Question

Find the approximate change in the surface area of a cube of side x metre caused by decreasing the side by 1%.

Answer 1

We know that the surface area of a cube is given by $S=6x^2$
$\Rightarrow$ $\frac{dS}{dx}=12x$ (Differentiate w.r.t,x)
Now, change in surface area, $\Delta S=\left(\frac{dS}{dx}\right)\Delta x$
$=\left(12x\right)\Delta x=12x(-0.01)x(as\Delta x=-1\%ofx=-0.01x)=-0.12x^2{m}^2$
Hence, the approximate change in the surface area of the cube is $0.12x^2{m}^2$
Note It the rate of change is positive, then it is increasing and if the rate of change is negative, then it is decreasing.