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 ) = 7 x ^ { 2 } + 8 x + 7 \\)
the difference quotient is \\( \square \\).
(simplify your 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 \( h
eq0 \), we can factor out \( h\) from the numerator: \(\frac{h(14x + 7h+8)}{h}\).
Cancel out the common factor \( h\): \(14x + 7h+8\).

Answer:

\(14x + 7h + 8\)