CK12 Calculus - Single Variable
We call this theTaylorPolynomialof fcenteredat a. For our discussion,we will focuson the quadraticcase.The TaylorPolynomialcorre ...
6.052.0125 2.0124 2.0124 As you can see from the graphbelow, is an excellentapproximationof near We get a slightlybetterapproxim ...
Verify the the followinglinearapproximationat Determinethe valuesof for whichthe linear approximationis accurateto Find the ...
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4. Integration.................................................................................................................. ...
With our definitionand initialexample,we now look to formalizethe definitionand developsomeusefulrules for computationalpurposes ...
The rule holdsfor Whathappensin the casewherewe havea powerfunctionto integratewith say . We can see that the rule doesnot works ...
Example2: Computethe followingindefiniteintegral. Solution: Usingour ruleswe have Sometimesour rulesneedto be modifiedslightlydu ...
This latterequationis calledadifferentialequation. This characterizationof the basicsituationfor which integrationappliesgivesri ...
6. 7. Solvethe differentialequation Find the antiderivative of the function that satisfies Evaluatethe indefiniteintegral (Hint ...
The InitialValue Problem LearningObjectives Find generalsolutionsof differentialequations Use initialconditionsto find particul ...
We can re-statethe problemin termsof a differentialequationthat satisfiesan initialcondition. with By integratingthe right side ...
and and Supposethe graphof includesthe point and that the slopeof the tangentline to at is Find In problems#9–10,find the func ...
The AreaProblem LearningObjectives Use sigmanotationto evaluatesumsof rectangularareas Find limitsof upperand lowersums Use the ...
We then summedthe areasof the rectanglesas follows: and We call this theuppersumsinceit is basedon takingthe maximumvalueof the ...
We can use the notationto indicateusefulformulasthat we will haveoccasionto use. For example,you may recallthat the sum of the f ...
We can then definethe lowerand uppersums,respectively, over partition , by where is the minimumvalueof in the intervalof length ...
We can re-writethis resultas: We observethat as We now are able to definethe area undera curveas a limit. Definition:Let be a co ...
Solution: If we partitionthe interval into equalsub-intervals,then each sub-intervalwill have length and height as variesfrom to ...
We usedthe limit definitionof area to solveproblems. ReviewQuestions In problems#1–2, find the summations. 1. 2. In problems#3 ...
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