Question:
The amount of pollution content added in air in a city due to $x$ diesel vehicles is given by $P(x)=0.005 x^{3}+0.02 x^{2}+30 x$. Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above questions.
Solution:
Since, marginal increase in the pollution content is the rate of change of total pollution with respect to the number of diesel vehicles, we have
Marginal increase in pollution $=\frac{\mathrm{d} P}{\mathrm{~d} x}=0.015 x^{2}+0.04 x+30$
When $x=3$, marginal increase in pollution $=0.015(9)+0.04(3)+30=0.135+0.12+30=30.255$
Hence, the required marginal increase in pollution is $30.255$ units.
It indicates the pollution level due to $x$ diesel vehicles.