NCERT Solutions Miscellaneous Exercise: Application of Derivative | Mathematics (Maths) Class 12 - JEE PDF Download

 Page 1


 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 6 Application of 
Derivatives 
 
 
Miscellaneous Exercise                                      Page No: 242 
 
1. Using differentials, find the approximate value of each of the following: 
(a)  
(b)  
Solution: (a)  
Consider  ……….(1) 
 
  
=  ……….(2) 
Changing  to  and  to  in equation (1), we have 
 ……….(3) 
Here  and  
So,  
From equation (3), 
Page 2


 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 6 Application of 
Derivatives 
 
 
Miscellaneous Exercise                                      Page No: 242 
 
1. Using differentials, find the approximate value of each of the following: 
(a)  
(b)  
Solution: (a)  
Consider  ……….(1) 
 
  
=  ……….(2) 
Changing  to  and  to  in equation (1), we have 
 ……….(3) 
Here  and  
So,  
From equation (3), 
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 6 Application of 
Derivatives 
 
   
   
  0.677 
(b)  
Consider  ……….(1) 
 
  
=  ……….(2) 
Changing  to  and  to  in equation (1), we have 
 …….(3) 
Here  and  
 
Page 3


 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 6 Application of 
Derivatives 
 
 
Miscellaneous Exercise                                      Page No: 242 
 
1. Using differentials, find the approximate value of each of the following: 
(a)  
(b)  
Solution: (a)  
Consider  ……….(1) 
 
  
=  ……….(2) 
Changing  to  and  to  in equation (1), we have 
 ……….(3) 
Here  and  
So,  
From equation (3), 
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 6 Application of 
Derivatives 
 
   
   
  0.677 
(b)  
Consider  ……….(1) 
 
  
=  ……….(2) 
Changing  to  and  to  in equation (1), we have 
 …….(3) 
Here  and  
 
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 6 Application of 
Derivatives 
 
From equation (3), 
 
   
    0.497 
2. Show that the function given by  has maximum value at x = e. 
Solution: Here  ……….(1) 
  ……..(2) 
And  
=  
Which implies,  ……….(3) 
Now  
  
  
Page 4


 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 6 Application of 
Derivatives 
 
 
Miscellaneous Exercise                                      Page No: 242 
 
1. Using differentials, find the approximate value of each of the following: 
(a)  
(b)  
Solution: (a)  
Consider  ……….(1) 
 
  
=  ……….(2) 
Changing  to  and  to  in equation (1), we have 
 ……….(3) 
Here  and  
So,  
From equation (3), 
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 6 Application of 
Derivatives 
 
   
   
  0.677 
(b)  
Consider  ……….(1) 
 
  
=  ……….(2) 
Changing  to  and  to  in equation (1), we have 
 …….(3) 
Here  and  
 
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 6 Application of 
Derivatives 
 
From equation (3), 
 
   
    0.497 
2. Show that the function given by  has maximum value at x = e. 
Solution: Here  ……….(1) 
  ……..(2) 
And  
=  
Which implies,  ……….(3) 
Now  
  
  
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 6 Application of 
Derivatives 
  
  
  
From equation (3), 
 =  =  < 0 
Point of local maxima and maximum value of f(x) at x = e. 
3. The two equal sides of an isosceles triangle with fixed base b are decreasing at the rate 
of 3 cm per second. How fast is the area decreasing when the two equal sides are equal 
to the base? 
Solution: Consider BC = b be the fixed base and  
AB = AC = x be the two sides of isosceles triangle. 
 
Since  cm/s ……….(1) 
Area of triangle = 1/2 x BC x AM 
=  
=  
 
 
Page 5


 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 6 Application of 
Derivatives 
 
 
Miscellaneous Exercise                                      Page No: 242 
 
1. Using differentials, find the approximate value of each of the following: 
(a)  
(b)  
Solution: (a)  
Consider  ……….(1) 
 
  
=  ……….(2) 
Changing  to  and  to  in equation (1), we have 
 ……….(3) 
Here  and  
So,  
From equation (3), 
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 6 Application of 
Derivatives 
 
   
   
  0.677 
(b)  
Consider  ……….(1) 
 
  
=  ……….(2) 
Changing  to  and  to  in equation (1), we have 
 …….(3) 
Here  and  
 
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 6 Application of 
Derivatives 
 
From equation (3), 
 
   
    0.497 
2. Show that the function given by  has maximum value at x = e. 
Solution: Here  ……….(1) 
  ……..(2) 
And  
=  
Which implies,  ……….(3) 
Now  
  
  
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 6 Application of 
Derivatives 
  
  
  
From equation (3), 
 =  =  < 0 
Point of local maxima and maximum value of f(x) at x = e. 
3. The two equal sides of an isosceles triangle with fixed base b are decreasing at the rate 
of 3 cm per second. How fast is the area decreasing when the two equal sides are equal 
to the base? 
Solution: Consider BC = b be the fixed base and  
AB = AC = x be the two sides of isosceles triangle. 
 
Since  cm/s ……….(1) 
Area of triangle = 1/2 x BC x AM 
=  
=  
 
 
 
 
 
 
 
NCERT Solutions for Class 12 Maths Chapter 6 Application of 
Derivatives 
=  [By chain rule] 
 
=  cm
2
/s 
 
Now, when  x = b. 
  cm
2
/s 
Therefore, the area is decreasing at the rate of  cm
2
/s. 
 
4. Find the equation of the normal to the curve x
2
 = 4y at the point (1, 2). 
 
Solution: Equation of the curve is x
2
 = 4y ……….(1) 
Differentiate w.r.t. x, 
 
2x = 4 dy/dx 
 
? 
????
????
=
?? 2
 = m (say) 
Slope of the normal to the curve at (1, 2) is -1/m = -2/x 
 Equation of the normal to the curve (1) at (1, 2) is  
x + y = 3 
5. Show that the normal at any point  to the curve  
  is at a constant distance from the origin. 
Solution: The parametric equations of the curve are