# 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 $$8{\text{i}}{{\text{n}}^{\text{2}}}$$.