Precalc0105to0111

Precalc0105to0111 - (Section 1.5: Piecewise-Defined...

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(Section 1.5: Piecewise-Defined Functions; Limits and Continuity in Calculus) 1.5.1 SECTION 1.5: PIECEWISE-DEFINED FUNCTIONS; LIMITS AND CONTINUITY IN CALCULUS LEARNING OBJECTIVES • Know how to evaluate and graph piecewise-defined functions. • Know how to evaluate and graph the greatest integer (or floor) function. • Preview limits and continuity from calculus. PART A: DISCUSSION • A piecewise-defined function applies different rules, usually as formulas, to disjoint (non-overlapping) subsets of its domain (subdomains ). • To evaluate such a function at a particular input value, we need to figure out which rule applies there. • To graph such a function, we need to know how to graph the pieces that correspond to the different rules on their subdomains. • The greatest integer (or floor) function and its graph, seen in calculus and computer science, exhibit similar features. • We will take a peek into calculus and preview the related topics of one- and two- sided limits and continuity . Piecewise-defined functions appear frequently in these sections of a calculus course.
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(Section 1.5: Piecewise-Defined Functions; Limits and Continuity in Calculus) 1.5.2 PART B: THE ABSOLUTE VALUE FUNCTION Let fx () = x . We discussed the absolute value function f in Section 1.3, Part N. The piecewise definition of f is given by: () = x = x ,i f x ± 0 ² x f x < 0 ³ ´ µ • For instance, f 3 = 3 = 3 , because 3 ± 0 , and we use the top rule, which applies to the subdomain 0, ± ² ³ ) . • However, f ± 3 = ± 3 = ±± 3 = 3 , because ± 3 < 0 , and we use the bottom rule, which applies to the subdomain ±² ,0 . We say that f is continuous at 0. That is, if we trace the graph of f with a pencil in the vicinity of the point 0, 0 , we do not have to lift our pencil. • We will provide a formal definition of continuity in Part G. WARNING 1 : Piecewise-defined functions are often discontinuous (i.e., they lose continuity) where they switch rules. §
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(Section 1.5: Piecewise-Defined Functions; Limits and Continuity in Calculus) 1.5.3 PART C: EVALUATING PIECEWISE-DEFINED FUNCTIONS Example 1 (Evaluating a Piecewise-Defined Function) Let the function f be defined by: fx () = x 2 , ± 2 ² x < 1 x + 1, 1 ² x ² 2 ³ ´ µ • To evaluate f ± 1 , we use the top rule, since ± 2 ²± 1 < 1 . f ± 1 = ± 1 2 = 1 • To evaluate f 1 , we use the bottom rule, since 1 ± 1 ± 2 . f 1 = 1 + 1 = 2 f 10 , for example, is undefined , because we have no rule for x = 10 . 10 is not in the domain of f , which is ± 2, 2 ² ³ ´ µ . § PART D: GRAPHING PIECEWISE-DEFINED FUNCTIONS Example 2 (Graphing a Piecewise-Defined Function with a Jump Discontinuity; Revisiting Example 1) Graph the function f from Example 1. § Solution WARNING 2 : Clearly indicate any endpoints and whether they are included in, or excluded from, the graph. Top rule: We will graph y = x 2 on the subdomain ± 2, 1 ² ³ ) .
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Precalc0105to0111 - (Section 1.5: Piecewise-Defined...

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