MHB -z.57 dx formula that estimates the change

[karush]

Write a differential formula that estimates the change in the volume
$V=\frac{4}{3}\pi{r}^{3}$ of a sphere when the radius changes from $r_0$ to $r_0+dr$

$$dV=4\pi{r}_{0}^{2}dr$$

this was one of the selections but I didn't know how to account for the
$r_0$ to $r_0+dr$

 

[Prove It]

Write a differential formula that estimates the change in the volume
$V=\frac{4}{3}\pi{r}^{3}$ of a sphere when the radius changes from $r_0$ to $r_0+dr$

$$dV=4\pi{r}_{0}^{2}dr$$

this was one of the selections but I didn't know how to account for the
$r_0$ to $r_0+dr$

Recall the formula for a derivative (instantaneous rate of change) from first principles:

$\displaystyle \begin{align*} f'(x) = \lim_{\Delta x \to 0} \frac{f \left( x + \Delta x \right) - f(x)}{\Delta x} \end{align*}$

so this means that as long as $\displaystyle \begin{align*} \Delta x \end{align*}$ is small

$\displaystyle \begin{align*} f'(x) &\approx \frac{f\left( x + \Delta x \right) - f(x)}{\Delta x} \\ f \left( x + \Delta x \right) - f(x) &\approx \Delta x \, f'(x) \\ \Delta f &\approx \Delta x \, f'(x) \end{align*}$

Here treat V as f and r as x. "dr" is just a small change in r.