DERIVATIVE (section 28)

-- definition of derivative and geometric meaning; be able to decide whether the function 
   is differentiableand compute derivative directly from definition 

-- Theorem: differentiablility implies continuity (know proof). Know why converse is not true 
   (counterexamples).

-- Basic differentiation rules (sum, product, quotient) as in Thm 28.3 (know proofs). Chain rule 
   (proof not required)

Practice: examples 1-3 p.224, questions 28.3, 28.7, 28.9, 28.11, homework 11 

MEAN VALUE THEOREM, RELATED RESULTS AND COROLLARIES (section 29)


-- Derivative is zero at max/min points (know precise statements
   and proofs). 
  
-- Statements and proofs of the MVT and Rolle's theorem, geometric meaning
  
-- a function with identically zero derivative is constant

-- increasing/decreasing functions, relation to positive/negative derivatives 
   (know precise statements and proofs for all cases: does increasing function 
    have positive derivative? Does a function with negative derivative strictly decrease?) 

Practice: homework 12

TAYLOR APPROXIMATION AND SERIES (part of section 31)

-- Definition for the Taylor approximation, remainder and Taylor series centered at c, 
   for a function which is differentiable the required number of times
   (be able to write out approximations for a given function, via direct computation)

 
-- Taylor's theorem on remainder estimates (Thm 31.3+corollary 31.4). Proofs are not required, 
   but you should know the statements and be able to use them to establish convergence of simple 
   Taylor series. 

-- Know Taylor series for functions such as sin x, cos x, exp(x), with *proof of convergence*.

Practice: examples 1.2 pp.252-253, questions 31.1, 31.2; also, write a Taylor series for 1/(1-x) 
           and study its convergence