MATH 133 – Calculus with Fundamentals 1 Practice on Absolute Values and Intervals September 4, 2015

Background

Recall from the video “lecture” for today that the absolute value of a real number is defined as

|x| = \begin{cases}x & if x ≥ 0 \\ −x & if x < 0. \end{cases}

Also, we have several different types of intervals in the real number system written as follows:

(a, b) = {x | a < x < b}  (1)
[a, b] = {x | a ≤ x ≤ b}  (2)
[a, b) = {x | a ≤ x < b}  (3)
(a, b] = {x | a < x ≤ b}.  (4)

Today we will practice on some problems related to these ideas.

Questions

0) (Warm-up) Real numbers can all be expressed by (possibly infinite or non-repeating) decimal expansions. But exactly what does a decimal expansion like

π = 3.1415926 ⋯

mean? What does a finite decimal expansion like \frac{15}{8} = 1.875 mean? (Hint: It’s really a way of writing the number as a sum of fractions with denominators of a particular form.)

1) The set of real numbers x satisfying |x| < 17 is an interval. Write it using the appropriate interval notation.

2) The set of real numbers x satisfying |x − 3| ≤ 1 is also an interval. Write it using the appropriate interval notation.

3) The set of real numbers x satisfying |3x + 1| < 4 is also an interval. Write it using the appropriate interval notation.

4) Write the interval (5, 9) in the form: “the set of all x such that |x − c| < r” for some real numbers c and r.

5) Same as question 4, but for the interval [−\frac{2}{5}, 3].

6) Using the graph of temperature versus time over a 24-hour period at a particular location, on the back of this page answer these questions:

Figure 1: Figure for Question 6 2 (a) Write the collection of all temperatures measured over the 24 hour period as an interval

[a, b] for suitable temperature values a, b in degrees Fahrenheit. (There’s an unstated assumption behind this question. Can you see what that is? Does it seem reasonable to make that assumption?)

(b) Estimate the interval [c, d] of times in hours for which the temperature was 16.8 degrees or less.

(c) At how many different times does it appear that the temperature was exactly 40 degrees F?

Note: The temperature values at the start of each hour are given in the table at the right.