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CBSE CLASS XII

Differentiability Applications (4 Mark)

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41. Find the intervals in which the function f(x) = 2x³ - 15x² + 36x + 1 is strictly increasing and decreasing. Also find the points on which the tangents are || to the x-axis.
42. Find the equations of the tangent and normal to the curve 16x² + 9y² = 144 at (2, y₁ > 0). Also find the point of intersection where both tangent and normal meet the x-axis.
43. A particle moves along 3y = 2x³ + 3. Find the points on the curve at which the y-coordinate changes twice as fast as x-coordinate.
44. Find the point on the parabola y = (x – 3)² where the tangent is parallel to chord joining the points (3, 0) and (4, 1).
45. The volume of a sphere is increasing at 3cm³/ s. what will be the rate at which the radius increases when radius is 2 cm
46. Verify Rolle’s theorem for the function f(x) = x²-5x+6 in the interval [2, 3].
47. Water is leaking from a conical funnel at the rate of 5cm³/Sec. If the radius of the base of funnel is 5 cm and height 10cm find the rate at which is water level dropping when it is 2.5 cm from the top.
48. The length x of a rectangle is decreasing at the rate of 2cm/s and the width y is increasing at the rate of 2cm/s. when x=12 cm and y=5cm, find the rate of change of
    (a) the perimeter and
    (b) the area of the rectangle.
49. Using differential, find the appropriate value of 3√29
50. Sand is being poured at the rate of 0.3 m³/sec into a conical pile. If the height of the conical pile is thrice the radius of the base, Find the rate of change of height when the pile is 5cm high.
51. Verify the condition of Mean Value Theorem and find a point c in the interval as statedby the MVT for the function given by f(x) = log_ex on [1, 2].
52. The two equal sides of isosceles triangle with a fixed base ‘b’ are increasing at the rate of 3 cm/sec. How fast is the area decreasing when the two equal sides are equal to the base?
53. Verify Rolle’s Theorem for the function f(x) = Sin x – Cos x in the interval [ π/4,5π/4]
54. The radius of a balloon is increasing at the rate of 10 cm/sec. At what rate is the volume of the balloon increasing when the radius is 10cm?
55. Find the interval in which the function f(x) = x³– 6x² – 36x + 2
56. Find the intervals in which the following function is increasing : f(x)=x⁴– 2x².
57. Using Rolles theorem, find the points on the curve y = 16 – x², x є [ -1,1] where the tangent is parallel to x-axis.
58. Show that the function f given by f(x)= tan^-1(sinx+cosx), is strictly decreasing function on (π/4,π/2).
59. Find the equation of the tangent to the curve x² + 3y = 3 which is parallel to the line y – 4x + 5 = 0.
60. A man 160 cm tall; walks away from a source of light situated at the top of a pole 6 m high, at the rate of 1.1 m/s. How fast is the length of his shadow increasing when he is 1 m away from the pole?
61. A point source of light along a straight road is at a height of ‘a’ metres. A boy ‘b’ metres in height is walking along the road. How fast is his shadow increasing if he is walking away from the light at the rate of ‘c’ m/min?
62. At what points of the ellipse 16x² + 9y² = 400 does the ordinate decrease at the same rate at which the abscissa increases?
63. The bottom of a rectangular swimming pool is 25 m by 40 m. Water is pumped out into the tank at the rate of 500 m³/min. Find the rate at which the level of the water in the tank rising.
64. An inverted cone has a depth of 40 cm and base of radius 5 cm. Water is poured into it at a rate of 1.5 cm³/ min. Find the rate at which the level of water in the cone is rising when the depth is 4 cm.
65. Water is dripping through a tiny whole at the vertex in the bottom of a conical funnel at a uniform rate of 4 cm³ / s. When the slant height of the water is 3 cm, find the rate of decrease of the slant height of the water, given that the vertical angle of the funnel is 120⁰.
66. Oil is leaking at the rate of 16 mL / s from a vertically kept cylindrical drum containing oil. It the radius of the drum is 7 cm and its height is 60 cm, find the rate at which the level of the oil is changing when the oil level is 18 cm.

 

Submitted By Mrs. E.Praveen
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