SM 3. Differentiate w.r.t x the following functions: (a) ( b ) x 1 ( x2 + 1 ) Vx (c) (d) *+1 x - 1 (e) *+1 3x - 5 3x - 1 x5 ( f ) 2x + 8 (g) 3x-11 (h) x 2 + x + 1 4. Differentiate w.r.t x the following functions: (a) Vx - 2 Vx+ 1 ( b ) 1 x2 + 1 (c) *+x+1 x2 - x+1 5. Let x = f(L) be the output when L units of labour are used as input. Assume that f(0) = 0 and that f' (L) > 0, f"(L) 0. Average productivity is defined by the formula g(L) = f(L) / L. (a) Let L* > 0. Indicate on a figure the values of f' (L*) and g(L*). Which is larger? (b) How does the average productivity change when labour input increases? SM 6. For each of the following functions, determine the intervals where it is increasing. (a) y = 3x2 - 12x + 13 (b) y = 4(x4- 6x2) (0) y = x2+ 2 2x x2-x3 revious Nex (d) y = 2 (x + 1) SM 7. Find the equations for the tangents to the graphs of the following functions at the specified points: x 2 - (a) y = 3- x-x atx = 1 ( b ) y = ; at x = 1 x2 + 1 (c) y = * 2 + 1 ) ( x 2 - 1 ) at x = 2 * *+ 1 (d) y = (x2 + 1) (x + 3) at x = 0 of the derivative