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

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)