Application of Differentials to Approximation

In this tutorial we shall be concerned with the application of differentials to approximate any real problem. Now we consider 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 increases?
(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, so equation (ii) gives

dx = \frac{1}{\pi }\,\,\,\,{\text{ - - - }}\left( {{\text{iii}}} \right)

This shows that the diameter of the tree is 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}}}.