Question

use the table at the right to find an approximate value of ( f(x) ) at ( x = 9 ).
use ( x = 8 ) and ( x = 10 ) in the difference quotient.
substitute ( f(10) ) and ( f(8) ). simplify the numerator.
simplify.
the approximate value of ( f(x) ) at ( x = 9 ) is 56.

( \frac { f ( x + delta x ) - f ( x ) } { delta x } = \frac { f ( 10 ) - f ( 8 ) } { 10 - 8 } )
( = \frac { 319 - 207 } { 2 } )
( = 56 )

Explanation:

Step1: Recall the difference quotient formula

The difference quotient formula for approximating \(f^{\prime}(x)\) is \(f^{\prime}(x)\approx\frac{f(x + \Delta x)-f(x)}{\Delta x}\). Here, \(x = 8\), \(x+\Delta x=10\), so \(\Delta x=10 - 8\).

Step2: Substitute values from the table

From the table, \(f(8) = 207\) and \(f(10)=319\). Then \(\frac{f(10)-f(8)}{10 - 8}=\frac{319-207}{2}\).

Step3: Simplify the fraction

\(\frac{319 - 207}{2}=\frac{112}{2}=56\).

Answer:

The approximate value of \(f^{\prime}(x)\) at \(x = 9\) is \(56\).