Rate Measure Approximations & Increasing-Decreasing Functions

1 Marks Questions

1. The amount of pollution content added in air in a city due to x diesel vehicles is given by

<math>P(x) = 0.005x^3 + 0.02x^2 + 30x</math>

Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above question? Delhi 2013; Value Based Question

Given, <math>P(x) = 0.005x^3 + 0.02x^2 + 30x</math>

On differentiating both sides w.r.t. x, we get

<math>P'(x) = 3 \times 0.005x^2 + 2(0.02)x + 30</math>

On putting <math>x = 3</math>, we get

<math>P'(3) = 3 \times 0.005 \times 9 + 2(0.02)(3) + 30</math> <math>= 0.135 + 0.12 + 30 = 30.255 (1/2)</math>

Pollution content in the city increases with the increase in number of diesel vehicles. (1/2)

2. The total cost C(x) associated with provision of free mid-day meals to x students of a school in primary classes is given by <math display="block">C(x) = 0.005x^3 - 0.02x^2 + 30x + 50.</math> If the marginal cost is given by rate of change <math>\frac{dC}{dx}</math> of total cost, then write the marginal cost of food for 300 students. What value is shown here? Delhi 2013C; Value Based Question

Given, <math>C(x) = 0.005x^3 - 0.02x^2 + 30x + 50</math>

On differentiating both sides w.r.t. x, we get

<math display="block">\frac{dC}{dx} = 0.005(3x^2) - 0.02(2x) + 30</math>

<math>=0.015x^2-0.04x+30</math>

On putting <math>x = 300</math>, we get

<math display="block">\frac{dC}{dx} = 0.015(300)^2 - 0.04(300) + 30</math>

<math>= 1350 - 12 + 30 = 1368</math> (1/2)

By providing free mid-day meals to students of primary classes, care and concern is shown towards their health and nutritional status. (1/2)

1. The money to be spent for the welfare of the employees of a firm is proportional to the rate of change of its total revenue (marginal revenue)

If the total revenue (in ₹) received from the sale of x units of a product is given by <math>R(x) = 3x^2 + 36x + 5</math>, then find the marginal revenue, when <math>x = 5</math> and write which value does the question indicate.

All India 2012; Value Based Question

We know that, marginal revenue is the rate of change of total revenue with respect to the number of units sold.

: Marginal revenue (MR) = <math>\frac{dR}{dx}</math>

<math display="block">= \frac{d}{dx} (3x^2 + 36x + 5) = 6x + 36</math>

When <math>x = 5</math>, then

<math>MR = 6(5) + 36 = 30 + 36 = 66</math>

Hence, the required marginal revenue is ₹ 66.

(1/2)

More amount of money spent for the welfare of the employees with the increase in marginal revenue. (1/2)