A stone is dropped into a quiet lake and | Class 12 Mathematics Chapter Application of Derivatives, Application of Derivatives NCERT Solutions

The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of increase of its circumference?

The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rate at which the area of the circle is increasing when the radius is 10 cm.

The total cost C (x) in Rupees associated with the production of x units of an item is given by

C(X) = 0.007 x³ - 0.003x² + 15x + 4000 

Find the marginal cost when 17 units are produced.

The total revenue in Rupees received from the sale of x units of a product is given by

R (x) = 3x² + 36x + 5. The marginal revenue, when x = 15 is

(A) 116 (B) 96 (C) 90 (D) 126

The total revenue in Rupees received from the sale of x units of a product is given by

R (x) = 13x² + 26x + 15

Find the marginal revenue when x = 7.

The radius of an air bubble is increasing at the rate of 1/2 cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?

The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x = 8 cm and y = 6 cm, find the rates of change of (a) the perimeter, and (b) the area of the rectangle.

An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the cube increasing when the edge is 10 cm long?

Sand is pouring from a pipe at the rate of 12 cm³/s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?

A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall?

In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.

(i) f : R → R defined by f(x) = 3 – 4x

(ii) f : R → R defined by f(x) = 1 + x² 

 Show that the Modulus Function f : R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative.

 Prove that the Greatest Integer Function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

Show that the function f : R_* → R_* defined by f(x) = 1/x is one-one and onto,where R_* is the set of all non-zero real numbers. Is the result true, if the domain R_* is replaced by N with co-domain being same as R_* ? 

In Figure, identify the following vectors.

(i) Coinitial (ii) Equal (iii) Collinear but not equal

 Let f : {1, 3, 4} → {1, 2, 5} and g : {1, 2, 5} → {1, 3} be given by f = {(1, 2), (3, 5), (4, 1)} and g = {(1, 3), (2, 3), (5, 1)}. Write down gof. 

Classify the following as scalar and vector quantities.

(i) time period (ii) distance (iii) force

(iv) velocity (v) work done