By A. Ginzburg
This article is helping scholars enhance their knowing and problem-solving talents in research, analytic geometry, and better algebra. Over 1,200 difficulties, with tricks and whole options. subject matters comprise sequences, features of a unmarried variable, restrict of a functionality, differential calculus for services of a unmarried variable, the differential, indefinite and yes integrals, extra. 1963 variation.
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Extra resources for Calculus: Problems and Solutions (Dover Books on Mathematics)
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.