1. Express the derivative dy/dx in terms of x if 4 y=u ( 1+u3 ) u=( 3 x2 )2 2. Use implicit differentiation to find an equation of the line tangent to the curve 1 1 + =2 at the point(2,1) 2 x +3 2 y3 3. Sketch the graph of f (x). Identify all extrema, inflection points, and intercepts, and asymptotes. Show the concave structure clearly and note any discontinuities. f ( x )= x x 1 2 4. Given f (x) = x2 + 3, find the exact area A of the region under y = f (x) on the interval [1, 3] by first computing n f ( x i) x i=1 and then taking the limit as. n 5. Sketch the region bounded by the graphs of: f ( x )=x 4 , y = 1 and then find its area. 6. Use the following conditions to sketch a graph of the function (1) f (x) . f ( 0 )=0, f (1 )=f ( 1 )=3, f (2 )=f ( 2 )=5, f ' (2 )=f ' ( 0 )=f ' ( 2 ) =0 (2) f ' ( x ) >0 on ( ,2 ) ( 0,2 ) , f ' ( x )< 0 on (2, 0 ) ( 2, ) (3) f ' ' ( x ) >0 on (1,1 ) , f ' ' ( x )< 0 on (,1 ) ( 1, )