Analytic inequalities by Nicholas D. Kazarinoff

By Nicholas D. Kazarinoff

Acceptable for undergraduates or even highschool scholars, this article introduces a couple of old theorems, explores inequalities and calculus, and covers smooth theorems, together with Bernstein's facts of the Weierstrass approximation theorem and the Cauchy, Bunyakovskii, Hölder, and Minkowski inequalities. 1961 version. contains 28 figures.

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X −3 −2 1 2 3 4 −2 −3 −4 1 2 3 4 −2 −3 −4 Numerical and Graphical Analysis In Exercises 79 and 80, (a) use a graphing utility to complete the table, (b) plot the points in the table and graph the function by hand, (c) use a graphing utility to graph the function and compare the result with your hand-drawn graph in part (b), and (d) determine any intercepts and symmetry of the graph. 2 3, ) ) 3 −π 2 83. arcsin 12 84. arcsin 0 1 2 86. arccos 0 85. arccos 87. arctan Ί3 3 89. arccsc͑Ϫ Ί2 ͒ 4 3 2 1 x −3 −2 −1 ,π 4 In Exercises 83–90, evaluate the expression without using a calculator.

675855 . . 666597 . . 665303 . . 665156 . . 665144 . . 66514. In practice, you can use a calculator to approximate numbers such as 2Ί2. In general, you can use any positive base a, a 1, to define an exponential function. Thus, the exponential function with base a is written as f ͑x͒ ϭ a x. Exponential functions, even those with irrational values of x, obey the familiar properties of exponents. Properties of Exponents Let a and b be positive real numbers, and let x and y be any real numbers.

2 ϭ 9a ϩ 7 Simplify. 2 b. f ͑b Ϫ 1͒ ϭ ͑b Ϫ 1͒ ϩ 7 Substitute b Ϫ 1 for x. ϭ b2 Ϫ 2b ϩ 1 ϩ 7 Expand binomial. 2 ϭ b Ϫ 2b ϩ 8 Simplify. 2 f ͑x ϩ ⌬x͒ Ϫ f ͑x͒ ͓͑x ϩ ⌬ x͒ ϩ 7͔ Ϫ ͑x 2 ϩ 7͒ c. ϭ ⌬x ⌬x 2 x ϩ 2x⌬x ϩ ͑⌬x͒ 2 ϩ 7 Ϫ x 2 Ϫ 7 ϭ ⌬x 2x⌬x ϩ ͑⌬x͒ 2 ϭ ⌬x ⌬x͑2x ϩ ⌬x͒ ϭ ⌬x ϭ 2x ϩ ⌬x, ⌬x 0 NOTE The expression in Example 1(c) is called a difference quotient and has a special significance in calculus. You will learn more about this in Chapter 2. 2 Functions and Their Graphs 11 The Domain and Range of a Function The domain of a function may be described explicitly, or it may be described implicitly by an equation used to define the function.

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