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Application of Derivatives

Short Answer Type

1. Find the rate of change of the area of a circle with respect to its radius r when r = 5 cm.

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Find the rate of change of the area of a circle with respect to its radius r when
(a) r = 3 cm (b) r = 4 cm

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Find the rate of change of the volume of a ball with respect to its radius r. How fast is the volume changing with respect to the radius when the radius is 2 m?

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How fast is the volume of a ball changing with respect to its radius when the radius is 3 m?

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A balloon which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the latter is 10 cm.

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6.
The cost function C(x), in rupees, of producing x items (x ≥ 15) in a certain factory is given by $straight C left parenthesis straight x right parenthesis space equals space 20 plus 10 straight x squared plus 15 over straight x.$ Determine the marginal cost function and the marginal cost of producing 100 items.

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Here, $straight C left parenthesis straight x right parenthesis space equals space 20 plus 10 straight x squared plus 15 over straight x space space space space rightwards double arrow space space space space space space straight C space equals space 20 plus 10 straight x squared plus 15 straight x to the power of negative 1 end exponent$
Marginal cost function $equals space dC over dx space equals space 0 plus 20 straight x minus 15 straight x to the power of negative 2 end exponent space equals space 20 straight x minus space 15 over straight x squared$
When x = 100, marginal cost = 20 (100) - $fraction numerator 15 over denominator left parenthesis 100 right parenthesis squared end fraction$
$equals space 2000 minus 15 over 10000 space equals space left parenthesis 2000 minus.001 right parenthesis space nearly space equals space Rs. space 2000 space nearly.$

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The total cost C(x) in Rupees, associated with the production of x units of an item is given by C(x) = 0.005 x³- 0.02 x²+ 30 x + 5000.
Find the marginal cost when 3 units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output.

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8. The total cost C(x) in Rupees associated with the production of x units of an item is given by C(x) = 0.007 x – 0.003 x² + 15 x + 4000.
Find the marginal cost when 17 units are produced.

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9. The total revenue in Rupees received from the sale of x units of a product is given by R(x) = 3x² +36x + 5. Find the marginal revenue, when x = 5, where by marginal revenue we mean the rate of change of total revenue with respect to the number of items sold at an instant.

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