The function f is defined for positive y and all x by f(x,y) = x2 lny−y lny. Find the critical (or stationary) points of f and determine whether each critical point is a local maximum, local minimum or saddle point.

A firm is the only producer of a particular good. The firm’s marginal revenue function is MR = 9−q, where q denotes the quantity of the good produced by the firm. The firm’s fixed costs are 12 and its average variable cost function is 1 + q/2. Find an expression for the firm’s profit function, Π(q). Find the value of the production, q, which maximises the firm’s profit, and hence calculate the firm’s maximum profit.
2. Using the method of row operations, solve the following system of linear equations to find x, y and z. 3x + y + 3z + 22 = x + 2y−2z + 150, x + y + 73 = 2x−2y + 2z + 44, x−2y + z + 102 = x−y + 130.
3. The function f is defined for positive y and all x by f(x,y) = x2 lny−y lny. Find the critical (or stationary) points of f and determine whether each critical point is a local maximum, local minimum or saddle point.
4. Two functions W(x,y) and U(x,y) are connected by the equation
W(x,y) = ex−4yU(x,y).
Find the partial derivatives
∂W ∂x
,
∂W ∂y
and
∂2W ∂x2
,
in terms of U and its partial derivatives. If W satisfies ∂W ∂y = ∂2W ∂x2 −2
∂W ∂x −3W, show that the function U then satisfies the equation
∂U ∂y
=
∂2U ∂x2
.

MT105a Mathematics 1 2016 Mock 5. (a) Determine the integralZ x−1dx (1 + lnx)lnx . (b) An arithmetic progression is such that its second term is 7 and its thirteenth term is ten times its first term. Determine the first term and the common difference.
6. An investor saves money in a bank account paying interest at a fixed rate of 100r%, where the interest is paid once per year, at the end of the year. She deposits an amount D at the beginning of each of the next N years. Show that she will then have saved an amount equal to D r(1 + r)N −1just after the last of these deposits.