Question

2.3 part ii homework - derivatives using limits
score: 10/80 answered: 1/8
question 2
use the four - step definition of the derivative to find $f(x)$ if $f(x)=6x^{2}-2x - 18$.
$f(x + h)=$
$f(x + h)-f(x)=$
$\frac{f(x + h)-f(x)}{h}=$
find $f(x)$ by determining $\lim\limits_{h\to0}\frac{f(x + h)-f(x)}{h}=$
question help: video

Explanation:

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

Substitute \( x + h \) into \( f(x) = 6x^2 - 2x - 18 \).
\[

$$\begin{align*} f(x + h) &= 6(x + h)^2 - 2(x + h) - 18 \\ &= 6(x^2 + 2xh + h^2) - 2x - 2h - 18 \\ &= 6x^2 + 12xh + 6h^2 - 2x - 2h - 18 \end{align*}$$

\]

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

Subtract \( f(x) = 6x^2 - 2x - 18 \) from \( f(x + h) \).
\[

$$\begin{align*} f(x + h) - f(x) &= (6x^2 + 12xh + 6h^2 - 2x - 2h - 18) - (6x^2 - 2x - 18) \\ &= 6x^2 + 12xh + 6h^2 - 2x - 2h - 18 - 6x^2 + 2x + 18 \\ &= 12xh + 6h^2 - 2h \end{align*}$$

\]

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

Divide \( f(x + h) - f(x) \) by \( h \).
\[
\frac{f(x + h) - f(x)}{h} = \frac{12xh + 6h^2 - 2h}{h} = 12x + 6h - 2
\]

Step4: Find \( \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \)

Take the limit as \( h \) approaches 0.
\[
\lim_{h \to 0} (12x + 6h - 2) = 12x - 2
\]

Answer:

• \( f(x + h) = 6x^2 + 12xh + 6h^2 - 2x - 2h - 18 \)
• \( f(x + h) - f(x) = 12xh + 6h^2 - 2h \)
• \( \frac{f(x + h) - f(x)}{h} = 12x + 6h - 2 \)
• \( f'(x) = 12x - 2 \)