Applications of Differentiation
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 a, for which f(x) = a(x+sinx)+a is increasing
6. Find the intervals in which the function f ( x ) = x3 - 6x2 + 9x + 15 is
(i) increasing (ii) decreasing. 7. Find the equation of the tangent line to the curve x = θ + sinθ, y = 1+cosθ a=π/4 8. Prove that curves y² = 4ax and xy = c² cut at right angles If c4 = 32 a4 9. 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. 10. Verify Lagrange’s mean value theorem for the function f(x) = x + 1/x in [1,3]. 11. Find the intervals in which f(x) = sin x + cos x , o ≤ x ≤ 2 π, is increasing or decreasing. 12. Find the interval in which the function given by f(x)= is increasing. 13. Find the local maximum & local minimum value of function x3– 12x2 + 36x – 4 14. For the curve y = 4x3 - 2x5, find all the points at which the tangent passes through the origin. 15. Find the interval in which the function f(x)= 2x3 -9x2 -24x-5 is Increasing or decreasing. 16. Find the equation of the tangents to the curve y = √3x-2 which is parallel to the line 4x-2y+5=0. 17. Find the interval in which the function f is given by f(x)=sin x – cos x ,o ≤ x ≤ 2 π (i) Increasing (ii) Decreasing.
18. Find the equation of the tangent to the curve x² + 3y – 3 = 0, which is perpendicular to the line y = 4x – 5.
19. Find the eq^ of of the tangent and normal to the curve x= a sin3t, y= b cos3t at the point t= π/4 20. It is given for the function f(x)=x3+ bx2–ax, x [1,3]Rolle’ theorem helds with c = ...
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