40. -/1 points find the difference quotient and simplify your answer. f(x)=x² - 2x + 4, (f(5 + h)-f(5))/h, h≠0 41. -/1 points find the difference quotient and simplify your answer. f(x)=x³ + 3x, (f(x + h)-f(x))/h, h≠0

Question

Accepted Answer

### Explanation:
#### Step1: Find $f(5 + h)$
$$
\begin{align*}
f(x)&=x^{2}-2x + 4\
f(5 + h)&=(5 + h)^{2}-2(5 + h)+4\
&=25+10h+h^{2}-10 - 2h+4\
&=h^{2}+8h + 19
\end{align*}
$$
#### Step2: Find $f(5)$
$$
\begin{align*}
f(5)&=5^{2}-2\times5 + 4\
&=25-10 + 4\
&=19
\end{align*}
$$
#### Step3: Calculate the difference - quotient
$$
\begin{align*}
\frac{f(5 + h)-f(5)}{h}&=\frac{(h^{2}+8h + 19)-19}{h}\
&=\frac{h^{2}+8h}{h}\
&=h + 8
\end{align*}
$$
### Answer:
$h + 8$

### Explanation for second - part:
#### Step1: Find $f(x + h)$
$$
\begin{align*}
f(x)&=x^{3}+3x\
f(x + h)&=(x + h)^{3}+3(x + h)\
&=x^{3}+3x^{2}h+3xh^{2}+h^{3}+3x+3h
\end{align*}
$$
#### Step2: Calculate the difference - quotient
$$
\begin{align*}
\frac{f(x + h)-f(x)}{h}&=\frac{(x^{3}+3x^{2}h+3xh^{2}+h^{3}+3x+3h)-(x^{3}+3x)}{h}\
&=\frac{x^{3}+3x^{2}h+3xh^{2}+h^{3}+3x+3h - x^{3}-3x}{h}\
&=\frac{3x^{2}h+3xh^{2}+h^{3}+3h}{h}\
&=3x^{2}+3xh+h^{2}+3
\end{align*}
$$
### Answer:
$3x^{2}+3xh+h^{2}+3$

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QUESTION IMAGE

Question

Explanation:

Step1: Find \(f(5 + h)\)

Step2: Find \(f(5)\)

Step3: Calculate the difference - quotient

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

Step2: Calculate the difference - quotient

Answer: