. [20 marks] Using the method outlined in lectures {and in Section 5.4 of the textbook), and show- ing all working, sketch the graph of each of the following functions. Note: you do not need to consider f\"(x), concavity and points of inection (step 6). (a) x) : 3x3 33c2 +x _ x/x + 3 (b) f(x) _ x + 1 . [3 marks] Suppose a function, f(x) has a vertical asymptote toward positive infinity at x = 1, a horizontal asymptote at y = 1/2 as x > :00, and a turning point at (1, 1). Sketch what you think the function would look like. . [8 marks] Find the absolute maximum and minimum of the function x2 x + 1 x2 + 1 Express your answers in simple exact form. g(x) : for x E [2, 2] . [7 marks] A farmer wishes to add a rectangular stockyard next to a shed (as shown). Using the shed wall as part of the enclosure can reduce the costs. Suppose the shed wall has length L 2 10m, and there are 50 metres of fencing, what are the dimensions that maximize the area of the enclosure, and what is the maximum area? (Remember to consider the limits on x) Shed .qpqp. 10 m x Yard . [7 marks] Consider the function defined by x2y3 x + 63' : 6 + e (a) Use implicit differentiation to find dy/ dx . (b) Show that the point (x,y) = (3,1) lies on the curve defined by the equation in part (a), and find the slope of the tangent line at this point. Give your answer exactly, not as an approximate decimal. Find the equation of the tangent line at this point