pn5 1
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h212.
For example, the chance of a “yes” on the third roll call vote is
Source: Mathematics in the Behavioral and Social Sciences.
Suppose that there is a chance of that Congressman Stephens will favor the budget appropriation bill before the first roll call, but only a probability of that he will change his mind on the subsequent vote. Find and interpret the following.
a. b.
c. d.
YOUR TURN ANSWERS
1. 3 2. 4 3. 5 4. Does not exist. 5. 49
6. 7 7. 1/2 8. 1/3
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Continuity
How does the average cost per day of a rental car change with the number of days the car is rented?
3.2
APPLY IT
We will answer this question in Exercise 38.
In 2009, Congress passed legislation raising the federal minimum wage for the third time in three years. Figure 14 below shows how that wage has varied since it was instituted in 1938.
We will denote this function by f (t), where tis the year. Source: U.S. Department of Labor.
Notice from the graph that and that so that
does not exist. Notice also that A point such as this, where a function has a sudden sharp break, is a point where the function is discontinuous.
In this case, the discontinuity is caused by the jump in the minimum wage from $4.75 per hour to $5.15 per hour in 1997.
f119972 55.15.
tl1997lim f1t2
tl1997lim1 f1t2 55.15,
tl1997lim 2 f1t2 54.75
FIGURE 14
1940 1950 1960 1970 1980 1990 2000 2010 1.00
2.00 3.00 4.00 7.00 6.00 5.00
Minimum wage
Year
Intuitively speaking, a function is continuousat a point if you can draw the graph of the function in the vicinity of that point without lifting your pencil from the paper. As we already mentioned, this would not be possible in Figure 14 if it were drawn correctly; there would be a break in the graph at for example. Conversely, a function is discon- tinuous at any x-value where the pencil mustbe lifted from the paper in order to draw the graph on both sides of the point. A more precise definition is as follows.
Continuity at
A functionf iscontinuousat if the following three conditions are satisfied:
1. is defined,
2. exists, and
3.
If fis not continuous at c, it is discontinuousthere.
The following example shows how to check a function for continuity at a specific point.
We use a three-step test, and if any step of the test fails, the function is not continuous at that point.
Continuity
Determine if each function is continuous at the indicated x-value.
(a) in Figure 15 at SOLUTION
Step 1 Does the function exist at ?
The open circle on the graph of Figure 15 at the point where means that does not exist at . Since the function does not pass the first test, it is discontinuous at x53, and there is no need to proceed to Step 2.
x53
f1x2 x53
x53 x53
f1x2
xlclim f1x2 5f1c2.
xlclim f1x2 f1c2
x5c
x 5 c
t51997,
3.2 Continuity 141
EXAMPLE 1
(b) in Figure 16 at SOLUTION
Step 1 Does the function exist at ?
According to the graph in Figure 16, exists and is equal to –1.
Step 2 Does the limit exist at ?
As xapproaches 0 from the left, is –1. As xapproaches 0 from the right, however, is 1. In other words,
while
xl0lim1 h1x2 51.
xl0lim2 h1x2 5 21, h1x2 h1x2
x50
h102 x50
x50 h1x2
0 3 x
f(x)
2 1
FIGURE 15
h(x)
0 x
1
–1
FIGURE 16
Since no single number is approached by the values of as x approaches 0, the limit does not exist. Since the function does not pass the second test, it is discontinuous at , and there is no need to proceed to Step 3.
(c) in Figure 17 at SOLUTION
Step 1 Is the function defined at ?
In Figure 17, the heavy dot above 4 shows that is defined. In fact, .
Step 2 Does the limit exist at ? The graph shows that
Therefore, the limit exists at and
Step 3 Does ?
Using the results of Step 1 and Step 2, we see that . Since the function does not pass the third test, it is discontinuous at x54.
xl4lim g1x2 g1422
xl4lim g1x2 g142 5
xl4lim g1x2 5 22.
x54
xl4lim2 g1x2 5 22, and lim
xl41 g1x2 5 22.
x54
g14251 g142
x54 x54
g1x2
x50
xl0 limh1x2 h1x2
0 2
–2 f(x)
x 2
–2
FIGURE 18
0 1 3 4 5
g(x)
x 3
2 1 –1 –2
FIGURE 17
(d) in Figure 18 at . SOLUTION
Step 1 Does the function exist at ?
The function fgraphed in Figure 18 is not defined at . Since the function does not pass the first test, it is discontinuous at . (Func- tion f is continuous at any value of xgreater than –2, however.)
Notice that the function in part (a) of Example 1 could be made continuous simply by defining Similarly, the function in part (c) could be made continuous by redefin- ing In such cases, when the function can be made continuous at a specific point simply by defining or redefining it at that point, the function is said to have a removable discontinuity.
A function is said to be continuous on an open intervalif it is continuous at every x-value in the interval. Continuity on a closed interval is slightly more complicated because
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