Calculus 2 Weiqi Zhou
if is differentiable in some neibourhood of ，then near we have
let ,
then ,
consequently
by MVT we get
if is twice differentiable in some neibourhood of , then near we have
if is -th differentiable in some neibourhood of , then near we have
by the L'Hospital's rule we get
if is differentiable in some neibourhood of , then there exists with , so that near
if is -th differentiable in some neibourhood of , then there exists with , so that
is called the -th order Taylor expasion of at (if , then it is sometimes also called the Maclaurin expansion)
find the -th order Taylor expansion at for the following functions, write out the Lagrange remainder and the Peano remainder respectively
,
the Lagrange Remainder:
, if and are bounded, then
often used when：, and is continuous on
example：； counterexample：, grows as fast as
find the -th order Taylor expansion of at
examples:
it need not always converge, e.g.,the expansion of at does not converge for
compute ，with precision
expand at : ，
take
，
compute
hence
compute , with precision
the capital doubles after about years if the annual (compound) interest rate is
（ has more divisors）