Section0_2

# Section0_2 - Section 0.2 Set notation and solving...

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Unformatted text preview: (5/31/07) Section 0.2 Set notation and solving inequalities Overview: Inequalities are almost as important as equations in calculus. Many functions’ domains are intervals, which are defined by inequalities. Inequalities are needed to study where functions have positive and negative values. They are also used in the definitions of limits and with derivatives to study where functions are increasing and decreasing and where their graphs are concave up and concave down. In this section we describe notation and terminoogy for intervals and other sets and discuss the rules for solving inequalities. These rules are similar to those for solving equations but are somewhat more difficult to apply. Topics: • Intervals and other sets of numbers • The absolute value function • Working with inequalities Intervals and other sets of numbers Intervals can be defined, as in the last section, by giving their defining inequalities. With this approach the interval in Figure 1 is called the interval 0 < x ≤ 2. The heavy line in the drawing indicates that the points x with 0 < x < 2 are in the interval; the dot shows that x = 2 is in the interval; and the open circle at x = 0 indicates that the point x = 0 is not. We can also define intervals with set-builder notation : The symbols { x : P } designate the set of numbers x that satisfy condition P . With this notation the interval in Figure 1 can be defined by { x : 0 < x ≤ 2 } , which reads “the set of those numbers x such that 0 < x ≤ 2.” We also refer to this interval as (0 , 2], where the parenthesis at the left indicates that the point x = 0 is not in the interval and the square bracket at the right indicates that the point x = 2 is in the interval. Similarly, the interval in Figure 2 can be referred to either as the interval x ≤ 2, as the interval { x : x ≤ 2 } , or as the interval (-∞ , 2]; and the interval in Figure 3 can be given by x > 1, by { x : x > 1 } , or by (1 , ∞ ). x- 1 1 2 3 x- 1 1 2 3 x- 1 1 2 3 The interval < x ≤ 2 or The interval x ≤ 2 or The interval x > 1 or { x : 0 < x ≤ 2 } = (0 , 2] { x : x ≤ 2 } = (-∞ , 2] { x : x > 1 } = (1 , ∞ ) FIGURE 1 FIGURE 2 FIGURE 3 To describe a set that consists of two intervals, we use the union symbol ∪ with the convention that A ∪ B designates the set consisting of the points in set A combined with the points in set B : † A ∪ B = { x : x is in A or x is in B } . Example 1 (a) Draw on an x-axis the set of points { x :- 5 ≤ x < 2 or x > 4 } . (b) Express the set in part (a) as a union of intervals. Solution (a) We show the intervals on an x-axis by drawing solid lines from x =- 5 to x = 2 and to the right of x = 4 and by putting a solid dot at x =- 5 and small open circles at x = 2 and x = 4, as in Figure 4....
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Section0_2 - Section 0.2 Set notation and solving...

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