Third Test:

  
 
 
1) Uniform Continuity : Definition 3.35 
   Important Theorem(Page 81 3.39) Continuous 
   functions are uniform continuous on closed bounded 
   Intervals. 
   Prove that f(x)=x^2+4x is not uniform on all of R. 
 
2) On Differentiation: Definition of derivative. 
   The Mean Value Theorem. 
    
 
3) Definition of Partitions, lower sum L(f,P) and upper sum  
   U(f,P). Remarks 5.6 and 5.7    
  
4) Definition of integrable, Theorem 5.10 (with proof) 
 
5) Definition (page 112 / 51.3) of integral  
   Theorem 5.15 (proof!) 
 
6) Riemann sums: Definition 5.17, Theorem 5.18 
   (an other way to define the integral) 
   Linearity of integral: Theorem 5.19 and 5.20    
   Comparison of integral: 5.21 
   Absolute Values: 5.22 
   The proof of one of the Theorems 5.18 - 5.22 
   will be asked in the test. 
 
7) The Fundamental Theorem of Calculus 
   Theorem 5.28  (know proof) 
 
8) Integration by parts (5.31) and by Change of 
   variables (5.34). Know how to prove it 
    using the Fundamental Theorem of Calculus. 
 
9)  Improper Riemann Integrals: Definition 5.38, 
    Theorems 5.42 and 5.43. 
        
         }