Taylor Series (Calculus II)Date: 2/14/96 at 18:57:10 From: Anonymous Subject: Taylor Series (Calculus II) I don't have a real specific question. I am having difficulty understanding the Taylor Series and Maclauren Series. I guess I am having trouble with all areas of the Thomas, Finney 8th edition Calculus textbook's eighth chapter. I was looking for some help on the theory behind the error estimate and "c" in particular - I am not understanding it. I am looking for a really super guide that would help me better understand Calc II and where I might get this. My professor just does examples and doesn't explain the theory. Thank you very much. I'm sure you won't know how much I appreciate this. Date: 7/29/96 at 13:41:47 From: Doctor Jerry Subject: Re: Taylor Series (Calculus II) The c in the Mean-value Theorem f(b)-f(a)=f'(c)(b-a) is a point between a and b, which (for smooth functions) is known to exist but is not necessarily easy to calculate. Geometrically, it's quite clear that there must be a value of x, call it c, between a and b for which the slope of the line joining (a,f(a)) and (b,f(b)) is exactly equal to the slope of the graph of f at (x,f(x)). The c in Maclaurin's formula f(a+h) = f(a)+f'(a)x/1!+f''(a)x^2/2!+...+fn'(a)x^n/n!+fn+1'(c)x^(n+1)/(n+1)! where fn' means the nth derivative and fn+1' means the (n+1)st derivative, is precisely analogous to the c in the Mean-value Theorem. The proof that it exists is more complicated, but not different in kind. -Doctor Jerry, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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