Math for Econ II, Written Assignment 3 (28 points) Due Monday, July 24, in class Summer 2017 Please write neat solutions for the problems below. Show all your work. If you only write the answer with no work, you will not be given any credit. Staple your homework if you have multiple pages! 1 Part I (28 points) 1. (2 points) Find A 1 \u0014 if A = 3 2 0 1 \u0015 . 2. Data interpolation has many applications. Suppose we have been given the task of finding the degree 3 polynomial P (x) = ax3 + bx2 + cx + d which satisfies P 0 (1) = 3 and passes through the points (1, 1), (0, 1), and (1, 3). (a) (3 points) Show that the coefficients of P satisfy the system of equations 3a 2b + c = 3 a + b c + d = 1 d=1 a + b + c + d = 3. a b = r so we could solve for a, b, c, and d. Write a matrix equation of the form A c d (b) (5 points) Find A1 , and use it to find a, b, c, and d. Write the formula for P (x). (c) (2 points) Your boss suddenly changes the problem at the last minute. You have now been asked to find the degree 3 polynomial f which satisfies f 0 (1) = 13 and passes through the points (1, 3), (0, 2), and (1, 3). Use your work from parts (a) through (c) to find the formula for f . a b c 1 1 0 (b) (2 pts) 0 d e 3. Find the following determinants: (a) (2 pts) 1 3 2 0 0 f 1 0 0 4. (3 points) Find all values of x so that the following matrix is invertible: 2x 0 3 0 1 2x 0 3 A= 0 0 2x 0 0 0 1 2x 5. (5 pts) Find 1 0 0 3 0 2 1 0 6 0 3 2 4 9 1 . 4 1 4 2 0 0 . 12 4 1 1 6. (4 pts) If A and B are invertible n n matrices, show that the inverse of AB is B 1 A1 . In other words, show (AB)1 = B 1 A1 . 2 Part II (28 points) 1. (5 points) An economy has three markets with supply and demand functions for the three goods given by q1S = 20 + 3p1 p2 q1D = 255 2p1 p3 q2S = 125 + 2p2 p1 p3 q3S = 4p3 2p1 q2D = 200 p2 q3D = 225 2p3 p1 Use Cramer's rule to solve for the equilibrium prices p1 , p2 , and p3 . 2. Assume an economics model where we have three industries with the following consumption matrix 0 0.1 0.4 4 C = 0.5 0.4 0.2 , and final demand d = 8 , 0.1 0.2 0.3 6 where the components of d are in millions of dollars. (a) (2 pts) What does the entry c12 = 0.1 mean? (b) (2 pts) Compute the output levels to meet the total demand. Note: You can use a calculator to do the computations for problem 2, but you must write every step of your work on paper. Do not skip any steps!!! 3. (6 points total; parts (a) and (b) are each 3 points) Estimate the area between the x-axis at the curve f (x) = 1 from x = 3 to x = 12 using: x+1 (a) a left-hand sum with n = 3. Is this estimate an overestimate or underestimate? (b) Write the formula for Rn (the right-endpoint sum with n rectangles) using sigma notation. 4. A function f is graphed below. y 8 y=6.2 7 6 5 4 3 2 1 -4 -3 -2 1 -1 2 x Estimate the area between the x-axis and the graph of f from x = 4 to x = 2 using: (a) (3 pts) L6 (b) (3 pts) R6 (c) (2 pts) Use the upper sum with n = 3 to give an upper bound to the area. 5. (5 points) Find the area between the x-axis and the curve y = x3 from x = 0 to x = 1 by finding the general formula for Rn and then taking lim Rn . n You may use the formula: (13 + 23 + 33 + + n3 ) = n X i=1 i3 = n2 (n + 1)2 . 4 Extra practice (not to be handed in): 1. Find the inverse of the coefficient matrix in problem 1 by using the classical adjoint formula. Use the inverse to find p1 , p2 , and p3 . 2. Find the area between the x-axis and the curve y = x3 from x = 0 to x = 2 by finding the general formula for Rn and then taking lim Rn . n You may use the formula: (13 + 23 + 33 + + n3 ) = n X i=1 i3 = n2 (n + 1)2 . 4