Practice Problems : Application Of Derivative

Level – I

1. Find the equation of normal to the curve y = (1 + x)^y + sin^-1(sin²x) at x = 0.

2. A point P is given on the circumference of circle of radius r. The chord QR is parallel to the tangent line at P. Find the maximum area of the triangle PQR.

3. Find the equations of the tangent drawn to the curve y² – 2x³ – 4y + 8 = 0 from the point (1, 2).

4(i) Show that the normal to the curve 5x⁵ – 10x³ + x + 2y + 6 = 0 at P(0, -3) meets the curve again at two points. Find the equations of the tangents to the curve at these points.

(ii). Find the length of the normal at ‘ t ‘ on the curve x = a ( t + sint ) , y = a ( 1 – cost ).

5. Prove that

(i) sin²θ < θsin(sinθ) for 0 < θ < π/2

(ii) cos(sinx) > sin(cosx), 0 < x < π/2

(iii) (a + b )^p ≤ a^p + b^p ; a , b > 0 , 0 < p < 1