Question 1 (17 marks (a) (i) Find lim x2 sin (Simply stating an answer will gain you no I-+0 credit) (3] (ii) Given that lim log f (x) = m and lim g (x) = n, and assuming the r-+0 functions are appropriatley defined, what is the value of lim If (x) 9(x) in terms of m and n ? 13 (ii) Show that lim does not exist. 2] (b) (i) The function f is continuous at 0. What does this mean? [1] (ii) Show that the function f : R - R such that f (x) = xx| is continuous at * = 0. [2] (c) (i) Give a definition of the statement "f is differentiable at c". [1] (ii) Use a limit definition of the derivative to find f' (x) , the derivative of f (x) = for x * -2. [5) 2+xQuestion 2 (19 marks) (a) Given that if a 2 is differentiable everywhere (on R), (i) find the values of m and b. (5] (ii) Give a sketch (NOT on graph paper) of f for the values of b and m found in part (i). (2] (b) Differentiate (with respect to x) : ' sin ? (i) y= for r E (1, T) ex log a (ii) ' + y' + ry = 20, where y is a function of r. (c) (i) Find the 27th derivative of cos ;x. (2] (ii) For x > 0, by first taking logarithm and assuming that the deriva- tive of log x is 1/x, show that the derivative of r" is no"-1, where n is a real number. (2]Question 3 (17 marks) (a) A rectangle is inscribed in a semicircle of radius R. If the rectangle has greatest area, what are its dimensions? (5] (b) Use the Mean Value Theorem to show that 1 - 0. [3] (b) Given that f (x) = cos (t? ) dt. Find f" (x) , the second derivative of f with respect to r. (c) Show that the arc length of the curve y = x3/ from a = 1 to r = 2 is where 22v22 - 13V13 e = 6 27