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.
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.
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