the expression \\( \\frac { f ( x + h ) - f ( x ) } { h } \\) for \\( h \\neq 0 \\) is called the difference…

the expression \\( \\frac { f ( x + h ) - f ( x ) } { h } \\) for \\( h \\neq 0 \\) is called the difference quotient. find and simplify the difference quotient for the following function. \\( f ( x ) = 8 x ^ { 2 } + 7 x + 5 \\) the difference quotient is \\( \\square \\). (simplify your answer.)

the expression \\( \\frac { f ( x + h ) - f ( x ) } { h } \\) for \\( h \\neq 0 \\) is called the difference quotient. find and simplify the difference quotient for the following function. \\( f ( x ) = 8 x ^ { 2 } + 7 x + 5 \\) the difference quotient is \\( \\square \\). (simplify your answer.)

Answer

Explanation:

Step1: Find ( f(x + h) )

Given ( f(x)=8x^{2}+7x + 5 ), then ( f(x + h)=8(x + h)^{2}+7(x + h)+5 ). Using the formula ( (a + b)^{2}=a^{2}+2ab + b^{2} ), we have ( f(x + h)=8(x^{2}+2xh+h^{2})+7x + 7h+5=8x^{2}+16xh+8h^{2}+7x + 7h+5 ).

Step2: Calculate ( f(x + h)-f(x) )

( f(x + h)-f(x)=(8x^{2}+16xh+8h^{2}+7x + 7h+5)-(8x^{2}+7x + 5) ) ( =8x^{2}+16xh+8h^{2}+7x + 7h+5 - 8x^{2}-7x - 5=16xh+8h^{2}+7h ).

Step3: Find the difference quotient (\frac{f(x + h)-f(x)}{h})

(\frac{f(x + h)-f(x)}{h}=\frac{16xh+8h^{2}+7h}{h}), since ( h\neq0 ), we can factor out ( h ) from the numerator: (\frac{h(16x + 8h+7)}{h}=16x + 8h+7).

Answer:

(16x + 8h+7)