use the table at the right to find an approximate value of ( f(x) ) at ( x = 9 ).\nuse ( x = 8 ) and ( x =…

use the table at the right to find an approximate value of ( f(x) ) at ( x = 9 ).\nuse ( x = 8 ) and ( x = 10 ) in the difference quotient.\nsubstitute ( f(10) ) and ( f(8) ). simplify the numerator.\nsimplify.\nthe approximate value of ( f(x) ) at ( x = 9 ) is 56.\n\n( \frac { f ( x + delta x ) - f ( x ) } { delta x } = \frac { f ( 10 ) - f ( 8 ) } { 10 - 8 } )\n( = \frac { 319 - 207 } { 2 } )\n( = 56 )

use the table at the right to find an approximate value of ( f(x) ) at ( x = 9 ).\nuse ( x = 8 ) and ( x = 10 ) in the difference quotient.\nsubstitute ( f(10) ) and ( f(8) ). simplify the numerator.\nsimplify.\nthe approximate value of ( f(x) ) at ( x = 9 ) is 56.\n\n( \frac { f ( x + delta x ) - f ( x ) } { delta x } = \frac { f ( 10 ) - f ( 8 ) } { 10 - 8 } )\n( = \frac { 319 - 207 } { 2 } )\n( = 56 )

Answer

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).