Question

simplify the difference quotient \\(\frac{f(x + h) - f(x)}{h}\\) for the given function. \\(f(x) = 7x^2 - 3x + 1\\) \\(f(x + h) = \square\\) \\(\frac{f(x + h) - f(x)}{h} = \square\\) (simplify your answer.)

Explanation:

Step1: Find \( f(x + h) \)

Substitute \( x + h \) into \( f(x) = 7x^2 - 3x + 1 \):
\[

$$\begin{align*} f(x + h)&= 7(x + h)^2 - 3(x + h) + 1\\ &= 7(x^2 + 2xh + h^2) - 3x - 3h + 1\\ &= 7x^2 + 14xh + 7h^2 - 3x - 3h + 1 \end{align*}$$

\]

Step2: Compute \( f(x + h) - f(x) \)

Subtract \( f(x) = 7x^2 - 3x + 1 \) from \( f(x + h) \):
\[

$$\begin{align*} f(x + h) - f(x)&= (7x^2 + 14xh + 7h^2 - 3x - 3h + 1) - (7x^2 - 3x + 1)\\ &= 7x^2 + 14xh + 7h^2 - 3x - 3h + 1 - 7x^2 + 3x - 1\\ &= 14xh + 7h^2 - 3h \end{align*}$$

\]

Step3: Simplify \( \frac{f(x + h) - f(x)}{h} \)

Divide the result by \( h \) ( \( h
eq 0 \) ):
\[

$$\begin{align*} \frac{f(x + h) - f(x)}{h}&=\frac{14xh + 7h^2 - 3h}{h}\\ &=\frac{h(14x + 7h - 3)}{h}\\ &= 14x + 7h - 3 \end{align*}$$

\]

Answer:

\( f(x + h) = 7x^2 + 14xh + 7h^2 - 3x - 3h + 1 \)
\( \frac{f(x + h) - f(x)}{h} = 14x + 7h - 3 \)