Test 1 September 20, 2002 1. Determine whether each of the following is a statement or not (answer yes or no): (a) Some sentences can be labelled true and false. (b) All students should study mathematics. (c) All consumers have transitive preferences. (d) This statement is not true. 2. Construct a truth table to prove each of the following tautologies: (a) (p q) [( p) ( q)] (b) (p q) ( q p) 3. Prove or disprove: The image of the function f : N N given by f(n) = n 2 + n + 13 is a subset of the the set of all prime numbers. 4. Let A be a subset of a universal set U. Prove that A (U \ A) = U. 5. Let f : R R and g : R R be given by f(x) = cos x and g(x) = x 2 + 1. Find the following: (a) g([ 1, 1]); (b) (g f)([0, π]); (c) (f g)([0, 2π]); (d) (g f 1 )([ π, π]).
Test 2 October 4, 2002 1. Suppose that x k and y k are sequences in R such that x k x and y k y. Show that x k + y k x + y. 2. Define the following terms in R n : (a) open ball, (b) open set, (c) closed set, (d) compact set. 3. Determine whether the following sets are (i) open, (ii) closed, (iii) both open and closed, or (iv) neither open nor closed: (a) B(x, r) R n, (b) Z R (c) {(x, y) R 2 : 1 < x < 2, y = x}, (d) {x R : 1 x 2, x Q} 4. Find the eigenvalues and eigenvectors of the matrix [ ] 3 4 A =. 4 3 5. Show that the inverse of the matrix A = 2 0 1 5 1 0 0 1 3 is B = 3 1 1 15 6 5 5 2 2.
Test 3 October 18, 2002 1. Find the derivative Df(x) of the function f : R 2 R 2, f(x) = (x 1 + sin x 2, x 2 e x 1 ). 2. Suppose f : R n R is a continuous function. Prove that the set is closed for any y R. {x R n : f(x) = y} 3. Prove that an n n symmetric matrix A is positive definite if and only if A is negative definite. 4. Find the hessian D 2 f(x) of the function f : R 2 ++ R at x = (2, 1) such that f(x) = x 1 + x 2. Determinate the definiteness of D 2 f(x) at that point. 5. Let f : R 2 ++ R, f(x) = (x 1 x 2 ) 1/2. (a) Find the derivative Df(x) at the point x = (1, 1). (b) What is the directional derivative Df(x; h) at (1, 1) in the direction of the vector h = (2/ 5, 1/ 5)?
Test 4 November 1, 2002 1. Find the second-order Taylor formula for f(x) = sin(x 1 + 2x 2 ) about the point x = (0, 0). 2. Prove that x = 2 sin x for some point in (π/2, π). 3. Give an example of a function f : [a, b] R, a, b R such that f[a, b] is an open set. 4. Let f : R 2 R 2 be given by f(x) = (x 1 cos x 2, x 1 sin x 2 ). Find the set of points that the inverse of f does not exist? 5. Consider the following Keynesian model in macroeconomics: Y = C[(1 t)y ] + I(r, Y ) + G, M/P = L(Y, r). In this model, Y (output) and r (interest rate) are endogenous and P (price), G (government expenditure), t (tax rate), and M (money demand) are exogenous. C, I, and L are functions for consumption, investment, and money demand respectively. Using Cramer s rule, find the output multiplier for income tax, Y/ t. What assumptions do you have to make in order for the model to work?
Test 5 November 15, 2002 1. Let f : D R be a nondecreasing function where D R is compact. Prove that f always has a maximum on D. 2. Find and classify the critical points of the function f(x, y) = x sin y. 3. Consider the cost minimization problem min z w 1 z 1 + w 2 z 2 subject to y = z 1 z 2. where z R 2 + and w R 2 ++. Show that the solution is f(z 1, z 2) = 2(w 1 w 2 y) 1/2 by representing it as an unconstrained minimization problem in one variable. 4. Let φ : R R be a strictly increasing C 2 function. Let f : D R be a C 2 function where D is an open subset of R n. Suppose that f has a strict local minimum at x D, and that D 2 f(x ) is positive definite. Prove that the composite function φ f also has a strict local minimum at x. 5. Let f : R n R be a C 2 function and let g = f. Given x R n, prove that the Hessian D 2 g(x) is negative definite if and only if D 2 f(x) is positive definite.
Test 6 November 29, 2002 1. Find the maxima and minima of the function f(x, y) = 1/x + 1/y subject to the constraint 1/x 2 + 1/y 2 = 1/4. 2. The utility function of a consumer is given by u(x 1, x 2 ) = (x 1 x 2 ) 1/2, where prices for goods 1 and 2 are p 1 > 0 and p 2 > 0. Given that the consumer has income I > 0, solve the utility maximization problem subject to the constraints, p 1 x 1 + p 2 x 2 I, x 1 0, x 2 0 as follows: (a) Show that this can be reduced to an equality-constrained problem. (b) Find the critical point(s) of the Lagrangean. (c) Check the second-order conditions. 3. A consumer who buys two goods has a utility function u(x 1, x 2 ) = min{x 1, x 2 }. Given income I > 0 and prices p 1 > 0 and p 2 > 0. (a) Describe the consumer s utility maximization problem. (b) Does the Weierstrass theorem apply to this problem? (c) Can the Kuhn-Tucker theorem be used to obtain a solution? (d) Solve the maximization problem. Explain your answers. 4. Solve the following maximization problem: Maximize x + log(1 + y) subject to x + 2y = 1, x 0, y 0. 5. Let {f 1, f 2,..., f n } be a set of concave functions from R n to R. Prove that the nonnegative linear combination is concave. f(x) = α 1 f 1 (x) + + α n f n (x), α 1,..., α n 0