Math Assignment Class XII Ch -06 | Application of Derivatives

 Math Assignment  Class XII  Ch - 06

 Application of Derivatives

Extra questions of chapter 06 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 1

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

 Question 2

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 3

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 4

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 5

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 6

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

Question 7  

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 8 

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 9:  

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

Answer:

Strictly increasing in 

Strictly Decreasing in  

Question 10

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]

Question 11 Let f(x) be a continuous function on [a, b] and differentiable on (a, b). Then, this function f(x) is strictly increasing in (a, b) if

Question 12 The function f(x) = x³ – 3x² + 12x – 18 is :
(D) strictly decreasing on (– ∞ , 0)
Answer bQuestion 13 Show that f(x) = e^x – e^–x + x – tan^–1 x is strictly increasing in its domain.
Solution
 
 
∴ f is strictly increasing over its domail RQuestion 14: Find the intervals in which the function  is strictly increasing or strictly decreasing.
Question 15:

(A) f’(x) < 0, ∀ x ∈ (a, b)

(B) f’(x) > 0, ∀ x ∈ (a, b)

(C) f’(x) = 0, ∀ x ∈ (a, b)

(D) f(x) > 0, ∀ x ∈ (a, b)

Ans: b

(A) strictly decreasing on R

(B) strictly increasing on R

(C) neither strictly increasing nor strictly decreasing on R

Solution:   

For strictly increasing / decreasing, put f’(x) = 0  ⇒  x = e,   x > 0

For strictly increasing, x ∈ (0, e ) and for strictly decreasing x ∈ (e, ∞ )

Find the absolute maximum and absolute minimum values of the function f given by   , on the interval [1, 2].

Answer: Absolute maximum value = 5/2 

               Absolute minimum value = 2

MAXIMA AND MINIMA

Question 1

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 2

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 3

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

Answer: 2

Question 4

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 5

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 6

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.

Question 7 Show that the function f(x) = 4x³ – 18x² + 27x – 7 has neither maxima nor minima.

Solution: f ′ (x) = 12x² − 36x + 27

          = 3 (2 x -3)² ≥ 0 for all x ∈ R

∴ f is increasing on R .

Hence f(x) does not have maxima or minima.

Question 8: It is given that function f(x) = x⁴ – 62x² + ax + 9 attains local maximum value at x = 1. Find the value of ‘a’, hence obtain all other points where the given function f(x) attains local maximum or local minimum values.

Solution: f’(x) = 4x³ – 124x + a

As at x = 1, 'f' attains local maximum value, f’(1) = 0  ⇒   a = 120

Now, f’(x) = 4x³ – 124x + 120

                 = 4(x - 1)(x² + x - 30)

                 = 4(x - 1)(x - 5)(x + 6)

Critical Points are x = - 6, 1, 5

     f’’(x) = 12x² – 124

  f’’(-6) > 0,  f’’(5) > 0  and f’’(1) < 0

So f attains local maximum value at x = 1 and local minimum value at x = - 6, 5

Question 9: 

The perimeter of a rectangular metallic sheet is 300 cm. It is rolled along one of its sides to form a cylinder. Find the dimensions of the rectangular sheet so that volume of cylinder so formed is maximum.

Solution:  Let length of rectangle be x cm and breadth be (150 – x) cm.

Let r be the radius of cyllander

Circumference of base of cyllander = 2πr = x

 

Volume of cyllinder = πr²h = 

     
     

Differentiating w r t  x

 

 

 

At x = 100 cm

 

⇒ V is maximum when x = 100 cm and breadth of rectangle is 50 cm