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Mean Value Theorem and Taylors Formula
We can easily find the Taylor polynomials of f(x) by differentiating. Let us now try to find a formula for the Taylor remainders. The Mean Value Theorem gives a formula for the Taylor remainder, R_{0}(x). MEAN VALUE THEOREM (Repeated) Suppose f(t) is differentiate at all t between c and d. Then for some point t_{0} strictly between c and d. When we replace d by x, this gives the formula f(x) = f(c) + R_{0}(x). R_{0}(x) = f'(t_{0})(x  c). Taylor's Formula is a generalization of the Mean Value Theorem which gives the nth Taylor remainder. TAYLOR'S FORMULA Suppose the (n + l)st derivative f^{(n+1)}(t) exists for all t between c and x. Then f(x) = f(c) + f'(c)(x  c) + (x  c)^{2} + ... +(x  c)^{n} + R_{n}(x) where R_{n}(x) = (x  c)^{n+1} for some point t_{n} strictly between c and x. Notice that the remainder term looks just like the (n + 1)st term of a Taylor polynomial except that f^{(n+1)}(c) is replaced by f^{(n+1)}(t_{n}). When r = 0 Taylor's Formula is sometimes called MacLaurin's Formula. Taylor's Formula can be used to get an estimate of the error R_{n}(x) between f(x) and the Taylor polynomial P_{n}(x). For example if f^{(n+1)}(t) ≤ M_{n+1 }for all t between c and x. then we obtain the error estimate R_{n}(x) ≤ x  c^{(n+1)}. Taylor polynomials with the error estimate are of great practical value in obtaining approximations. In the next example we use Taylor's Formula to approximate the value of e.


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