2.4.Think of a linear equation system 2.3.Evaluate 2.2.Recall that the discount factor of continuous componding is 2.1. 2.Computations and Proofs ( 1.Elementary math check (No steps needed!) 1.1. Simplify the expressions. Example: limx!1 ex2 = 1; eln x = x 1.1.1. limx!0+ ln ?? 1x(x > 0) 1.1.2. limx!0+ eln(x) (x > 0) 1.1.3. e??ln(x2) 3 (x > 0) 1.2. True or false: 1.2.1. exy = exey 1.2.2. ln (xy) = ln (x) ln (y) 1.2.3. abc d = ad bc 1.2.4. ex+y = ex + ey 1.2.5. ln (x + y) = ln (x) + ln (y) Show your steps): limx!0 2x+2??x??2 x2 (Hint: Use LHopitals Rule.) er; which is widely used for continuous time models in nance, macroeconomics, labor economics, etc. In macro models, sometimes, it is convenient to get a polynomial approximation of er: Calculate the rst order and second order Taylor approximation of er around r = 0. R 1 0 ex+2 (x + 1)2 dx: Use substitution u = x + 1, EXPLAIN why d (x + 1) = du = dx: (You can solve a simpler integral R 1 0 ex+1 (x + 1)2 dx, but will only receive half credit for it.) AX = b. The column rank of A always equal to the row rank of A. (a) If A has 20 rows and 20 columns and all elements in b are 0, can we concluded that the system has a unique solution? Can we conclude that the system has solution(s), either unique or innitely many? You must explain your answer. 1
to compute for the Hessian. 2.7.For the utility maximization problem with 2.6. 2.5.Compute the total di造eretial of (b) If A has 20 rows and 20 columns and NOT all elements in b are 0, can we concluded that the system has solution(s), either unique or innitely many? You must explain your answer. (c) If A has 20 rows and 1 column and NOT all elements in b are 0, can we concluded that the system has no solution? You must explain your answer. (d) An invertible squared n-dimension matrix A always have full rank. That is, r (A) = n: If det (A) = 9; and all elements in b are 0, can we concluded that the system has a unique solution? You must explain your answer. (e) If A has 20 rows…