-z.57 dx formula that estimates the change

In summary, the differential formula for estimating the change in volume of a sphere with radius $r_0$ to $r_0+dr$ is $dV = 4\pi r_0^2 dr$, which can be derived using first principles of derivatives. This allows us to approximate the change in volume as $\Delta V \approx \Delta x \, f'(x)$.
  • #1
karush
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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 $r_0+dr$
 
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  • #2
karush said:
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.
 

FAQ: -z.57 dx formula that estimates the change

1. What is the -z.57 dx formula used for?

The -z.57 dx formula is used to estimate the change in a variable based on the change in another variable. It is commonly used in scientific studies to analyze data and make predictions.

2. How is the -z.57 dx formula calculated?

The -z.57 dx formula is calculated by multiplying the change in the independent variable (dx) by a constant value of -z.57. This value is based on the standard normal distribution and is used to estimate the change in the dependent variable.

3. What does the -z.57 value represent in the formula?

The -z.57 value in the formula represents the standard deviation of the normal distribution. It is used to determine the probability of an event occurring within a certain range of values.

4. Can the -z.57 dx formula be used for any type of data?

The -z.57 dx formula is most commonly used for continuous data, but it can also be applied to discrete data. However, the data should follow a normal distribution for the formula to provide accurate estimates.

5. How accurate are the estimates provided by the -z.57 dx formula?

The accuracy of the estimates provided by the -z.57 dx formula depends on the data and assumptions made. It is important to consider the limitations and potential sources of error when using this formula for analysis.

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