By Michael J. Cloud
A operating wisdom of inequalities could be valuable to the training engineer, and inequalities are significant to the definitions of all proscribing techniques, together with differentiation and integration. whilst particular ideas are unavailable, inconvenient, or pointless, inequalities can be utilized to procure mistakes bounds for numerical approximation. they could additionally bring about an realizing of the qualitative habit of ideas. This advisor to inequalities was once written in particular with engineers and different utilized scientists in brain, and is helping fill the space among university algebra-level remedies, and the bold treatise at the topic that exist within the arithmetic literature. To consolidate the educational strategy, each bankruptcy ends with a wealthy selection of routines.
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Extra info for Inequalities : With Applications to Engineering
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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 .