ANSWER. THESE QUESTIONS AS GIVEN IN THE ATTACHMENT.

(a) Use implicit differentiation to compute and where z is defined implicitly by y . cos(I' 2) = 1. [3 marks] (b) Suppose that a function f(r, y) has constant gradient Vf = (3, 1). (i) Give two examples of a function with this property. (ii) Consider the graphs of z =f(r, y) for functions with this property. Describe an infinite geometric family which includes all of these graphs. [2 marks] (c) Let f be a function of two variables that has continuous partial derivatives and consider the points A(1,2), B(3, 2), C(1, 5) and D(4, 6). The directional derivative of f at A in the direction of the vector AB is 10 and the directional derivative of f at A in the direction of the vector AC is 15. Find the directional derivative of f at A in the direction of the vector AD [5 marks] Show transcribe1. * A f100 bond is redeemable at par in 27 years. It pays semiannual coupons at a rate of 6% per annum payable semiannually. This means that the bondholder receives $3 (half of 6% of [100) every half a year for 27 years, and a further payment of $100 after 27 years. (a) Compute the price of the bond which achieves a yield of 5% per annum effective by adding the value of the coupons to the value of the redemption payment. (b) Repeat the same with the premium / discount formula. (c) What is the price (still assuming a yield of 5%) if the maturity term is 17 years? 7 years? (d) What is the price if the maturity term is 27 years and the yield is 4% p.a. ef- fective? If the yield is 3% p.a. effective? (e) Compute the price of a 27-year bond with annual coupons at a rate of 6% p.a. effective bought to yield 5% p.a. effective.*_9 The intent of this problem is that you explore the action of orthogonal matrices on the vectors the}r multiply. (a) In general, a vector in R2 selected at random has two degrees of freedom. Make a case that, if that vector is constrained to be of unit length, then there is only one degree of freedom, which can be taken to be 6, the angle with the positive horizontal axis. What is the form of this vector ube specic about its componentsonce 6 is selected. (b) Using u from part (a), describe [ul the orthogonal complement of 11. Though this orthogonal complement has infinitel}r many bases, each consisting of a single vector, there are far fewer bases consisting of a unit vector. Give all of these. (c) Describe all the (real) orthogonal 2-by2 matrices. [Hintz For each choice of angle 9, there are precisely two.] (d) Fix 6 = f3, and consider the two orthogonal matrices of part (c) associated to 6. For each one, compute both the determinant and the eigenvalues. You should observe that one has real eigenvalues, and the other has nonreal ones. Which of these has determinant equal to 1? Do these observations carry over to other choices of 6? (e) For the two matrices of part (d) tie, the ones associated with 6 = HIS), compute Ai and Aj, where i = (1,0), j = (0,1). Which of these two matrices would rightly be called orientation-preserving, the one whose determinant is positive, or the one with a negative determinant? My? (f) Again with 6 = 11/3, take A to be the matrix from part (d) with positive determinant. What does the function x) = A): do? Is thereaneasily-described relationship between input vector x and output vector Ax? [I-Iint: Compute the angle between x and Ax] What changes if we use AT in place of A? (g) Now, with 9 = 1r! 3, take A to be the matrix from part (:1) with negative determinant. What does the function f(x) = Ax do? Is there an easily-described relationship between input vector x and output vector Ax? [Hint The eigenvectors v1, V2 of A form an orthonormal basis of IRZ, so any x E R2 can be expressed as a linear combination X = $1171 + C2V2. Compute A(c1v1 + czvz), and compare with the original 1: = clvl + czvz. What angle does the eigenvector associated with eigenvalue 1 make with the x-axis?]