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9.10 TAYLOR's FORMULA If we wish to express f(x) as a power series in x  c, we need two things: (1) A sequence of polynomials which approximate f(x) near x = c, (1) a_{0}, a_{0} + a_{1}(x  c), ...,a_{0} + a_{1}(x  c) + ... + a_{n}(x  c)^{n},.... (2) An estimate for the error E_{n} between f(x) and the nth polynomial, (2) f(x) = a_{0} + a_{1}(x  c) + ... + a_{n}(x  c)^{n} + E_{n}. In the last section the formula was used to obtain power series approximations. A much more general formula of this type is Taylor's Formula. In Taylor's Formula the nth polynomial P_{n}(x) is chosen so that its value and first n derivatives agree with f(x) at x = c. The tangent line at x = c, P_{1}(x) = f(c) + f'(c)(x  c), has the same value and first derivative as f(x) at x = c. A polynomial of degree two with the same value and first two derivatives as f(x) at c is P_{1}(x) and P_{2}(x) are the first and second Taylor polynomials of f(x) (see Figure 9.10.1). Figure 9.10.1 First and Second Taylor Polynomials To continue the procedure we need a formula for the nth derivative of a polynomial. LEMMA 1 Let P(x) be a polynomial in x  c of degree n. P(x) = a_{0} + a_{1}(x  c) + a_{2}(x  c) + ... + a_{n}(x  c)^{n}. For each m ≤ n, the mth derivative of P(x) at x = c divided by m! is equal to the coefficient a_{m}, PROOF Consider one term a_{k}(x  c)^{k}. Its mth derivative is
At x = c, the mth derivative of a_{k}(x  c)f is: 0 if in < k, m! a_{m} if in = k, 0 if in > k. It follows that P^{(m)}(c) = m! a_{m}. This lemma shows us how to find a polynomial P(x) whose value and first n derivatives agree with f(x) at x = c. The mth coefficient of P(x) must be DEFINITION Let f(x) have derivatives of all orders at x = c. The nth Taylor polynomial of f(x) at x = c is the polynomial By Lemma 1, P_{n}(x) is the unique polynomial of degree n whose value and first n derivatives at x = c agree with f(x), P_{n}(c) = f(c), P_{n}'(c) = f'(c),.... P_{n}^{(n)}(c) = f^{(n)}(c). The difference between f(x) and the nth Taylor polynomial is called the nth Taylor remainder, R_{n}(x) = f(x)  P_{n}(x). Thus f(x) = f(c) + f'(c)(x  c) +(x  c)^{2} + ... + (x  c)^{n} + R_{n}(x).


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