Chapter 03. Applications of Derivatives

APPLICATIONS OF DERIVATIVES

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CHAPTER 03. APPLICATIONS OF DERIVATIVES QUESTIONS FROM BOARD PAPERS[1992- ]

Sec1.Rate of change of quantities

1. A stone is dropped into a quiet lake and waves move in circles at a speed of 3.5 cm per second. At the instant when the radius of the circular wave is 7.5cm, how far is the enclosed area increasing?

2. The radius of a balloon is increasing at the rate of 10cm/sec. At what

rate is the surface area of the balloon increasing when the radius is 15 cm long.

3. An edge of a variable cube is increasing at the rate of 5 cm per second.

How fast is the volume of the cube increasing when the radius is 15cm?

4. A particle moves along the curve y=4 x³+5. Find the points on the

curve at which the y-coordinate 3 changes twice as fast as x-coordinate?.

5. The radius of a circular soap bubble is increasing at the rate of

0.2cm/s. Find the rate of increase of the volume when the radius is 5cm.

6. At what points of the ellipse 16x²+9y²= 400 does the ordinate decrease at the same rate at which the abscissa increase?

7. Find the point on the curve y²= 8x for which the abscissa and ordinate

change at the same rate. 8. A balloon, which always remains spherical, is being inflated by

pumping in gas at the rate of 900 cu cm/s. Find the rate at which the radius of the balloon is increasing when the radius of the balloon is 15cm.

9. The volume of a spherical balloon is increasing at the rate of

25cm³/sec. Find the rate of change of its surface area at the instant when its radius is 5cm. [CBSE 2004]


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Ch2.Increasing and Decreasing functions

1. Find the intervals in which the following function is decreasing: (i). f(x) = 2x 3 - 15x 2+36x+1 (ii) f(x) = x³ - 12x

2. Find the intervals in which the following function is increasing or decreasing

(i) f(x) = x³ - 6x² +9x + 15 (ii) f(x) = sinx – cos x , o< x< 2 (iii) f(x) = log(1+x) – x (iv) f(x) = (x+3) e-x (v) f(x)=2x³- 9x²+12x+ 30

1+x (vi) f(x) = (x+1)²(x-3)² (vii) f(x) = x

1 + x² 3. Find the intervals in which the function f(x) is increasing or decreasing. (i) f(x) = x³- 6x³ - 36x + 2, (ii) f(x) = 5x³- 15x² -120 x +3 (iii)f(x) = 2x³ + 9x² + 12 x + 20 4. State when a function is said to be an increasing function in [a, b]. Test whether the function f(x) = x³- 8 is increasing in [1,2]. 5. For the function f(x) = 2x³- 24x + 5 , find a. the interval(s) where it is increasing b. the interval(s) where it is decreasing

6. Show that the function f(x) = x³ - 6x² +12x –18 is increasing on R. 7. Find the intervals in which the function f(x) = 2x³ - 15x² + 36x +1 is

strictly increasing or decreasing. Also, find the points on which the tangents are parallel to the x-axis. [ CBSE 2005]

8. Find the intervals in which the function f(x) = x³ - 12x² + 36x +17 is

a. increasing b. decreasing [ CBSE 2006]

Ch03. Tangents and Normals 1. Find the equation of the tangent line to the curve x =(1 –cos θ) ,

y= (θ - sin θ) at θ= /4.

2. Find a point on the parabola y=(x-2)2 where the tangent is parallel to the chord joining the points (2,0) & (4,4). 3. Find the equation of the tangent to the curve x+y= a at the point

(a²/4, a²/4).


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4. Find the equation of the tangent to the curve y=-5x²+6x+7 at the

point (1/2, 35/4).

5. Find the point on the curve y = 2x² - 6x- 4 at which the tangent is parallel to the x- axis.

6. Find the points on the curve y= x³- 2x²- 2x at which the tangent lines

are parallel to the line y = 2x –3.

7. Show that the curve x = y² and xy= k cut at right angles if 8k² = 1 8. At what point on the curve x² + y² - 2x - 4y + 1 = 0 is the tangent parallel to the y- axis. 09.Show that the curve xy= a² and x²+ y² = 2a² touch each other.

10.Find the equation of the tangent to the curve y = √(3x –2) which is parallel to the line 4x –2y + 5 = 0.[CBSE 2005]

Ch04.Maxima and Minima

1. Find the radius of a closed right circular cylinder of volume 100 cubic

centimeters, which has the minimum total surface area. 2. Show that a closed right circular cylinder of given total surface area and maximum volume is such that its height is equal to the diameter of its base.

3. Show that of all rectangles with a given perimeter the square has the largest area.

4.Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area. 5.Show that the rectangle of maximum area that can be inscribed in a

circle of radius r is a square of side 2 r. 6. Find all the points of local maxima and minima and the corresponding

maximum and minimum values of the function f(x)=-3 x4–8x³-45x²+ 105.

4 2 7. The combined resistance R of two resistors, R1 and R2 (R1 + R2 > 0) is

given by 1 = 1 + 1 . If R1 + R2 = C (Constant), find R1 and R2 so that


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R R1 R2 . R is maximum.

