goldch7 - Contents VII Taylor Approximation 1 VII.A...

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Unformatted text preview: Contents VII Taylor Approximation 1 VII.A LHospitals Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 VII.B Taylor Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 VII.C Taylors Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 VII.D The Remainder in Taylors Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 VII.E Applications of Taylors Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 VII.E.1 lHospitals Rule Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 VII.E.2 Approximating in a Given Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 VII.E.3 Binomial Theorem, Taylors Version . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 VII Taylor Approximation VII.1 VII Taylor Approximation VII.A LHospitals Rule Assume f and g are functions defined near some a R . If lim x a f ( x ) and lim x a g ( x ) exist and lim x a g ( x ) 6 = 0, then one has lim x a f ( x ) g ( x ) = lim x a f ( x ) lim x a g ( x ) , the limit of a quotient is the quotient of the limits provided the limit of the denominator is not zero . What about a limit like lim x sin x x ? If one tries to take limits separately in numerator and denominator one ends up with an indeterminate form which means that both numerator and denominator tend to 0 with x 0. LHospitals rule sometimes allows one to determine limits of such indeterminate forms by differentiation. In the example at hand it says lim x sin x x = lim x (sin x ) lim x ( x ) = lim x cos x lim x 1 = cos0 1 = 1 1 = 1 , so we obtain the limit by taking the quotient of the limits of the derivatives of numerator and denominator. The precise formulation is the following. Theorem: (lHospitals rule for ): Let f,g be functions and a R . If (i) f and g are differentiable in some interval ( a- h,a + h ) with h > 0, and (ii) lim x a f ( x ) = 0 and lim x a g ( x ) = 0, and (iii) lim x a f ( x ) g ( x ) exists (allowing the limits + or- ), then the limit lim x a f ( x ) g ( x ) exists as well and lim x a f ( x ) g ( x ) = lim x a f ( x ) g ( x ) . The rule works also for limits of the indeterminate form . Theorem (lHospital for ): If f and g are functions that satisfy (i) and (iii) above together with (ii ) lim x a | f ( x ) | = and lim x a | g ( x ) | = , then lim x a f ( x ) g ( x ) = lim x a f ( x ) g ( x ) . VII.A LHospitals Rule VII.2 To repeat it: lHospitals rule works for those indeterminate forms where either both numerator and de- nominator tend to 0 or both tend to ....
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goldch7 - Contents VII Taylor Approximation 1 VII.A...

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