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CBSE CLASS XII

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Differentiability Applications (4 Mark)

Page 3 of 6

  1. A ladder 5m. long is leaning against a wall . The bottom of the ladder is pulled along the ground, away from the wall at the rate of 2 m./sec. How fast is its height decreasing on the wall,when the foot of the ladder is 4 m away from the wall.
  2. Show that the function f given by f(x) = tan-1(sinx+cosx), is strictly decreasing function on (π/4,π/2)
  3. Find the equation of the tangents to the curve y = √3x-2 which is parallel to the line 4x-2y+5=0.
  4. Find the intervals in which the function f(x) = x3+1/ x3 increasing or decreasing.
  5. Verify Rolle’s theorem for the function f(x)=x3+2x-8 ,
  6. Find the equation of the tangent and normal to the parabola y2 = 4 a x at ( at2, 2at).
  7. Using LMVT, find a point on the parabola y = ( x – 3 )2 , where the tangent is parallel to the chord (3,0) and (4,1).
  8. Verify Rolle’s theorem for f (x) = x3- 2x2- x + 3 in [0,1] .
  9. Find the intervals in which f(x) = x3+ 2x2–1 decreasing or increasing
  10. It is given for the function of defines by f(x)=x3+ bx2–ax, Rolle’ theorem helds with
  11. Find the intervals in which f(x) = - 2 x3+15x2- 36x + 1 is increasing or decreasing.
  12. Show that the parabola y2 = 4 x + 4 & y2 = 4 - 4 x intersect at right angle.
  13. Find the interval in which the function f is given by f(x)=sinx-cosx,o ≤ x ≤ 2 π
    (i) Increasing
    (ii) Decreasing.
  14. It is given that for the function f given by f(x)=x3+ bx2–axRolle’s theorem hold .Find the values of a and b
  15. Find the equation of the tangent and normal to the curve: x = acost +at sint , y = asint - atcot at any point ‘t’. Also show that the normal to the Curve is at a constants distance from the origin.
  16. Using differentiate find approximate value of √51.
  17. The surface area of a spherical bubble is increasing at the rate of 2m²/sec. Find the rate of at which the volume of the bubble is increasing at the instant its radius is 6cm.
  18. Prove that x/a + y/b = 1 is a tangent to the curve y = be-x/a at the point where the curve cuts y-axis.
  19. Find the equation of the tangent to the curve x² + 3y – 3 = 0, which is perpendicular to the line y = 4x – 5.
  20. Find the approximate change in the volume V of cube of side x mts caused by increasing the side 2 %.

 

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Submitted By Mrs. E.Praveen
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