8. A figure consists of a semi circle with a rectangle on its diameter.

Given the perimeter of the figure, find its dimensions in order that the area may be maximum.

9. A square piece of tin of side 18cm is to be made into a box without top

by cutting a square from each corner and folding up the flaps to form a box. What should be the side of the square to be cut off so that the volume of the box is maximum? Also, find the maximum volume.

10. Find the volume of the largest cylinder that can be inscribed in a

sphere of radius r cm. 12. A wire of length 25m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the lengths of the two pieces so that the combined area of the square and the circle is minimum? 13. Find the point on the curve y² = 4x which is nearest to the point (2,-8). [ CBSE D ’06]

14. A rectangle is inscribed in a semi-circle of radius r with one of its sides on the diameter of the semi-circle. Find the dimensions of the rectangle so that its area is maximum. Find also its area.

11. Show that the surface area of a closed cuboids with square base

& given volume is minimum when it is a cube.

12. A right circular cylinder is inscribed in a cone. Show that the curved surface area of the cylinder is maximum when the diameter of the cylinder is equal to the radius of the base of the cone.

13. Show that the height of a closed circular cylinder of given total

surface area and maximum volume is equal to the diameter of the base.

14. An open tank with a square base and vertical sides is to be

constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of the material will be least when the depth of the tank is half of its width.

15. A window is in the form of a rectangle surmounted by a semi-

circular opening. If the perimeter of the window is 20m,find the


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dimensions of the window so that the maximum possible light is admitted through the whole opening.

16. Find the largest possible area of a right angled triangle whose

hypotenuse is 5cm long.

17. A closed right circular cylinder has a volume of 2156 cu cm.What will be the radius of its base so that total surface area is minimum.

(Take = 22/7)

18. Show that the height of a cylinder of maximum volume that can

be inscribed in a sphere of radius R is 2R/3. [ CBSE D ‘06]

19. An open box with a square base is to be made out of a given quantity of sheet of area a². Show that the maximum volume of the box

is a³/ 63.

20. Show that the volume of the greatest cylinder, which can be

inscribed in a cone of height h and semi-vertical angle 30, is 4 h³/81.

21. Two sides of triangle have length ‘a’ and ‘b’ and the angle between them is ‘θ’. What value of θ will maximize the area of the triangle? Find the maximum area of the triangle also.

22. A rectangular window is surmounted by an equilateral triangle.

Given that the perimeter is 16m, find the width of the window so that maximum amount of light may enter.

23. Show that the maximum volume of the cylinder, which can be

inscribed in a sphere of radius 5/√3 cm, is 500 cm³.

24. Show that the height of the cone of maximum volume that can

be inscribed in a sphere of radius 12 cm is 16cm.[ CBSE 2005]

25. Prove that the curves x= y² and xy = k cut at right angles if 8k² = 1.

[ CBSE 2005] 26. An open box with a square base is to be made out of a given

quantity of sheet of area c². Show that the maximum volume of the box is

c³/ 63. [CBSE 06]


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31. A window is in the form of a rectangle surmounted by a semi-circle. If the total perimeter of the window is 30m,find the dimensions of the window so that the maximum light is admitted [CBSE 2006].

32. Find the point on the curve x² = 4y which is nearest to the point

(-1,2). [ CBSE ‘07]

33. A wire of length 28m is to be cut into two pieces. One of the two pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and these is minimum? [ CBSE ‘07]

Ch 05. Rolle’s theorem and Lagrange’s Mean value theorem 01. Verify the applicability of Rolle’s theorem for the function: (i)f(x)=x(x-1)2in[0,1] (ii) f(x)=x(x-3)³,0≤x≤3. (iii)f(x)=(x-1)(x-2)(x-3),0≤x≤4 (iv) f(x) = x² - 4x + 3, [1,3] (v)f(x)= x²- x – 12,[-3,4] (vi) f(x)= 4x² -12x+9 in [0,3] (vi) f(x) = sin x + cos x. x ε [ 0. ∏/2] (vii) f(x)=[x-1][x-2]2, [1,2] (viii) f(x) = x2 – 4x +3 on [ 1,3]

02.Verify Lagrange’s Mean value theorem for the function (i) f(x) = (x-1)(x-2)(x-3), [1,4] (ii) f(x)= x² +x –1 in the interval [0,4]. (iii) f(x) = x + 1 on [1,3]. iv) f(x)= x2 + 2x +3, [ 4, 6]

x 03. Using Rolle’s theorem, find the points on the curve y= x², x[-2,2],

where the tangent is parallel to the x-axis.

Ch 06. Errors and Approximations

01.Using differentials, find the approximate value of (i)(29)1/3 (ii)(0.48)

02.If y= x4 – 10 and x change from 2 to 1.97,using differentials, find the approximate change in y.

03. Using differentials, find the approximate value of

(i).2 (ii)0.37 (iii)37

04. Using differentials find the approximate value of (82)1/4 up to 3 places of decimal.


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Chapter 03. Applications of Derivatives