3. The Taylor Expansion

Calculus 2



Weiqi Zhou

Motivating Example：Differentiation

if is differentiable in some neibourhood of ，then near we have

Error Analysis: With Smoothness

  • let ,

  • then ,

  • consequently

  • by MVT we get

The Lagrange Remainder

if is twice differentiable in some neibourhood of , then near we have

The Lagrange Remainder

if is -th differentiable in some neibourhood of , then near we have

Error Analysis: Without Smoothness

  • let ,

  • then ,

  • consequently

  • by the L'Hospital's rule we get

The Peano Remainder

if is differentiable in some neibourhood of , then there exists with , so that near

The Peano Remainder

if is -th differentiable in some neibourhood of , then there exists with , so that

The Taylor Expansion

is called the -th order Taylor expasion of at
(if , then it is sometimes also called the Maclaurin expansion)

Exercise

  • find the -th order Taylor expansion at for the following functions, write out the Lagrange remainder and the Peano remainder respectively



  • ,

Estimate of the Remainder

  • the Lagrange Remainder:

  • , if and are bounded, then

  • often used when：, and is continuous on

  • example：；
    counterexample：, grows as fast as

Exercise

find the -th order Taylor expansion of at

The Taylor Series



  • examples:

  • it need not always converge, e.g.,the expansion of at does not converge for

Applications

  • compute ，with precision

  • expand at :
    ，

  • take

Applications

  • ，

  • ，

  • ，

Applications

  • compute

  • ,

  • hence

Exercise

  • compute , with precision

  • compute , with precision

  • compute

  • compute

The Rule of

  • the capital doubles after about years if the annual (compound) interest rate is





  • （ has more divisors）

Summary

  • The Taylor Expansion

  • The Remainders

  • Applications