3. Consider the initial valve problem for U = U(t, x) for XER given by Ut + UU x = 0, U (0,x) = 1+ x² Write down a formula for the characteristic line that goes through the point y on the x-axis. sketch some of the characteristic lines in the (t,x)- plane calculate the critical time, t*, at which the solution, u, develops a vertical tangent line. Use the method of characteristics to sketch the Solution as a function of X just before and just after the critical time (Just after the critical time interpret the solution as a multivalved function)

Repeat the instruction of Exercise 11 for the function. f(x)=x3+x For part d, use i. a1=0.1 ii a1=0.1 11. Consider the function f(x)=4x2(1x) a. Find any equilibrium points where f(x)=x. b. Determine the derivative at each of the equilibrium points found in part a. c. What does the theorem on the Stability of Equilibrium points tell us about each of the equilibrium points found in part a? d. Find the next four iterations of the function for the following starting values. i. a1=0.4. ii. a2=0.7 e. Describe the behavior of successive iteration found in part d. f. Discuss how the behavior found in part d relates to the results from part c.