To calculate and compare the rates of change on the intervals \([1,3]\) and \([4,6]\), we need a function to work with. For the sake of example, let's consider a simple linear function:
\[ f(x) = x^2 \]
Step 1: Calculate the rate of change on the interval \([1,3]\)
The formula for the rate of change between two points \(x_1\) and \(x_2\) for a function \(f\) is given by:
\[ \text{Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \]
For the interval \([1, 3]\):
- \(x_1 = 1\)
- \(x_2 = 3\)
Calculating \(f(1)\) and \(f(3)\):
\[ f(1) = 1^2 = 1 \] \[ f(3) = 3^2 = 9 \]
Now insert the values into the rate of change formula:
\[ \text{Rate of Change} = \frac{f(3) - f(1)}{3 - 1} = \frac{9 - 1}{2} = \frac{8}{2} = 4 \]
Step 2: Calculate the rate of change on the interval \([4,6]\)
Now for the interval \([4, 6]\):
- \(x_1 = 4\)
- \(x_2 = 6\)
Calculating \(f(4)\) and \(f(6)\):
\[ f(4) = 4^2 = 16 \] \[ f(6) = 6^2 = 36 \]
Now insert the values into the rate of change formula:
\[ \text{Rate of Change} = \frac{f(6) - f(4)}{6 - 4} = \frac{36 - 16}{2} = \frac{20}{2} = 10 \]
Comparison of Rates of Change
- Rate of Change on \([1, 3]\): \(4\)
- Rate of Change on \([4, 6]\): \(10\)
Observation
- The rate of change increased from the interval \([1, 3]\) to \([4, 6]\). This indicates that the function is increasing at a faster rate in the interval \([4, 6]\) compared to \([1, 3]\).
- For the function \(f(x) = x^2\), this behavior aligns with the fact that it is a quadratic function, where the slope (rate of change) increases as \(x\) increases. This is due to the second derivative being positive, indicating upward concavity.
In conclusion, as we move to higher values of \(x\) in this specific function, the rate of change (slope) increases.
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