Answer :
To find the average rate of change of the function [tex]\( f(x) = \sqrt{3x + 1} \)[/tex] from [tex]\( x_1 = 2 \)[/tex] to [tex]\( x_2 = 4 \)[/tex], we can follow these steps:
1. Evaluate the function at [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex] to find [tex]\( f(x_1) \)[/tex] and [tex]\( f(x_2) \)[/tex]:
- [tex]\( f(x_1) = f(2) \)[/tex]
[tex]\[ f(2) = \sqrt{3 \cdot 2 + 1} = \sqrt{6 + 1} = \sqrt{7} \approx 2.6457513110645907 \][/tex]
- [tex]\( f(x_2) = f(4) \)[/tex]
[tex]\[ f(4) = \sqrt{3 \cdot 4 + 1} = \sqrt{12 + 1} = \sqrt{13} \approx 3.605551275463989 \][/tex]
2. Calculate the difference in the function values:
[tex]\[ f(x_2) - f(x_1) = f(4) - f(2) \approx 3.605551275463989 - 2.6457513110645907 = 0.9597999643993983 \][/tex]
3. Calculate the difference in the input values:
[tex]\[ x_2 - x_1 = 4 - 2 = 2 \][/tex]
4. Compute the average rate of change:
[tex]\[ \text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \approx \frac{0.9597999643993983}{2} = 0.4798999821996992 \][/tex]
5. Round the average rate of change to the nearest hundredth:
The average rate of change to the nearest hundredth is:
[tex]\[ \approx 0.48 \][/tex]
Thus, the average rate of change of [tex]\( f(x) = \sqrt{3x + 1} \)[/tex] from [tex]\( x_1=2 \)[/tex] to [tex]\( x_2=4 \)[/tex], rounded to the nearest hundredth, is [tex]\( 0.48 \)[/tex].
1. Evaluate the function at [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex] to find [tex]\( f(x_1) \)[/tex] and [tex]\( f(x_2) \)[/tex]:
- [tex]\( f(x_1) = f(2) \)[/tex]
[tex]\[ f(2) = \sqrt{3 \cdot 2 + 1} = \sqrt{6 + 1} = \sqrt{7} \approx 2.6457513110645907 \][/tex]
- [tex]\( f(x_2) = f(4) \)[/tex]
[tex]\[ f(4) = \sqrt{3 \cdot 4 + 1} = \sqrt{12 + 1} = \sqrt{13} \approx 3.605551275463989 \][/tex]
2. Calculate the difference in the function values:
[tex]\[ f(x_2) - f(x_1) = f(4) - f(2) \approx 3.605551275463989 - 2.6457513110645907 = 0.9597999643993983 \][/tex]
3. Calculate the difference in the input values:
[tex]\[ x_2 - x_1 = 4 - 2 = 2 \][/tex]
4. Compute the average rate of change:
[tex]\[ \text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \approx \frac{0.9597999643993983}{2} = 0.4798999821996992 \][/tex]
5. Round the average rate of change to the nearest hundredth:
The average rate of change to the nearest hundredth is:
[tex]\[ \approx 0.48 \][/tex]
Thus, the average rate of change of [tex]\( f(x) = \sqrt{3x + 1} \)[/tex] from [tex]\( x_1=2 \)[/tex] to [tex]\( x_2=4 \)[/tex], rounded to the nearest hundredth, is [tex]\( 0.48 \)[/tex].
