Post on 21-Feb-2016
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MAT 1235Calculus II
Section 6.3* The Natural Exponential
Functions
http://myhome.spu.edu/lauw
Caro... Caro will not be here today, tomorrow,
and Monday She will participate in an international
math competition (Mathematical Contest in Modeling 2016)
Homework and … WebAssign HW 6.3*
• Part 1 (16 problems, 95 min., due tomorrow)• Part 2 (8 problems, 28 min., due Monday)• Both parts available after class• Note that there will be HW for 6.4* that due on
Monday also (7 problems, 30 min.)
Preview
xln
xe
xalog
tivesAntiderivaesDerivateiv
Properties
xa
Recall 6.1
6.2*
))((1)(agf
ag
x
dtt
x1
1ln
1 fg
Definition (6.2*) Let be the number such that 1ln e
x
y
1
e
lny x
The number e... Let be the number such that
………
1ln e
On the other hand...Let be the inverse function of
In particular, we have
xx ))exp(ln(
At this point,...We have not establish any relationship between the number and the function .
So, how are they related?
, i.e. , i.e. , where is an rational number, we get
Then,… xx ))exp(ln( 1ln e
r
r
rr
er
eer
ee
)exp(
))ln(exp(
))exp(ln(
, i.e. , i.e. , where is an rational number, we get
Then,… xx ))exp(ln(
r
r
rr
er
eer
ee
)exp(
))ln(exp(
))exp(ln(
1ln e
So,… and give the same values for rational
numbers However, is undefined for irrational
number (not yet!)
It makes sense to define…For all ,
exp( )x
x
e x
Properties
no. rational,
,
,)ln( ,ln
ree
eeeeee
xexe
rxrx
y
xyxyxyx
xx
Properties (limits at infinities)
lim
0lim
x
x
x
x
e
e
xy ln
xey
xy
Properties (limits at infinities)0lim
x
xe
xey
Example 1 (a)2
2
1lim1
x
xx
ee
Example 1 (a)
lim
0lim
x
x
x
x
e
e
0lim
x
xe
2
2
1lim1
x
xx
ee
Example 1 (b)
lim
0lim
x
x
x
x
e
e
0lim
x
xe
2
lim x
xe
Example 1 (c)
lim
0lim
x
x
x
x
e
e
0lim
x
xe
12
2lim x
xe
Remarks is not a number, it is a concept. In particular, it does not make sense to
write: Again, you can blame whoever told you
that this is ok. But it is your responsibility to do the right thing from now on.
, 5,e
Derivatives & Antiderivatives
Cedxe
dxduee
dxdee
dxd
xx
uuxx
,
Derivatives & Antiderivatives
x xd e edx
Example 2. find ,12
yxey x
u ud due edx dx
Example 32xxe dx
x xe dx e C