a) Evaluate lim sin? (kx) . k#0 without using L'Hopital's Rule. [4] b) Evaluate lim x+a using L'Hopital's Rule. [6] Xb) d) Produce the Taylor series approximating f near 1 for the function, f (x) 23in ax. Find the derivative y'{x) implicitly given y = sin(3x+4y). Find the coordinates of the points on the curve 3:1 -y which are nearest to the point {0.6). 2 16 a) Find the following limit if it exists: (1) lim X X (ii) For what value of the constant c is the function f continuous on (-0o,00) ? f(x)=. cx2 + 2x if x0 for 0 4 (iii) f"(x)> 0 for x2 C) A piece of wire 10m long is used to form a square and an equilateral triangle. Find the dimensions, which give a minimum area.b) The graph of a function f is given in Figure 1. Use it to sketch the derivative function, f'. [4] Figure 1 C) Two vertical poles PO and ST are secured by a rope RPS going from the top of the first pole to a point R on the ground between the poles and then to the top of the second pole as in Figure 2. Show that the shortest length of such a rope occurs when 0, = 02. R T(i) Obtain the fifth-order Taylor series for e about x=0.a) (1) State the Mean Value Theorem. (ii) With the aid of the Mean Value Theorem prove the following theorem: If f'(x)=0 for all x in an interval (a,b) then f is constant on (a, b)