JEE CONCEPTS BOOSTER CONTINUITY AND DIFFERENTIABILITY ... · CONCEPT FOR JEE || Chapter CONTINUITY...
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JEE CONCEPTS BOOSTER CONTINUITY AND DIFFERENTIABILITY Download Doubtnut Today Ques No. Question 1 CONCEPT FOR JEE || Chapter CONTINUITY AND DIFFERENTIABILITY 1. INTRODUCTION 1. Revision of limits Click to LEARN this concept/topic on Doubtnut 2 CONCEPT FOR JEE || Chapter CONTINUITY AND DIFFERENTIABILITY 2. CONTINUITY 1. What is continuity what do we mean by it? Click to LEARN this concept/topic on Doubtnut 3 CONCEPT FOR JEE || Chapter CONTINUITY AND DIFFERENTIABILITY 2. CONTINUITY 2. Continuous graphs or non continuous graphs Click to LEARN this concept/topic on Doubtnut
Transcript of JEE CONCEPTS BOOSTER CONTINUITY AND DIFFERENTIABILITY ... · CONCEPT FOR JEE || Chapter CONTINUITY...
JEECONCEPTSBOOSTER
CONTINUITYANDDIFFERENTIABILITY
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QuesNo. Question
1
CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
1.INTRODUCTION
1.Revisionoflimits
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2
CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
2.CONTINUITY
1.Whatiscontinuitywhatdowemeanbyit?
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3
CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
2.CONTINUITY
2.Continuousgraphsornoncontinuousgraphs
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
2.CONTINUITY
3.DefinitionConditionforcontinuity
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
3.TYPESOFDISCONTINUITY
1.TypeofDiscontinuity-removableorirremovable
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
3.TYPESOFDISCONTINUITY
2.Findthepointsofdiscontinuityoff(x) =1
2sinx - 1
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
3.TYPESOFDISCONTINUITY
3.TypeofDiscontinuity
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
3.TYPESOFDISCONTINUITY
4.Numberofpointofdiscontinitypointininterval
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
3.TYPESOFDISCONTINUITY
5.Discussthedifferentiabilityoff(x) = arcsin2x
1 + x2
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
4.ALGEBRAOFCONTINUOUSFUNCTION
1. Let f and g be two real functions; continuous at x = a Then f + g and f - g iscontinuousatx = a
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
4.ALGEBRAOFCONTINUOUSFUNCTION
2.Letfandgbetworealfunctions;continuousatx = aThenfgandfgiacontinousat
x = a;providedg(a) ≠ 0
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
4.ALGEBRAOFCONTINUOUSFUNCTION
3.Letfandgbetworealfunctions;continuousatx = aThenαfand1fiscontinuousat
x = aprovidedf(a) ≠ 0andαisarealnumber
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
4.ALGEBRAOFCONTINUOUSFUNCTION
4.Letfandgbetworealfunctions;suchthatfogisdefined.Ifgiscontinuousatx = aandfiscontinuousatg(a);showthatfogiscontinuousatx = a
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
5.CONTINUITYOFFUNCTION
1.Continuityoffunctionwhichhavedifferentdefinitionsindifferentinterval
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
5.CONTINUITYOFFUNCTION
2.Showthatthefunctionf(x) = 2x - |x|iscontinuousatx = 0
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
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5.CONTINUITYOFFUNCTION
3.Continuityofcompositefunctionatapoint
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
5.CONTINUITYOFFUNCTION
4.Continuityinopenintervalandclosedinterval
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
6.PROPERTIESOFCONTINUOUSFUNCTIONS
1.If fandgaretwocontinuousfunctionsontheircommondomainD thenf + g andf - giscontinuousonD.
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
6.PROPERTIESOFCONTINUOUSFUNCTIONS
2.IffandgaretwocontinuousfunctionsontheircommondomainDthenf. gandfgis
continuousonDprovidedg(x)isdefinedatthatpoint.
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
6.PROPERTIESOFCONTINUOUSFUNCTIONS
3.Thecompositionoftwocontinuousfunctionsisacontinuousfunction.
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
6.PROPERTIESOFCONTINUOUSFUNCTIONS
4.IffiscontinuousonitsdomainD;then|f|isalsocontinuousonD
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
6.PROPERTIESOFCONTINUOUSFUNCTIONS
5.Aconstantandidentityfunctioniseverywherecontinuous.
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
6.PROPERTIESOFCONTINUOUSFUNCTIONS
6.Apolynomialfunctioniseveryeherecontinuous.
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
6.PROPERTIESOFCONTINUOUSFUNCTIONS
7.Amodulusfunctioniseverywherecontinuous.
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
6.PROPERTIESOFCONTINUOUSFUNCTIONS
258.Theexponentialfunctionax; a > 0iseverywherecontinuous.
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
6.PROPERTIESOFCONTINUOUSFUNCTIONS
9.Alogarithmicfunctioniscontinuousinitsdomain.
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
6.PROPERTIESOFCONTINUOUSFUNCTIONS
10.Thesine;cosineand(tangent)functioniscontinuouseverywhere(ondomain).
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
6.PROPERTIESOFCONTINUOUSFUNCTIONS
11.Continuityoffunctiondefineddifferentlyforrationalandirrationalvalueofx
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
6.PROPERTIESOFCONTINUOUSFUNCTIONS
12.Thecosecant;secantandcotangentfunctioniscontinuousinitsdomain.
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
307.CONTINUITYOFSPECIALTYPEOFFUNCTIONS
1.Continuityoffunctionscontaininggreatestintegerfunction
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
7.CONTINUITYOFSPECIALTYPEOFFUNCTIONS
2.Continuityoffunctionscontainingsignumfunction
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
8.MISCELLANEOUS
1.Ifthefunction`f(x)={Ax-B,xlt=13x,1ClicktoLEARNthisconcept/topiconDoubtnut
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
9.DIFFERENTIABILITY
1.Meaningofdifferentiability
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
9.DIFFERENTIABILITY
2.Ifafunctionisdifferentiableatapoint;itisnecessarilycontinuousatthatpoint.But;theconverseisnotnecessarilytrue.
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
9.DIFFERENTIABILITY
3.Showthaty = |x|isnotdifferentiableatx = 0
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
10.DERIVATIVEOFSOMESTANDARDFUNCTIONS
1.Definitionandexamplesofsomedifferentiablefunctions
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
11.PROPERTIESOFDIFFERENTIABILITY
1. the addition of differentiable and non-differentiable functions is always non-differentiable.
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
11.PROPERTIESOFDIFFERENTIABILITY
2.Discussthedifferentiabilityoff(x) = |log|x ∣ ∣andf(x) = arcsin|sinx|
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
11.PROPERTIESOFDIFFERENTIABILITY
3.Theproductofdifferentiableandnon-differentiablefunctionsmaybedifferentiable.
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
11.PROPERTIESOFDIFFERENTIABILITY
4. If both f(x) and g(x) are non-differentiable at x = a then f(x) + g(x) may bedifferentiableatx = a
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
11.PROPERTIESOFDIFFERENTIABILITY
5. If y = f(x) is differentiable at x = a; then it is not necessary that the derivative iscontinuousatx = a
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CONCEPTFORJEE||ChapterCONTINUITYANDDIFFERENTIABILITY
12.INTERMEDIATEVALUETHEOREM
1.Iffiscontinuouson[a; b]andf(a) ≠ f(b)thenforanyvaluec ∈ (f(a); f(b))thereisat
leastonenumberx0in(a; b)forwhichf x0 = c
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