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# Class Note for MATH 1431 with Professor Caboussat at UH

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This 22 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 17 views.

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Date Created: 02/06/15

MATH 1431 19328 Alexandre Caboussat Lecture MWF 11AM 12PM Office hours MWF 12PM 1PM httpwwwmathuheduNcaboussatmath1431 J HWTH 1431 1 Newton Method Geometrically Approximation of roots by using the tangent lines i Chap Newton Method Formula Let f be a twice differentiable function and suppose a is a real number at which fa 0 If f a 7 O and X0 is sufficiently close to a then the iteration fx Xn1Xn Fix n0123 n will converge rapidly to the root 3 Newton method Newton Raphson method METH 1411 Newton method can go bad 41 The Mean Value Theorem METH 1411 afa How many points are there between 71 and 3 where the tangent line is parallel to the secant line How many points are there between 71 and 3 where the tange t line is parallel to the secant line Mean Value Theorern If f is differentiable on the open interval 37 b and continuous on the closed interval 37 b then there is at least one number 6 in 37 b for which 1 07 i 3 fl C b 7 a or equivalently Verify the mean value theorem for fx 3X7 X2 on the interval 717 Particular Case Maw 1411 11 Rolle s Theorem Let f be differentiable on the open interval 37 b and continuous on the closed interval 37 b If fa O and fb 0 then there is at least one number 6 in 37 b for which fc 0 iniiTH 1431 1 Are there values between a and b where he derivative is equal to zero How many Are there values between a and b where the derivative is equal to zero How many But here fa fb my 4 41 Rolle s Theorem Generalization Let f be differentiable on the open interval 37 b and continuous on the closed interval 37 b If fa fb not necessarily equal to zero then there is at least one number 6 in 37 b for which fc 0 lwliTH 1431 1 Example Let fx X3 7 4X on O7 3 Verify the mean value theorem Let fx X3 7 4X on 02 Verify the Rolle39s theorem METH 1411 Show that the function fx 2X3 5X 1 has exactly one zero Example Show that the function fx 2X3 5X 71 has exactly one zero One Last Remark Let f be differentiable at X0 0 If f x0 gt 0 then fX0 7 h lt fX0 lt fX0 h7 for all positive h sufficiently small 0 If f x0 lt 0 then fX0 7 h gt fX0 gt fX0 h7 for all positive h sufficiently small METH 1411 Example fxx2X7 X02 MATH 1431 19328 Alexandre Caboussat Lecture MWF 11AM 12PM Office hours MWF 12PM 1PM httpwwwmathuheduNcaboussatmath1431 J HWTH 1431 1

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