this is wrong. but can you explain how to correct it? thanks
use differential of x^-1/3 to approximate solution to (1/cube root(7.952))
differential: df= -1/3(x^(-4/3)) * dx
df= -1/48 * 0.0048
df= -0.001
0.500 - 0.001 = 0.499
but this is wrong.
dx = -0.048
df = 0.001
This is also what I entered, yet it is wrong
To correct the mistake in your calculation, let's start by reviewing the process.
You want to use the differential of \(x^{-1/3}\) to approximate the solution to \(\frac{1}{\sqrt[3]{7.952}}\).
First, let's find the differential of \(x^{-1/3}\):
\(df = -\frac{1}{3}x^{-4/3}dx\).
Now, to approximate the solution, we need to substitute the values into the differential equation. However, it seems like you made a mistake in the substitution. Let's correct it:
\(df = -\frac{1}{3}(0.0048)^{-4/3} \cdot 0.0048\)
To simplify this:
\(df = -\frac{1}{3}(\frac{1}{0.0048^{4/3}}) \cdot 0.0048\)
\(df = -\frac{1}{3}(\frac{1}{0.1153}) \cdot 0.0048\)
\(df = -\frac{1}{3}(8.67) \cdot 0.0048\)
\(df \approx -0.0144\) (rounded to four decimal places)
Now, to find the approximation, we subtract the differential from the original value:
\(0.500 - 0.0144 \approx 0.4856\) (rounded to four decimal places)
Hence, the corrected approximation is approximately 0.4856.