Answer :
To approximate the change in the function [tex]\( f(x) = x^{3/4} \)[/tex] as [tex]\( x \)[/tex] changes from 81 to 83, we need to find the difference in the function's value at these two points.
1. Evaluate [tex]\( f(81) \)[/tex]:
[tex]\[
f(81) = 81^{3/4}
\][/tex]
Calculating [tex]\( 81^{3/4} \)[/tex], we find that:
[tex]\[
81^{3/4} = (81^{1/4})^3
\][/tex]
Here, [tex]\( 81^{1/4} \)[/tex] is the fourth root of 81, which equals 3, because [tex]\( 3^4 = 81 \)[/tex]. Therefore:
[tex]\[
(81^{1/4})^3 = 3^3 = 27
\][/tex]
Hence, [tex]\( f(81) = 27 \)[/tex].
2. Evaluate [tex]\( f(83) \)[/tex]:
[tex]\[
f(83) = 83^{3/4}
\][/tex]
Calculating [tex]\( 83^{3/4} \)[/tex] is slightly more complex because 83 is not a perfect fourth power, but with some numerical methods or approximation techniques, we determine:
[tex]\[
83^{3/4} \approx 27.4985
\][/tex]
So, [tex]\( f(83) \approx 27.4985 \)[/tex].
3. Calculate the change in [tex]\( f \)[/tex]:
The change in the function [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] changes from 81 to 83 is:
[tex]\[
\Delta f = f(83) - f(81)
\][/tex]
[tex]\[
\Delta f \approx 27.4985 - 27 = 0.4985
\][/tex]
Therefore, the approximate change in the function [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] changes from 81 to 83 is approximately [tex]\( 0.4985 \)[/tex], which is closest to the option [tex]\(\frac{1}{2}\)[/tex] among the given choices.
1. Evaluate [tex]\( f(81) \)[/tex]:
[tex]\[
f(81) = 81^{3/4}
\][/tex]
Calculating [tex]\( 81^{3/4} \)[/tex], we find that:
[tex]\[
81^{3/4} = (81^{1/4})^3
\][/tex]
Here, [tex]\( 81^{1/4} \)[/tex] is the fourth root of 81, which equals 3, because [tex]\( 3^4 = 81 \)[/tex]. Therefore:
[tex]\[
(81^{1/4})^3 = 3^3 = 27
\][/tex]
Hence, [tex]\( f(81) = 27 \)[/tex].
2. Evaluate [tex]\( f(83) \)[/tex]:
[tex]\[
f(83) = 83^{3/4}
\][/tex]
Calculating [tex]\( 83^{3/4} \)[/tex] is slightly more complex because 83 is not a perfect fourth power, but with some numerical methods or approximation techniques, we determine:
[tex]\[
83^{3/4} \approx 27.4985
\][/tex]
So, [tex]\( f(83) \approx 27.4985 \)[/tex].
3. Calculate the change in [tex]\( f \)[/tex]:
The change in the function [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] changes from 81 to 83 is:
[tex]\[
\Delta f = f(83) - f(81)
\][/tex]
[tex]\[
\Delta f \approx 27.4985 - 27 = 0.4985
\][/tex]
Therefore, the approximate change in the function [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] changes from 81 to 83 is approximately [tex]\( 0.4985 \)[/tex], which is closest to the option [tex]\(\frac{1}{2}\)[/tex] among the given choices.
