math152_9.7_9.10.pdf - Math 152 Jagodina Chapter 9.7-9.9...

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Math 152, JagodinaChapter 9.7-9.9 PacketTutors may help.9.7 Taylor Polynomials and Approxi-mationsDefinitionIffhasnderivatives atc, then the polynomialPn(x) =f(c) +f0(c)(x-c) +fPn(x) =f(0) +f0(0)x+f00(0)2!x2+· · ·+f(n)(0)n!xnis called thenthMaclaurin Polynomialforf.ExampleFind thenthMaclaurin Polynomial forf(x) =exifn= 6.00(c)2!(x-c)2+· · ·+f(n)(c)n!(x-c)nis called thenthTaylor Polynomialforfatc. Ifc= 0, then
Math 152, JagodinaChapter 9.7-9.9 PacketTutors may help.ExampleFind thenthMaclaurin Polynomial forf(x) = cosxifn= 6. UseP6to approx-imate cos(0.1).Solution:P6(x) =f(0) +f0(0)x+f00(0)x22!+f000(0)x33!+f(4)(0)x44!+f(5)(0)x55!+f(6)(0)x66!f(x) =f(0) =f0(x) =f0(0) =f00(x) =f00(0) =f000(x) =f000(0) =f(4)(x) =f(4)(0) =f(5)(x) =f(5)(0) =f(6)(x) =f(6)(0) =SoP6(x) =cos(0.1)P6(0.1) =2
Math 152, JagodinaChapter 9.7-9.9 PacketTutors may help.ExampleFind the 3rd degree Taylor Polynomial forf(x) =xcentered atc= 4.
Math 152, JagodinaChapter 9.7-9.9 PacketTutors may help.ExampleFind the 3rd degree Taylor polynomial forf(x) = sinx, expanded aboutc=π6.Solution:T3(x) =f(c) +f0(c)(x-c) +f00(c)(x-c)22!+f000(c)(x-c)33!f(x) =fπ6
Math 152, JagodinaChapter 9.7-9.9 PacketTutors may help.

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