# 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

Taking the differential of the above equation (i), we have

Since the change in the circumference is $dC = 2$, equation (ii) gives

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

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