Question

1. -/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

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

\[

$$\begin{align*} \frac{f(5 + h)-f(5)}{h}&=\frac{(h^{2}+8h + 19)-19}{h}\\ &=\frac{h^{2}+8h}{h}\\ &=h + 8 \end{align*}$$

\]

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:

\(h + 8\)