Question

Find approximate value of the $(4.04)^3$

Answer 1

65.92

Answer 2

To find the approximate value of $(4.04)^3$, we use the concept of approximation using differentials.

Let $f(x) = x^3$. We want to find the approximate value of $f(4.04)$.
We can write $4.04$ as $x + \Delta x$, where $x = 4$ and $\Delta x = 0.04$.

The formula for approximation using differentials is:
$f(x + \Delta x) \approx f(x) + f'(x) \Delta x$

First, find $f(x)$ at $x=4$:
$f(4) = 4^3 = 64$.

Next, find the derivative of $f(x)$:
$f'(x) = \frac{d}{dx}(x^3) = 3x^2$.

Now, find $f'(x)$ at $x=4$:
$f'(4) = 3(4^2) = 3(16) = 48$.

Substitute these values into the approximation formula:
$f(4.04) \approx f(4) + f'(4) \times 0.04$
$(4.04)^3 \approx 64 + 48 \times 0.04$

Calculate the product $48 \times 0.04$:
$48 \times 0.04 = 48 \times \frac{4}{100} = \frac{192}{100} = 1.92$.

Now, add this to $f(4)$:
$(4.04)^3 \approx 64 + 1.92 = 65.92$.

Thus, the approximate value of $(4.04)^3$ is $65.92$.