Homework 1 due Tues.Sept2

Read quickly Notations section, Appendix A of Evans.
 
Evans exercises: 1.5 #1(a)(b), 

#4 Just do the case n=2 . Also you can use the standard Taylor's Theorem for 1 variable that you find in any beginning calculus book. 

2.5 #1,#2 



Extra exercise. (now due Thurs.Sept.4)  Find infinitely many distinct continuously differentiable nonnegative functions  
u  on the real line satisfying the two conditions:

     u'(t) = 2 sqrt[u(t)] for all t  and  u(0) = 0  .

[Here: "continuusly differentiable" means that the derivative  u'(t) exists and is continuous.]
Hints:

1. When you integrate to find solutions, there is a constant of integration.

2. Note that you can make a continuously differentiable differentiable function by putting together 2  adjacent C^1 graphs which touch at the common endpoint and share the same tangent line there. For example, |x^3| comes from -x^3 on the left and x^3 on the right.