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Taylor's Theorem

We are familiar with polynomial equations of the form f(x) = a0 + a1x + . . . + an xn where a0, a1, . . . , an are real numbers. We can easily compute the value of a polynomial at any point x = a by using the four basic operations of addition, multiplication, subtraction and division. On the other hand there are functions like ex, cos x, ln x etc. which occur frequently in all branches of mathematics which cannot be evaluated in the same manner. For example, evaluating the function f(x) = cos x at 0.524 is not so simple. Now, to evaluate such functions we try to approximate them by polynomials which are easier to evaluate. Taylor's theorem gives a simple method for approximating functions f(x) by polynomials.



Let f(x) be a real-valued function defined on R which is n-times differentiable.


Consider the function where x0 is any given real number. Now Pl(x) is a polynomial in x of degree 1 and P1 (x0) = f(x0) and P1 '(x0) = f '(x0). The polynomial P1(x) is called the first Taylor polynomial of f(x) at x0. Now consider another function


Then P2(x) is a polynomial in x of degree 2 and P2(x0) = f(x0), P '2(x0) = f'(x0) and P ''2(x0) = f '(x0). P2(x) is called the second Taylor polynomial of f(x) at x0.

Similarly we can define the rth Taylor polynomial of f(x) at x0 where 1 ≤ r ≤ n. The rth Taylor polynomial at x0 is given by


Theorem

Let f be a real valued function having (n + 1) continuous derivatives on ]a, b[ for some n ≥ 0. Let x0 be any point in the interval ]a, b[. Then for any x ∈ ]a, b[, we have


where c is a point between x0 and x.

The series is called the nth Taylor's expansion of f(x) at x0.

Remainder or error

We can write the above equation as


where Pn(x) is the nth Taylor polynomial of f(x) about x0 and


Rn+1(x) depends on x, x0 and n. Rn+1(x) is called the remainder (or error) of the nth Taylor's expansion after n + 1 terms.


Suppose we put x0 = a and x = a + h where h > 0 . Then any point between a and a+h will be of the form a+θh,O<θ<1.

Therefore we can write the equation as



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