(c) A 10 ft plank is leaning against a wall. If at a certain instant the bottom of the plank is 2 ft from the wall and is being pushed towards the wall at the rate of 1/2 ft/s, how fast is the acute angle that the plank makes with the ground increasing? (d) The volume of a sphere is to be computed from a measured value of its radius. Estimate the maximum permissible error in the measurement if the percentage error in the volume must be kept within 19%. (e) Use an appropriate local linear approximation to estimate the value of 673. Question 2 [ 2 + 5 + 5 + 5 + 5 + (3 + 5)= 30 Marks] (a) Evaluate 2+ +2 dt 1+3x2 (b) Evaluate the integral Vitx 3 dx using the substitution u = 1 + x3. (c) Find the area of the region enclosed by x = y2 - 3y + 2 and x = y - 1. (d) Find the volume of the solid that results when the region enclosed by y = x2 + 4, y = x3, and x = 0 is revolved about the x axis. (e) Use cylindrical shells to find the volume of the torus obtained by revolving the circle x2 + y? = a" about the line x = b, where b > a > 0. (f) For the curve x = + from y = 2 and y = 3 find the (i) arc length and (ii) surface area generated when the curve is revolved about the y-axis. THE ENDQuestion 1 [ (5 + 3) + (4 +4) + 5 +5+4=30 Marks] (a) Use the rigorous definition of limits to prove the following lim 28 (i) : 4 x-1 4x+3 (ii) lim x + 3 = +00 X-+00 (b) Evaluate the following limits: 1 + 22 + 32 + ... + 12 lim (i) lim (0 + e)e (ii) n-+00 n3 8-0