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

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

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

Answer

Explanation:

Step1: Find ( f(x + h) )

Given ( f(x)=7x^{2}+8x + 7 ), then ( f(x + h)=7(x + h)^{2}+8(x + h)+7 ). Using the formula ( (a + b)^{2}=a^{2}+2ab + b^{2} ), we have ( f(x + h)=7(x^{2}+2xh+h^{2})+8x + 8h+7=7x^{2}+14xh+7h^{2}+8x + 8h+7 ).

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

( f(x + h)-f(x)=(7x^{2}+14xh+7h^{2}+8x + 8h+7)-(7x^{2}+8x + 7) ). Expand the expression: ( f(x + h)-f(x)=7x^{2}+14xh+7h^{2}+8x + 8h+7 - 7x^{2}-8x - 7 ). Simplify by combining like - terms: ( f(x + h)-f(x)=14xh+7h^{2}+8h ).

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

(\frac{f(x + h)-f(x)}{h}=\frac{14xh+7h^{2}+8h}{h}). Since ( h\neq0 ), we can factor out ( h) from the numerator: (\frac{h(14x + 7h+8)}{h}). Cancel out the common factor ( h): (14x + 7h+8).

Answer:

(14x + 7h + 8)