MAT 1235Calculus II

Section 6.3* The Natural Exponential

Functions

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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