CK12 Calculus - Single Variable
We are familiarwith the formulasfor Perimeterand Area: Supposewe knowthat at an instantof time,the lengthis changingat the rate ...
We havea watertank shapedas an invertedright circularcone.Supposethat waterflowsinto the tank at the rate of At whatrate is the ...
Now we can solvethe problemby substituting into the originalequation: Hence and by substitution, LessonSummary We learnedto sol ...
Supposethat the lengthof a rectangleis increasingat the rate of ft/minand the widthis increasingat a rate of ft/min.At whatrate ...
Usingthe followingdiagram, Usingthe followingdiagram, 7. The demandis increasingat a rate of per thousandunits,or 250 unitspe ...
9. a. b. Extremaand the MeanValue Theorem LearningObjectives A studentwill be able to: Solveproblemsthat involveextrema. Stud ...
Min-MaxTheorem:If a function is continuousin a closedinterval then f(x) has both a maximum valueand a minimumvaluein In orderto ...
Let’s observethat the converseof the last theoremis not necessarilytrue: If we consider and its graph,then we see that while at ...
Notethat Since is continuousin and existsin then Rolle’s Theorem applies.Hencethereexists in with So for every in In particular, ...
Sincewe needto have in the interval the positiveroot is the solution, . LessonSummary We learnedto solveproblemsthat involveext ...
In problems#4–6,find the extremaand sketchthe graph. Verify Rolle’s Theoremby findingvaluesof for which and Verify ...
Absolutemaximumat absoluteminimumat relativeminimumat Note:thereis no relativeminimumat becausethereis no openintervalaround si ...
Absolutemaximumat absoluteminimumat Absoluteminimumat at (by Rolle’s Theorem,thereis a criticalvaluein each of the intervals an ...
The FirstDerivativeTest LearningObjectives A studentwill be able to: Find intervalswherea functionis increasingand decreasing. ...
The functionindicatedhere is strictlyincreasingon and and strictlydecreasingon and We can now statethe theoremsthat relatederiva ...
Notefirst that we havea relativemaximumat and a relativeminimumat The slopesof the tangentlineschangefrom positivefor to negativ ...
We observethat the slopesof the tangentlines to the graphchangefrom negativeto positiveat The first derivativetest verifiesthis ...
2. Give the sign of the followingquantitiesfor the graphin #2. a. b. c. d. For problems#4–6,determinethe intervalsin whichthe ...
10. Answers Increasingon decreasingon constanton Increasingon and decreasingon Relative minimum at ; increasing on and , ...
Absolutemaximumat ; increasingon decreasingon Relativemaximumat ; relativeminimumat ; increasingon and decreasingon Relative ma ...
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