CK12 Calculus - Single Variable
Thereare no maximumsor minimums;no relativemaximumsor minimums. The SecondDerivativeTest LearningObjectives A studentwill be a ...
Example1: Considerthe function : The functionhas zerosat and has a relativemaximumat and a relativeminimum at. Notethat the grap ...
Supposethat is continuouson and that is someopenintervalin the domainof If for all then the graphof is concaveupwardon 2.If fo ...
Example3: Let’s examinethe function Find the criticalvaluesfor which or at Notethat when Applythe First and SecondDerivativeT ...
Consider with a. Determine and so that is a criticalvalueof the function b. Is the point a maximum,a minimumor neither? In pro ...
Relativemaximumat , locatedat ; relativeminimumat , located at. Thereare no inflectionpoints. Relativemaximumat relativeminimum ...
Relativemaximumsat relativeminimumat ; the relativemaximumsare locatedat and ; the relativeminimumis locatedat Thereare two inf ...
9. on ...
Also, Limitsat Infinity LearningObjectives A studentwill be able to: Examineend behaviorof functionson infiniteintervals. Deter ...
Examineinfinitelimitsat infinity. Introduction In this lessonwe will returnto the topicsof infinitelimitsand end behaviorof fu ...
Let’s examineboth the graphand valuesof the functionfor appropriate values,to see if they cluster aroundparticular values.Hereis ...
So we infer that. For the infinitelimit, , the inferenceof the limit is not as obvious.The functionappearsto approachthe value b ...
Likewise, Now let’s look at somemoreexamples. Example3: Evaluate Solution: Since , L’Hospital’s Rule appliesand we have Let’s lo ...
Herewe observethat we still havethe indeterminateform. So we applyL’Hospital’s Rule againto find the limit as follows: L'Hospita ...
10. Answers Hint:Let ,so Analyzingthe Graphof a Function LearningObjectives A studentwill be a ...
Lessonon Relationsand Functions,particularlyrational,polynomial,radical,and trigonometricfunctions. Beforewe beginour workon the ...
. Hencethe functionis differentiableat everypoint of its domain, and since on its domain,then is decreasingon its domain, . . in ...
Example2: AnalyzingPolynomialFunctions Considerthe function GeneralProperties The domainof is and the -interceptat The functionc ...
on and ; hencethe functionis increasingin these intervals. Similarly, on and thus is decreasingthere. if wherethereis an inflect ...
relative minimum at and located at the point concaveup in. Concavity concavedownin. , locatedat the point Inflectionpoints Herei ...
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