an isotope.Sincehalf-lifeis defi ...
This is a famousexpressionin physicsfor measuringthe half-lifeof a substanceif the decayconstant is known.It can also be usedto ...
Fromreadingthe first sentencein the problem,we learnthat the bacteriais increasingexponentially. Therefore,the exponentialgrowth ...
This tells us that after about hours(around minutes)the populationof the bacteriawill doublein number. CompoundInterest Investor ...
OtherExponentialModelsand Examples Not all exponentialgrowthsand decaysare modeledin the naturalbase or by Actually, in everyday ...
Accordingto the model, studentswill becomeinfectedwith the flu virus.Assumefurtherthat the re- searcherwantsto knowhow long it w ...
Otherapplicationsare introducedin the exercises. ReviewQuestions In 1990,the populationof the USAwas million.Assumethat the ann ...
b. Whatis the soundlevel (in ) of a soundwhoseintensityis? Referringto problem#6, a singlemosquito metersawayfrom a personmake ...
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