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Grafice ale functiilor si Rolul derivatelor

Grafice ale functiilor si Rolul derivatelor. Tiselice Dragos. Rolul derivatei intai in studiu functiilor. Derivata intai a unei functii ne da informatii despre monotonia functiei si despre eventualele puncte de extrem ale acesteia .

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Grafice ale functiilor si Rolul derivatelor

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  1. Grafice ale functiilorsiRolulderivatelor TiseliceDragos

  2. Rolulderivateiintai in studiufunctiilor • Derivataintai a uneifunctii ne da informatiidespremonotoniafunctieisidespreeventualelepuncte de extrem ale acesteia. • Functia f(x)=sin(x) are maxim in si minim in - ( f’()=0 si f’(-)=0).

  3. ConsecintaTeoremeiluilagrange • Consecinta: • daca f’(x)>0, x I, f este strict crescatoarepe I • dacaf’(x)<0, x I, f este strict descrescatoarepe I

  4. Exemplumonotonie • f: [0,2]R, f(x)=+x • f’(x)=2x+1, f’(x)>0 • f este strict crescatoarepe [0,2]

  5. Rolulderivatei a doua in studiufunctiilor • Intervale de convexitatesiconcavitate ale uneifunctii • f:IR • f esteconvexape I, daca, si [0,1] • f((1-)+)(1-)f()+f() adicadaca f’’(x)>0 • f esteconcavape I, daca, si [0,1] • f((1-)+)(1-)f()+f()adicadaca f’’(x)<0 • Concava: Convexa:

  6. Punctele de inflexiune ale uneifunctii • estepunct de inflexiune al functiei f daca f are derivata in sidacape I, de o perte a luifunctiaesteconvexa, iar de cealalta parte a luifunctiaesteconcava. • =0 estepunct de inflexiune • pentru f(x)=adica f’’(x)=0

  7. Reprezentareagrafica a functiilor • IDomeniul de definitie (determinare, interesectii cu axele, calcularea la capetesiasimptote) • II Derivataintai(rezolvareaecuatiei f’(x)=0, intervale de monotonie) • IIDerivata a doua (rezolvareaecuatiei f’’(x), intervale cu semn constant) • IV Tabelul de variatie (valoriremarcabile, f’(x), f’’(x), f(x)) • V Trasareagraficului

  8. Exemplu • f(x)= • I D=(-∞,-1]U[1, +∞) • Intersecteaza Ox in (-1,0) si (1,0) dar nu siOypentruca x0. • Asimptoteoblice y=si y= • II f’(x)=0 nu are solutii. • III f’’(x)=0 nu are solutii. IV

  9. grafic

  10. Bibliografie • Matematica – Manual pentruclasa a XI-a Editura Sigma 2003 • Matematicaclasaa XI-a “Elemente de analizamatematica” EdituraCarminis • Manual pentruclasa a XI-a “Elemente de analizamatematica” EdituraMathpress2003 • Exercitiisiprobleme de clasa a XI-a (si nu numai) EdituraBirchi • GazetaMatematicaEditie Electronica 1895-2004 Intuitext • Revista de Matematica din Timisoara Editie Electronica 1921-2006 Intuitext • http://rechneronline.de/function-graphs/

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