By E.T. Robinson, G. Whittaker
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Extra resources for The calculus of observations: a treatise on numerical mathematics
Example text
21 for an application. 24. 1. Construct an analytical proof of Young’s inequality. 2. Assume a, b, c, d > 0 and prove the following: (a) a2 + b2 ≥ 2ab. (b) a4 + b4 ≥ 2a2 b2 . (c) a2 + b2 + c2 ≥ ab + bc + ca. (d) a4 + b4 + c4 + d4 ≥ 4abcd. (e) (a + b)(b + c)(c + a) ≥ 8abc. 3. Use AM–GM to show that n! < n+1 2 n . Prove also that if n > 1 is a natural number, then [45] (2n − 1)!! , where (2n − 1)!! = (2n − 1) · (2n − 3) · (2n − 5) · · · · · 5 · 3 · 1 (2n)!! = (2n) · (2n − 2) · (2n − 4) · · · · · 4 · 2.
3. If f (x) and g(x) are integrable on [a, b] with f (x) ≤ g(x), then b a b f (x) dx ≤ g(x) dx. a 22 2. Methods from the Calculus Proof. With the notation described above, we form Riemann sums: n n f (xi )∆x ≤ i=1 g(xi )∆x. 1. 1 (Simple Estimate). If f (x) is integrable on [a, b] with m ≤ f (x) ≤ M , then b m(b − a) ≤ f (x) dx ≤ M (b − a). 2 (Modulus Inequality). If f (x) is integrable on [a, b], then b b f (x) dx ≤ a |f (x)| dx. a The second corollary follows from the inequalities −|f (x)| ≤ f (x) ≤ |f (x)| and plays the role of the triangle inequality for integrals.
Underestimating an integral. By the quotient rule for differentiation, d dx 1 x x f (u) du 0 ≥0 as required. It is possible to obtain other inequalities involving integrals through an ad hoc consideration of areas bounded by various plane curves. This simple process, reminiscent of the integral test from calculus, is probably best described in an example. 12. The function f (x) = xp , where −1 < p < 0, is strictly decreasing. 4 it is apparent that n+1 1 n xp dx < n kp < xp dx. 0 k=1 Hence, after carrying out the integrations, (n + 1)p+1 − 1 < p+1 n kp < k=1 np+1 .