Question

for $y = f(x)=2x^{2}$, find $delta x,delta y$, and $\frac{delta y}{delta x}$, given $x_1 = 2$ and $x_2 = 4$.

Explanation:

Step1: Calculate $\Delta x$

$\Delta x=x_2 - x_1$. Given $x_1 = 2$ and $x_2=4$, so $\Delta x=4 - 2=2$.

Step2: Calculate $y_1$ and $y_2$

First, find $y_1$ when $x = x_1=2$. Substitute $x = 2$ into $y = f(x)=2x^{2}$, then $y_1=2\times2^{2}=2\times4 = 8$. Second, find $y_2$ when $x = x_2 = 4$. Substitute $x = 4$ into $y = f(x)=2x^{2}$, then $y_2=2\times4^{2}=2\times16 = 32$.

Step3: Calculate $\Delta y$

$\Delta y=y_2 - y_1$. Since $y_1 = 8$ and $y_2 = 32$, then $\Delta y=32 - 8=24$.

Step4: Calculate $\frac{\Delta y}{\Delta x}$

$\frac{\Delta y}{\Delta x}=\frac{y_2 - y_1}{x_2 - x_1}$. We know $\Delta y = 24$ and $\Delta x=2$, so $\frac{\Delta y}{\Delta x}=\frac{24}{2}=12$.

Answer:

$\Delta x = 2$, $\Delta y=24$, $\frac{\Delta y}{\Delta x}=12$