CK12 Calculus - Single Variable

Anisocline(for constantk) is the line alongwhichthe solutioncurveshavethe samegradient(k). By calcu- latingthis gradientfor each ...

Exercise Sketchthe slopefield of the differentialequation. Sketchthe solutioncurvesbasedon it. Sketchthe slopefield of the diff ...

Example 2 Solvethe differentialequation. Solution. We have and a substitution gives . Exercise Solvethe differentialequation wi ...

Resolvingthe partialfraction giveslinearequationsA+B= 2 andA-B= 0. So .Integratingbothsides,wehave or with. Then , i.e. whereD&g ...

Thenbackin the first equation, 10000 =P 0 (2). SoP 0 = 5000.Thereare 5000initialimmigrants. Example 2 (Logistic Growth) The popu ...

The Runge-Kuttamethodsare an importantfamilyof implicitand explicititerativemethodsfor the approxi- mationof solutionsof our ODE ...

8. InfiniteSeries............................................................................................................... ...

We can generatethe termsfor this rule as follows: n 1 2 3 4 ... ... Example 2 Considerthe sequencerule. The termsof the sequence ...

Solution We can graphthe correspondingfunction forn= 1, 2, 3,.... The graphof is similarto the con- tinuousfunction for the doma ...

Recallthat |an-L| <εmeansthe valuesofansuchthatL-ε<an<L+ε. Whatdoesthe definitionof the limit of a sequencemean?Hereis ...

Considerthe sequence{n+ 1} in Figure4. Asngets largerand goesto infinity, the termsofan=n+ 1 becomelargerand larger. The sequenc ...

TheoremIf a sequenceis convergent,then its limit is unique. Keepin mindthat beingdivergentis not the sameas not havinga limit. L ...

Let’s applytheserulesto help us find limits. Example 11 Find. Solution We coulduse L’Hôpital’s rule or we coulduse someof the ru ...

You can see how the nameof the theoremmakessensefromthe statement.Aftera certainpointin the sequences,the termsof a sequencecnar ...

Sandwich/SqueezeTheorem,. Picard’s Method The followingmethodappearedin 1891by EmilePicard,a famousFrenchmathematician.It is a m ...

Let be sequencedefinedsuccessivelyby, for The sequenceof approximationsconvergesto the solution , i.e. Now that we havedefinedPi ...

Step 3 – Substitute fory(t) in the integrandas above: Step 4 – Substitute fory(t) in the integrandas donepreviously: Thus,the in ...

Clearlythis solutionsatisfiesy(x) = xy(x) and y(-1) = 1. ReviewQuestions Find the rule for the sequencean. n 1234 ... an= ?-2 ...

y(x) = 1 + y^2 with y(0) = 0. Find the first threefunctions definedby Picard’s methodfor the solutionto the initialvalue proble ...

Levelsof Difficulty Beginning Beginning Beginning Beginning Beginning Intermediate Intermediate Intermediate Challenging Challe ...