Introduction to the Applications of Graphs
Graphs are useful tools to present a lot of information in a small space. Being able to read a graph and draw conclusions from it are important in many subjects in addition to mathematics. In the example below, we will practice drawing conclusions based on information given in the graph in Figure 3.12. This graph shows the daily balance of a checking account for about two weeks. No more than one transaction (a deposit or a check written) is made in one day. For example, the balance at the end of the second day is $350 and $300 at the end of the third day, so a $50 check was written on the third day.
- On what day was a check for $200 written? On the 12th day when the balance dropped from $150 to −$50.
- What is the largest deposit? The largest increase was $200, on the 8th day when the balance increased from $200 to $400.
- What is the largest check written?
The largest check was written on the tenth day when the balance dropped from $400 to $150.
Fig. 3.12
4. When was the account overdrawn?
The balance was negative on the 12th day.
Average Rate of Change Examples
Calculus deals with the rate of change. A familiar example of a rate of change is speed (or more accurately, velocity). Velocity is the rate of change of distance per unit of time. A car traveling in city traffic will generally have a lower rate of change of distance per hour than a car traveling on an interstate freeway. A glass of water placed in a refrigerator will have a lower rate of temperature change than a glass of water placed in a freezer. In calculus, you will study instantaneous rates of change of functions at different values of x . We will study the average rate of change in this book. As you will see in the following examples, the average rate of change can hide a lot of variation.
Examples
Example 1:
- Suppose $1000 was invested in company stock of some manufacturing company. The value of the investment at the beginning of each year is given in Table 3.1.
Table 3.1
|
Year |
Value (in dollars) |
Change from the previous year |
|
1 |
1000 |
New investment of $1000 |
|
2 |
1205 |
Gain of $205 |
|
3 |
1162 |
Loss of $43 |
|
4 |
1025 |
Loss of $137 |
|
5 |
1190 |
Gain of $165 |
|
6 |
1252 |
Gain of $62 |
|
7 |
1434 |
Gain of $182 |
|
8 |
1621 |
Gain of $187 |
|
9 |
2015 |
Gain of $394 |
|
10 |
2845 |
Gain of $830 |
1. How much did the stock increase per year on average from the beginning of Year 3 to the beginning of Year 6?
For this three-year period the investment increased in value from $1162 to $1252. The average rate of change is
2. What was the average annual loss from the beginning of Year 2 to the beginning of Year 5?
The average rate of change during this three-year period is
The negative symbol means that this change is a loss, not a gain.
3. What was the average annual increase over the full period?
The average increase in the investment over the full nine years is
Example 2:
- Find the average rate of change between (−3, 9) and (−1,3) and between (1, 1.5) and (3, 1.125) for the function whose graph is given in Figure 3.13.
-
The average rate of change of a function between two points on the graph is the slope of the line containing the two points. For the points (−3,9) and (−1, 3), x 1 = −3, y 1 = 9 and x 2 = −1 and y 2 = 3.
Average rate of change 
Fig. 3.13
Between x = −3 and x = −1, the y -values of this function decrease by 3 as x increases by 1, on average.
For the points (1, 1.5) and (3, 1.125) x 1 = 1, y 1 = 1.5 and x 2 = 3, y 2 = 1.125.
Average rate of change 
Between x = 1 and x = 3, the y -values of this function decrease on average by 0.1875 as x increases by 1.
Example 3:
- Find the average rate of change of f(x) = −3 x 2 + 10 between x = −1 and x = 2.
-
Once we have found the y -values by putting these x -values into the function, we will find the slope of the line containing these two points.
y 1 = f ( x 1 ) = f (−1) = −3(−1) 2 + 10 = 7
y 2 = f ( x 2 ) = f (2) = −3(2) 2 + 10 = −2
Average rate of change 
Between x = −1 and x = 2, this function decreases on average by 3 as x increases by 1.
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