Answer :
To estimate the average rate of change of the function [tex]\( f(x) = 1.6875x \)[/tex] between 3.9 seconds and 8.2 seconds, you can follow these steps:
1. Evaluate the function at the given points:
- Find [tex]\( f(3.9) \)[/tex]. Plug 3.9 into the function [tex]\( f(x) = 1.6875x \)[/tex]:
[tex]\[
f(3.9) = 1.6875 \times 3.9 = 6.58125
\][/tex]
- Find [tex]\( f(8.2) \)[/tex]. Plug 8.2 into the function:
[tex]\[
f(8.2) = 1.6875 \times 8.2 = 13.8375
\][/tex]
2. Calculate the average rate of change:
The average rate of change of a function between two points [tex]\( a \)[/tex] and [tex]\( b \)[/tex] is given by:
[tex]\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\][/tex]
- Substitute the values for [tex]\( f(3.9) \)[/tex] and [tex]\( f(8.2) \)[/tex]:
[tex]\[
\text{Average Rate of Change} = \frac{13.8375 - 6.58125}{8.2 - 3.9} = \frac{7.25625}{4.3} = 1.6875
\][/tex]
3. Round the result:
Round the average rate of change to two decimal places:
[tex]\[
1.6875 \approx 1.69
\][/tex]
So, the average rate of change of the elevator's speed between 3.9 seconds and 8.2 seconds is approximately 1.69 feet per second. This matches with the option "about 1.69 feet/second".
1. Evaluate the function at the given points:
- Find [tex]\( f(3.9) \)[/tex]. Plug 3.9 into the function [tex]\( f(x) = 1.6875x \)[/tex]:
[tex]\[
f(3.9) = 1.6875 \times 3.9 = 6.58125
\][/tex]
- Find [tex]\( f(8.2) \)[/tex]. Plug 8.2 into the function:
[tex]\[
f(8.2) = 1.6875 \times 8.2 = 13.8375
\][/tex]
2. Calculate the average rate of change:
The average rate of change of a function between two points [tex]\( a \)[/tex] and [tex]\( b \)[/tex] is given by:
[tex]\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\][/tex]
- Substitute the values for [tex]\( f(3.9) \)[/tex] and [tex]\( f(8.2) \)[/tex]:
[tex]\[
\text{Average Rate of Change} = \frac{13.8375 - 6.58125}{8.2 - 3.9} = \frac{7.25625}{4.3} = 1.6875
\][/tex]
3. Round the result:
Round the average rate of change to two decimal places:
[tex]\[
1.6875 \approx 1.69
\][/tex]
So, the average rate of change of the elevator's speed between 3.9 seconds and 8.2 seconds is approximately 1.69 feet per second. This matches with the option "about 1.69 feet/second".
