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 be the radius of the tree, then its circumference is
Taking the differential of the above equation (i), we have
Since the change in the circumference is , equation (ii) gives
This shows that the diameter of the tree has increased by inches.
If the cross-section area of the tree is , then
This shows that the change in the cross-section area of the tree is .