CK12 Calculus - Single Variable
3. 4. providedthat With thesepropertieswe can evaluatea widerangeof polynomialand radicalfunctions.Recallingour ex- ampleabove, ...
In makingthis simplification,we are indicatingthat the originalfunctioncan be viewedas a linearfunction for x valuescloseto but ...
Consider. Find. We see that and note that property#5 doeshold.Henceby directsubstitutionwe have Example5: Consider Thenwe haveth ...
Fromthe graphwe note that: The functionis boundedby the graphsof and 2.. Hencethe SqueezeTheoremappliesand we concludethat Les ...
7. Consider. We found Find Considerfunction suchthat for Use the SqueezeTheoremto find. Use the SqueezeTheoremto showthat ...
Continuity LearningObjectives A studentwill be able to: Learnto examinecontinuityof functions. Find one-sidedlimits. Understand ...
But whatwe actuallyhaveis the originalfunction, that we knowis not defined at At we havea hole in the graph,or a discontinuityof ...
In this examplewe have exists,x= 1 is in the domainoff(x), but. One-SidedLimitsand ClosedIntervals Let’s recallour basicsquarero ...
We see that and also that. Propertiesof ContinuousFunctions Let’s recallour exampleof the limit of compositefunctions: We saw th ...
We will concludethis lessonwith a theoremthat will enableus to solvemanypracticalproblemssuchas findingzerosof functionsand root ...
Of coursewe couldzoomin on the graphto see that the lowestpointon the graphlies withinthe fourth quadrant,but let’s use theCALCV ...
, in the interval Statewhetherthe indicatedx-valuescorrespondto maximumor minimumvaluesof the functiondepicted below. Provethe ...
reals,Shas an upperbound,c. 3) Thereare then threepossibilitiesto explore:f(c) <d,f(c) =d, orf(c) >d. Exploretheseand show ...
and. Moreformally, we definetheseas follows: Definition:The right-handlimit of the functionf(x) atx=ais infinite,and we write , ...
We observe that the limits f(x) and both exist. In particular, and . HenceProperty#5 now appliesand we have . LessonSummary We ...
9. 10. Answers Zero atx= -4; verticalasymptotesatx= 3, 5; Zero atx= 1; no verticalasymptotes; as ; ...
2. Derivatives.................................................................................................................. ...
Example1: Find the slopeof the tangentline to the curvef(x) =x^3 passingthroughpointP(2, 8). Solution: SinceP(x 0 ,y 0 ) = (2, 8 ...
Since then To find the slope,we simplysubstitute into the result , and Thusslopesof the tangentlines at and are and respectively ...
Substituting , Thusthe slopeof the tangentline at for the curve is To find the equationof the tangentline, we simplyuse the poin ...
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