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CBSE CLASS XII

Differentiability Applications (4 Mark)

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1. Find the intevals in which the function f(x) = 2log(x-2) - x² + 4x + 1is increasing or decreasing.
2. Find the intervals in which the function f ( x ) = x³ - 6x² + 9x + 15 is
   (i) increasing
   (ii) decreasing.
3. Find the equation of the tangent line to the curve x = θ + sinθ, y = 1+cosθ a=π/4
4. Prove that is increasing in [o, π/2]
5. Prove that curves y² = 4ax and xy = c² cut at right angles If c⁴ = 32 a⁴
6. A water tank has the shape of an inverted right circular cone with its axis vertical and vertex lower most. Its semi vertical angle is . Water is poured into it at a constant rate of 5 cubic meter per minute. Find the rate at which the level of the water is rising at the instant when the depth of water in the tank is 10m. Find the point on the curve y =x²-7x+12, where the tangent is parallel to x-axis.
7. Discuss applicability Rolle’s Theorem for the function f(x) = cosx + sinx in [0,2π ] and hence find a point at which tangent is parallel to X axis.
8. Verify Lagrange’s mean value theorem for the function f(x) = x + 1/x in [1,3].
9. Find the intervals in which f(x) = sinx + cosx , o ≤ x ≤ 2 π, is increasing or decreasing.
10. Use differentials to find the approximate value of √25.2
11. Find the interval in which the function given by f(x)= (4sinx – 2x – x cosx) / (2+cos x) is increasing.
12. Find the local maximum & local minimum value of function x³– 12x² + 36x – 4
13. For the curve y = 4x³ - 2x⁵, find all the points at which the tangent passes throughthe origin.
14. Show that the curves 2x = y² and 2xy = k cut at right angles if k² = 8.
15. Find the interval in which the function f(x)= 2x³ -9x² -24x-5 is Increasing or decreasing.
16. Find the interval in which the function is increasing or decreasing.
17. Prove that the curves x = y² and xy = k cut at right angle if 8k² = 1.
18. If f(x) = 3x² + 15x + 5, then find the approximate value of f(3.02), using diffrentials.
19. Find the local maximum and minimum values of function: f(x) = sin 2x – x,-π/2 < x < π/2
20. Find the interval in which f(x) =sin 3x is increasing or decreasing in [0, π/2].

 

Submitted By Mrs. E.Praveen
Email Id : [email protected]