By I. Craw

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20. Exercise. The function f satisfies f (x) = 1 and f (0) = 0. Use the Mean 5 + sin x Value theorem to estimate f (π/2). Note the “common sense” description of what we have done. If the derivative doesn’t change much, the function will behave linearly. Note also that this gives meaning to the approximation f (a + h) ≈ f (a) + hf (a). We now see that the accurate version of this replaces f (a) by f (c) for some c between a and a + h. 21. Theorem. (The Cauchy Mean Value Theorem) Let f and g be both continuous on [a, b] and differentiable on (a, b).

Thus we can only have limits 2 or −3, and since an ≥ 0 for all n, necessarily l > 0. Hence l = 2. Warning: There is a difference between showing that f is increasing, and showing that the sequence is increasing. There is of course a relationship between the function f and the sequence an ; it is precisely that f (an ) = an+1 . What we know is that if f is increasing, then the sequence carries on going the way it starts; if it starts by increasing, as in the above example, then it continues to increase.

Solution. We have f (x) = (1 + x)3 f (x) = 3(1 + x)2 f (x) = 6(1 + x) f (x) = 6 f (n) (x) = 0 if n > 3. f (0) = 1 f (x) = 3 f (x) = 6 f (x) = 6 CHAPTER 5. DIFFERENTIABILITY 52 and so, by Taylor’s theorem (1 + x)3 = 1 + 3x + 6 2 6 x + x3 , 2! 3! a result we could have got directly, but which is at least reassuring. 33. Example. Find the Taylor polynomial of order n about 0 for f (x) = sin x, and write down the corresponding remainder term. Solution. There is no difficulty here in calculating derivatives — we have f (x) = sin x f (x) = cos x f (x) = − sin x f (x) = − cos x f (4) (x) = sin x and so on.

### Advanced Calculus And Analysis by I. Craw

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