# Application of Differentials to Approximation

In this tutorial we shall discuss the application of differentials to approximate any real problem. Let’s look at an example.

The diameter of a tree was 8 inches. After one year the circumference of the tree increased by 2 inches. How much did:

(i) the diameter of the tree increase?
(ii) the cross-section area of the tree change?

Let $x$ be the radius of the tree, then its circumference $C$ is
$C = 2\pi x\,\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right)$

Taking the differential of the above equation (i), we have
$dC = 2\pi dx\,\,\,\,\,{\text{ – – – }}\left( {{\text{ii}}} \right)$

Since the change in the circumference is $dC = 2$, equation (ii) gives
$dx = \frac{1}{\pi }\,\,\,\,{\text{ – – – }}\left( {{\text{iii}}} \right)$

This shows that the diameter of the tree has increased by $\frac{2}{\pi }$ inches.

If the cross-section area of the tree is $A$, then
$\begin{gathered} A = \pi {x^2} \\ \Rightarrow dA = 2\pi xdx \\ \Rightarrow dA = 2\pi \left( 4 \right)\left( {\frac{1}{\pi }} \right) = 8 \\ \end{gathered}$

This shows that the change in the cross-section area of the tree is $8{\text{i}}{{\text{n}}^{\text{2}}}$.