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By I. Craw

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In all of what follows, we will assume the domain of all of our functions is suitably restricted. We won’t spend time in this course discussing standard functions. It is assumed that you know about functions such as sin x, cos x, tan x, log x, exp x, tan−1 x and sin−1 x, as well as the “obvious” ones like polynomials and rational functions — those functions of the form p(x)/q(x), where p and q are polynomials. In particular, it is assumed that you know these are differentiable everywhere they are defined.

1: Graph of the function (x2 − 4)/(x − 2) The automatic graphing routine does not even notice the singularity at x = 2. In this example, we can argue that the use of the (x2 − a2 )/(x − a) was perverse; there was a more natural definition of the function which gave the “right” answer. But in the case of sin x/x, example 4, there was no such definition; we are forced to make the two part definition, in order to define the function “properly” everywhere. So we again have to be careful near a particular point in this case, near x = 0.

An−1 xn−1 + an xn . Note that f (0) = a0 . Differentiating gives f (x) = a1 + 2a2 x + 3a3 x2 . . + (n − 1)an−1 xn−2 + nan xn−1 , and so f (0) = a1 . 2a3 x . . + (n − 1)(n − 2)an−1 xn−3 + n(n − 1)an xn−2 , CHAPTER 5. DIFFERENTIABILITY 50 and f (0) = 2a2 . ak , provided k ≤ n. Thus we can rewrite the polynomial, using its value, and the value of its derivatives at 0, as f (x) = f (0) + f (0)x + f (0) 2 f (0) 3 f (n−1) (0) n−1 f (n) (0) n + x + x + ... + x x . 2! 3! (n − 1)! n! This opens up the possibility of representing more general functions than polynomials in this way, and so getting a generalisation of the Mean Value Theorem.

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Advanced Calculus And Analysis by I. Craw

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