Math Assignment Class XII Ch -06 | Application of Derivatives

 Math Assignment  Class XII  Ch - 09 

 Differential Equations

Extra questions of chapter 09 Applications of Derivatives, class XII  with answers and  hints to the difficult questions, strictly according to the CBSE Board . Important and useful math. assignment for the students of class XII

MATHEMATICS ASSIGNMENT OF CHAPTER 06

STRICTLY ACCORDING TO THE PREVIOUS CBSE SAMPLE QUESTION PAPERS AND CBSE BOARD PAPERS

RATE OF CHANGE OF QUANTITIES

Question 1

A particle moves along the curve x2 = 2y. At what point, ordinate increases at the same rate as abscissa increases?

Answer (1, 1/2)

Solution Hint:

Differentiating the given equation w. r. t. t

Now putting dy/dt = dx/dt and find the value of x.

Putting the value of x in the given equation and find the value of y

Question 2

The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of triangle is  20 cm?

Answer: 20√3 cm²/s

Question 3

Volume of sphere is increasing at the rate of 3cm³/s. Find the rate of increase of surface area, when radius is 2 cm

Answer: 3cm²/s

Question 4

For the curve y = 5x – 2x³. If x increases at the rate of 2 unit/sec., find the rate of change of the slope of the curve when x = 3.

Answer: -72 unit /sec

Question 5

A man 2m high walks at a uniform speed of 6km/h away from the lamp post 6m high. Find the rate at which the length of his shadow increases.

Answer : 3km/h

Solution Hint

AB = 6m,  CD = 2m  dy / dt = 6 km/h,  

Let AC = y,  CE = x,  AE = x + y,  

To find : dx/dt = ?

△ABE ~ △ CDE
∴

⇒ y = 2x
Differentiating on both side w.r.t. t we get

INCREASING & DECREASING OF FUNCTIONS

Question 6

Without using derivative show that the function f(x) = 4x³ – 18x² + 27x – 7 is always increasing in R

 Question 7

Find the interval in which the function f(x)  is (i) Strictly Increasing (ii) Strictly Decreasing

Answer

Strictly increasing in (-∞, -2)⋃(6, ∞).

Strictly Decreasing in (-2, 6)

Question 8

Find the intervals in which the function f(x) = x⁴ / 4 – x³ – 5x² + 24x + 12 is

i) Strictly increasing   ii) Strictly decreasing

Answer: 

Strictly increasing in (-3, 2)⋃(4, ∞).

Strictly Decreasing in (-∞, 3)⋃(2, 4)

Question 9

Find the interval in which the function:

f(x) = sin⁴x + cos⁴x,   0 ≤ x < π/2

is strictly increasing or strictly decreasing.

Answer:

Strictly increasing in (π/4, π/2)

Strictly decreasing in (0, π/4)

Question 10

Find the intervals in which  f(x) = sin3x - cos3x, 0<x<π, is strictly increasing or strictly decreasing.

Answer:

Strictly increasing in (0, π/4) ⋃ (7π/12, 11π/12)

Strictly decreasing in (π/4, 7π/12) ⋃ (11π/12, π)

Question 11

Prove that f(x) = x² -x + 1 is neither increasing nor decreasing strictly on (-1, 1)

Question 12  

Find the interval in which the following functions are strictly increasing and strictly decreasing :  f(x) = 4x³ – 6x² - 72x + 30

Answer: 

Strictly increasing in (-∞, -2)⋃(3, ∞).

Strictly Decreasing in (-2, 3)

Question 13 

Find the interval in which the following functions are strictly increasing and strictly decreasing :  f(x) = 2x³ – 12x² + 18x + 5

Answer:

Strictly increasing in (-∞, 1)⋃(3, ∞).

Strictly Decreasing in (1, 3)

Question 14:  

Find the interval in which the following functions are strictly increasing and strictly decreasing :   

Answer:

Strictly increasing in 

Strictly Decreasing in  

Question 15

Find the intervals in which the function given by f(x) = sin3x,  x∈ [0, π/2] is increasing and decreasing.

Answer

Increasing on [0, π/6]

Decreasing on [π/6, π/2]

MAXIMA AND MINIMA

Question 16

If      , then find the maximum value of f(x)

Ans:  3/4

Solution Hint:

f(x) is maximum if   4x² + 2x + 1  is minimum

g(x) = 4x² + 2x + 1 

g'(x) = 8x + 2

For critical point  g'(x) =0  ⇒ x = -1/4

g''(x) = 8 > 0

⇒ g(x) is minimum at x = -1/4

Minimum value of g(x) = 3/4

⇒ f(x) is maximum at x = 3/4

Question 17

Find the maximum value of    

Ans:  1/e

Solution Hint

Find the derivative of f(x) we get   

f ' (x) = 0 ⇒  x = 0 and e (critical points)

But at x = 0 logx is not defined so we have x = e only

Now find f '' (x) we get 

 

At x = e, f ''(x) < 0 ⇒ f(x) is maximum at x = e

Max. value of f(x) = f(e) = 1/e   (∵ Loge = 1)

Question 18

Find the least value of f(x) = e^x – e^-x 

Answer: 2

Question 19

Without using the derivatives, find the maximum and minimum values if any for the function f(x) = sin2x + 5.

Answer: Max. value = 6, Min Value = 4

Question 20

Of all the rectangles each of which has perimeter 40 meters, find one which has maximum area. Also find the maximum area?

Solution Hint:

Let sides of rectangle = x and y

Perimeter = 2(x + y) 

ATQ :  2(x + y) = 40 ⇒ x + y = 20 ⇒ y = 20 - x

A = Area = xy = x(20 - x) ⇒ 20x - x²

A' = 20 - 2x 

A' = 0 ⇒ 20 - 2x = 0 ⇒ x = 10

A'' = -2 < 0 ⇒ A is maximum at x = 10

If x = 10 then y = 10 ⇒ Rectangle is a square

Maximum area = xy = 10 x 10 = 100 sq m

Question 21

At what point, the slope of the curve y = - x³ + 3x² + 9x - 27 is maximum? Also find the maximum slope.

Answer: Maximum slope = m = 12 at x = 1

Solution Hint

y = - x³ + 3x² + 9x - 27

Slope = m = dy/dx = -3x² + 6x + 9

Now using second derivative test to find the maxima and minima.