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Question:

The amount of pollution content added in air in a city due to diesel vehicles is given by P(x) = 0.005x³ + 0.02x² + 30x. Find the marginal increase in pollution content when 3 diesel vehicles are added.

Solution:

P(x) = 0.005x³ + 0.02x² + 30x
Differentiating w.r.t to x we get,
P'(x) = 3(0.005)x² + 2(0.02)x + 30
P'(3) = 3 × 0.005(3)² + 2(0.02)(3) + 30
P'(3) = 0.135 + 0.12 + 30
P'(3) = 30.255