Question

2.3 part ii homework - derivatives using limits
score: 30/80 answered: 3/8
question 4
use the limit definition of the derivative to answer the following questions about $f(x) = 2x^2$.
find the difference quotient $\frac{f(x + h) - f(x)}{h}$
find $f(x)$ by determining $\lim\limits_{h \to 0} \frac{f(x + h) - f(x)}{h}$.
find $f(-2)$
find $f(0)$
find $f(1)$

Explanation:

1. Find the difference quotient $\frac{f(x + h) - f(x)}{h}$

Step 1: Compute $f(x + h)$

Given $f(x)=2x^{2}$, substitute $x$ with $x + h$:
$f(x + h)=2(x + h)^{2}=2(x^{2}+2xh + h^{2}) = 2x^{2}+4xh + 2h^{2}$

Step 2: Compute $f(x + h)-f(x)$

Subtract $f(x)$ from $f(x + h)$:
$f(x + h)-f(x)=(2x^{2}+4xh + 2h^{2})-2x^{2}=4xh + 2h^{2}$

Step 3: Divide by $h$ ( $h

eq0$)
$\frac{f(x + h)-f(x)}{h}=\frac{4xh + 2h^{2}}{h}=\frac{h(4x + 2h)}{h}=4x + 2h$

2. Find $f'(x)$ by $\lim\limits_{h

ightarrow0}\frac{f(x + h)-f(x)}{h}$

Step 1: Use the difference quotient result

From part 1, $\frac{f(x + h)-f(x)}{h}=4x + 2h$.

Step 2: Take the limit as $h

ightarrow0$
$\lim\limits_{h
ightarrow0}(4x + 2h)=4x+2(0)=4x$

3. Find $f'(-2)$

Step 1: Use $f'(x)=4x$

Substitute $x = -2$ into $f'(x)$:
$f'(-2)=4(-2)=-8$

4. Find $f'(0)$

Answer:

s:

• Difference quotient: $\boldsymbol{4x + 2h}$
• $f'(x)$: $\boldsymbol{4x}$
• $f'(-2)$: $\boldsymbol{-8}$
• $f'(0)$: $\boldsymbol{0}$
• $f'(1)$: $\boldsymbol{4}$