Question

if $f(x)=\frac{g(x)}{h(x)}$, then $f(2)=$ question help: video

Explanation:

Step1: Find the slope - intercept form of $g(x)$

The line $g(x)$ passes through $(0,0)$ and $(4,8)$. The slope $m_1=\frac{8 - 0}{4-0}=2$ and $y - intercept = 0$, so $g(x)=2x$. Then $g'(x)=2$.

Step2: Find the slope - intercept form of $h(x)$

For $h(x)$, in the interval $[0,3]$, it passes through $(0,2.5)$ and $(3,5)$. The slope $m_{21}=\frac{5 - 2.5}{3-0}=\frac{2.5}{3}=\frac{5}{6}$. So for $x\in[0,3]$, $h(x)=\frac{5}{6}x + 2.5$. Then $h'(x)=\frac{5}{6}$ for $x\in[0,3]$.

Step3: Use the quotient rule

The quotient rule states that if $f(x)=\frac{g(x)}{h(x)}$, then $f'(x)=\frac{g'(x)h(x)-g(x)h'(x)}{h(x)^2}$.
When $x = 2$:
$g(2)=2\times2 = 4$, $g'(2)=2$, $h(2)=\frac{5}{6}\times2+2.5=\frac{5}{3}+\frac{5}{2}=\frac{10 + 15}{6}=\frac{25}{6}$, $h'(2)=\frac{5}{6}$

Step4: Calculate $f'(2)$

$f'(2)=\frac{g'(2)h(2)-g(2)h'(2)}{h(2)^2}=\frac{2\times\frac{25}{6}-4\times\frac{5}{6}}{(\frac{25}{6})^2}=\frac{\frac{50 - 20}{6}}{\frac{625}{36}}=\frac{\frac{30}{6}}{\frac{625}{36}}=\frac{5}{\frac{625}{36}}=5\times\frac{36}{625}=\frac{36}{125}=0.288$

Answer:

$0.288$