If data are evenly spaced, we need only calculate differences to see whether the data are linear. But if data are not evenly spaced, then we must calculate the average rate of change over each interval to see whether the data are linear. If the average rate of change is constant, it is the slope of the linear function.In the following table, show that the average rate of change is the same over each interval. This shows the data are linear, even though the differences in y are not constant. **Understanding Linear Data through Average Rate of Change**When data points are evenly spaced along the x-axis, determining if the data follows a linear trend is simplified by calculating the differences between y-values. For unevenly spaced data, calculating the average rate of change over each interval helps analyze linearity. This average rate of change equates to the slope of the linear function when constant.**Example Table: Linear Data Verification**The table below demonstrates that the average rate of change remains consistent across various intervals, which confirms the data's linear nature, even if y-value differences vary.| x | 1 | 2 | 5 | 6 ||---|---|---|---|---|| y | 5 | 9 | 21 | 25 |**Average Rates of Change:**- From 1 to 2: 4 ✔️- From 2 to 5: 4 ✖️- From 5 to 6: 4 ✔️The consistent average rate of change of 4 indicates that the y-values increase uniformly, confirming a linear relationship.**Defining the Linear Model for the Data:**A linear equation represents this relationship. Based on the average rate of change (slope), the equation of the line can be expressed as:\[ y = \]*Conclusion:* Data with a consistent average rate of change across intervals is linear. Utilize this method to validate linearity when differences in y-values are inconsistent.

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If data are evenly spaced, we need only calculate differences to see whether the data are linear. But if data are not evenly spaced, then we must calculate the average rate of change over each interval to see whether the data are linear. If the average rate of change is constant, it is the slope of the linear function. In the following table, show that the average rate of change is the same over each interval. This shows the data are linear, even though the differences in y are not constant. x125 59 21 25 average rate of change from 1 to 2 average rate of change from 2 to 5 average rate of change from 5 to 6 Find a linear model for the data. y 4 X

If data are evenly spaced, we need only calculate differences to see whether the data are linear. But if data are not evenly spaced, then we must calculate the average rate of change over each interval to see whether the data are linear. If the average rate of change is constant, it is the slope of the linear function. In the following table, show that the average rate of change is the same over each interval. This shows the data are linear, even though the differences in y are not constant. x125 59 21 25 average rate of change from 1 to 2 average rate of change from 2 to 5 average rate of change from 5 to 6 Find a linear model for the data. y 4 X

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

If data are evenly spaced, we need only calculate differences to see whether the data are linear. But if data are not evenly spaced, then we must calculate the average rate of change over each interval to see whether the data are linear. If the average rate of change is constant, it is the slope of the linear function.

In the following table, show that the average rate of change is the same over each interval. This shows the data are linear, even though the differences in y are not constant.

**Understanding Linear Data through Average Rate of Change**

When data points are evenly spaced along the x-axis, determining if the data follows a linear trend is simplified by calculating the differences between y-values. For unevenly spaced data, calculating the average rate of change over each interval helps analyze linearity. This average rate of change equates to the slope of the linear function when constant.

**Example Table: Linear Data Verification**

The table below demonstrates that the average rate of change remains consistent across various intervals, which confirms the data's linear nature, even if y-value differences vary.

| x | 1 | 2 | 5 | 6 |
|---|---|---|---|---|
| y | 5 | 9 | 21 | 25 |

**Average Rates of Change:**
- From 1 to 2: 4 ✔️
- From 2 to 5: 4 ✖️
- From 5 to 6: 4 ✔️

The consistent average rate of change of 4 indicates that the y-values increase uniformly, confirming a linear relationship.

**Defining the Linear Model for the Data:**

A linear equation represents this relationship. Based on the average rate of change (slope), the equation of the line can be expressed as:

\[ y = \]

*Conclusion:*

Data with a consistent average rate of change across intervals is linear. Utilize this method to validate linearity when differences in y-values are inconsistent.

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