1. If a line y = x + 1 is a tangent to the curve y2= 4x ,find the point the of contact ? 2. Find the point on the curve y = 2x2– 6x – 4 at which the tangent is parallel to the x – axis 3. Find the slope of tangent for y = tan x + sec x at x = π/4 4. Show that the function f(x) == x3– 6x2 +12x -99 is increasing for all x. 5. Find the maximum and minimum values, if any of
6. For the curve y = 3x² + 4x, find the slope of the tangent to the curve at the point x = -2. 7. Find a point on the curve y = x2– 4x -32 at which tangent is parallel to x-axis. 8. Find a, for which f(x) = a(x+sinx)+a is increasing .
9. The side of a square is increasing at 4 cm/minute. At what rate is the area increasing when the side is 8 cm long? 10. Find the point on the curve y =x2-7x+12, where the tangent is parallel to x-axis. 11. Find the intevals in which the function f(x) = 2log(x-2) - x2 + 4x + 1is increasing or decreasing. 12. Find the intervals in which the function f ( x ) = x3 - 6x2 + 9x + 15 is (i) increasing
13. Find the equation of the tangent line to the curve x = θ + sinθ, y = 1+cosθ a=π/4 14. Prove that is increasing in [o, π/2]
15. Prove that curves y² = 4ax and xy = c² cut at right angles If c4 = 32 a4 16. 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. 17. 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. 18. Verify Lagrange’s mean value theorem for the function f(x) = x + 1/x in [1,3]. 19. Find the intervals in which f(x) = sinx + cosx , o ≤ x ≤ 2 π, is increasing or decreasing. 20. Use differentials to find the approximate value of √25.2 21. Find the interval in which the function given by f(x)= (4sinx – 2x – x cosx) / (2+cos x) is increasing. 22. Find the local maximum & local minimum value of function x3– 12x2 + 36x – 4 23. For the curve y = 4x3 - 2x5, find all the points at which the tangent passes throughthe origin. 24. Show that the curves 2x = y2 and 2xy = k cut at right angles if k2 = 8. 25. Find the interval in which the function f(x)= 2x3 -9x2 -24x-5 is Increasing or decreasing. 26. Find the interval in which the function is increasing or decreasing. 27. Prove that the curves x = y² and xy = k cut at right angle if 8k2 = 1. 28. If f(x) = 3x² + 15x + 5, then find the approximate value of f(3.02), using diffrentials. 29. Find the local maximum and minimum values of function: f(x) = sin 2x – x,-π/2 < x < π/2 30. Find the interval in which f(x) =sin 3x is increasing or decreasing in [0, π/2]. 31. 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. 32. Show that the function f given by f(x) = tan-1(sinx+cosx), is strictly decreasing function on (π/4,π/2) 33. Find the equation of the tangents to the curve y = √3x-2 which is parallel to the line 4x-2y+5=0. 34. Find the intervals in which the function f(x) = x3+1/ x3 increasing or decreasing. 35. Verify Rolle’s theorem for the function f(x)=x3+2x-8 , 36. Find the equation of the tangent and normal to the parabola y2 = 4 a x at ( at2, 2at). 37. 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). 38. Verify Rolle’s theorem for f (x) = x3- 2x2- x + 3 in [0,1] . 39. Find the intervals in which f(x) = x3+ 2x2–1 decreasing or increasing...
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