Question no#2

EXERCISE 2 (17 points) Consider the following system of two non-linear equations in two unknowns: -2x 0 | (1 ) -exp(x) - 12 = -c (In) where y and r are the unknowns and ce R is some erogenous variable. 1. Assume that c = 1 and answer the following questions: (a) (3 points) How many solutions does the system have? You do not need to calculate the solution(s), but please explain your answer. Hint: Try to sketch the equations in a two dimensional coordinate system. To sketch the second equation, you might want to use a table. (b) (1 point) Verify that (r*, y*) = (0, 0) is one solution. 2. Assume now that c increases by the small amount Ac > 0 from 1 to Act 1 (i.e., the LHS of equation (1/) becomes -Ac-1). Use first-order approximation to linearise the system and find the new solution in the neighborhood of (0,0). That is, (a) (1 point) Write each equation as a function, y(r), with r being the argument. Don't forget to replace -c with -(1 + Ac) in equation (II). (b) (4 points) Approximate the function that corresponds to equation (1/) around the point (r, y) = (0, y(0)) by using the first-order approximation method from the lecture. (c) (4 points) Use the linear function you derived in part (b) to set up a system of two linear equations in two unknowns. Solve this system to obtain an approximate solution, say (r, y). This solution will depend on Ac. (d) (4 points) Let (i, y) be the exact solution to the system given that c = 1 + Ac. Without calculating the exact solution, is + larger or lower than 27 Please explain your answer. Hint: A graphical answer (or explanation) is fine