A circular metal plate expands under heating so that its radius increases by k%. Find the approximate increase in the area of the plate, if the radius of the plate before heating is 10 cm.
Given the radius of a circular plate initially is 10 cm and it increases by k%.
Let x be the radius of the circular plate, and Δx is the change in the value of x.
Hence, we have x = 10 and
∴ Δx = 0.1k
The area of a circular plate of radius x is given by
A = πx2
On differentiating A with respect to x, we get
We know
When x = 10, we have
.
Recall that if y = f(x) and Δx is a small increment in x, then the corresponding increment in y, Δy = f(x + Δx) – f(x), is approximately given as
Here, 
and Δx = 0.1k
⇒ ΔA = (20π)(0.1k)
∴ ΔA = 2kπ
Thus, the approximate increase in the area of the circular plate is 2kπ cm2.