By Vladimir Kozlov

The asymptotic research of boundary worth difficulties in parameter-dependent domain names is a quickly constructing box of analysis within the thought of partial differential equations, with vital purposes in electrostatics, elasticity, hydrodynamics and fracture mechanics. development at the paintings of Ciarlet and Destuynder, this e-book offers a scientific assurance of those equipment in multi-structures, i.e. domain names that are depending on a small parameter *e* in this kind of manner that the restrict zone involves subsets of alternative area dimensions. An undergraduate wisdom of partial differential equations and useful research is believed.

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2 Ϫ2 41. For what values of x is f (x) > 0? 1 Ϫ2 Ϫ2 4 x 52. On what interval(s) is the function f constant? 53. On what interval(s) is the function f nonincreasing? 54. On what interval(s) is the function f nondecreasing? 13 14 Chapter P • Preparing for Calculus In Problems 55–60, answer the questions about the function 69. 70. x +2 . g(x) = x −6 y 55. What is the domain of g? 1 58. If g(x) = 2, what is x? What is(are) the corresponding point(s) on the graph of g? (1, 0) Ϫ2 57. If x = 4, what is g(x)?

F (x) = x = smallest integer greater than or equal to x The domain of the ceiling function x is the set of all real numbers; the range is the set of integers. The y-intercept of x is 0, and the x-intercepts are the numbers in the interval (−1, 0]. The ceiling function is constant on every interval of the form (k, k + 1], where k is an integer, and is nondecreasing on its domain. See Figure 28. y (3, 3) (2, 2) 2 (1, 1) 2 Ϫ2 x <0 The domain of the floor function x is the set of all real numbers; the range is the set of all integers.

DEFINITION Polynomial Function A polynomial function is a function of the form f (x) = an x n + an−1 x n−1 + · · · + a1 x + a0 where an , an−1 , . . , a1 , a0 are real numbers and n is a nonnegative integer. The domain of a polynomial function is the set of all real numbers. 18 Chapter P • Preparing for Calculus If an = 0, then an is called the leading coefficient of f , and the polynomial has degree n. The constant function f (x) = A, where A = 0, is a polynomial function of degree 0. The constant function f (x) = 0 is the zero polynomial function and has no degree.