Question

the expression \\( \frac { f ( x + h ) - f ( x ) } { h } \\) for \\( h \
eq 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.)

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 \( h
eq0 \), we can factor out \( h \) from the numerator: \(\frac{h(16x + 8h+7)}{h}=16x + 8h+7\).

Answer:

\(16x + 8h+7\)