Limitas y Derivadas: Conceptos Básicos, Ejercicios de Cálculo

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Unit1:

Limits

LIMIT

PROPERTIES

Occustnt

rule

2 Identity

Rule: Coefficient

Rule.

limn

=

u

I=

(im[bof(x]

=b.

liMf(x)

x

x

⑪

motuctrule:

X

• >

6

x

• b

/im

(f(x). 9(X)]

=

lim

f(x)

·

(i,

a

X-

>

( x- C

⑤ Quotient

Due:

xi(x)

=

liM

g(t)

FO

/im

G(x)

i

X- C

⑥

pover

Rule:

x

C

⑦

composite

Rule:

/im

[f(x)

=

(life)

im

f(gcx)_

f((img(t)

X

n x

x

x

C

I experent

rule: x =In

limits

W/Y

SO

lig

big

bigur=0,

Srale=0,

gt=

y,

all--

Discontinuities

②

nole ③

Infinite

① JUMP

·

wen

f(x)

lim f(x)

· CANNOT we

·

2

different

liMil ·

can

he remues

remued

from (t)

& (-) when he elation. At least

as

· CANNOT

be

rende is

fuctured size limit

DNE

to

0

emer-

u

key:

continuous

if f(x)

=

limfCH

X-x

·

buth sizes

must

be

eal

(t) a

C-

Icukes

a saves

a

63

=

(a

x)(a

• ab

a

• 62

=

(a

b)(a

b)

a

63

=

(a

• ()(a

ab

+b2)

92

62

= (a

x)(a

x)

properties

sinx

lim-=

1

SIN

lim

P

= 0

CQS

X

/im

Fux

=

1 lim

(

0

X

vertical asumies

lascites

· the sizes limits atV.A

Small

-o =

y

approach

y,

D

very

lig

Small

f(x)

=

x

2

-x

• 12

weig faster

x

7

-->

Stuer

=

NO

H.A

X

V.A X =

7

lig

42

5x

in

the

H.

A

y

=

3

ems

Oscleeze

iovem:

2

Intermediate

race never

If f(x)

(q(X)

(h(x) (IVT):

Men,

lim f(x

$ ·If

a

function

is itines

/im h(x)

=

2 S

der a

clues

interval,

new

he

im

G(N)=

L

Factin

mustpass

tech

· De sNch

in Gemen

every

x a

y

rule

between

will

be he

she as

He initial

XsY

as he

he

over I

if

my

final

x

sy

values

hue

the

she

lim

at

mat

point.

I

velocity,

acceration...

· G(t)

=

displacement

·d(t)

=

v(t)

·vCH)

=

a(t)

A

speeding

up

->

sal

sic

USIA

A

slowing

tam

-> apposite

sans

USA

· if

VCH) 0

->

RIGHT,

UCHC

-> left

·IF rct)

=

0,

Stup chances

direct

· If act) (0-V

is increasing, act)

c0->

V is

recensing

· IfacH

= 0

->

V

is consent

A

difrentiability

DS

imply

IDifferentialeility continuity

·

at

cusps or covers

he serivate might

not

exist,

so

he

function

is not dirential.

3 V.

A

men discontinues

herivates of equentials

a lous

f(ex)

=

ex

flugax)

= aox

f

(ax)

=

a* olna

f((nx)

=x

rems

· Extere Vale

inem: If

Mech

rue

recen (Evil:

If

continues of a

clues

He fuction is on

a

clues

interval,

he suction interal, continues,

ans

*I

nue

an

absolve differentiable

here is sue

Minimum

a

absolute

Its

race

in he interal

maxin

he interval. at while he

instuturucy

·Ethere

roles:

mee f(x)

=

0 ar rate

of chance

is

equal to

Umb

or

he enjoints

he

cry.

Cunce

of

ray.

f(x)

=

(f(u)

6

• 4

Wit 3:

ORrotct

Due:

f(x).q(X)

f(x)g(x)

i

hitin e

in

a

ne

immic

rus

64

· Jxe=e

(x)

=

of(lugax)

=

max

terrace

of x

Alne*

=

1

· f(x)

=

axolna.serivative

ofX

EX.

1093

(x2)

=

ins2x

5

=

1n5.

62x

3

= 62x

In 6

2

Implicit

direntiation

is

Meet

to

get

all he

y'

in one

& f(x)

=

1 f((4) =y sizes

he rest on he over a

fivise.

Mit

4:

Economy

review

· cost

functiv->((X=total cost

of

producing X-mils

·

Arg

cost

fencin-

[(x1=

·

necome function

• RCX)

=

rutal

recence

received

on

he sale

Of X-mil

·

Marginal

cust=

((X) Produce 1 mue

item

derivative of

((x+1)

• ((X)

costfunctio

· MAX

PROFITS

I

getmust

fucin M(X)=R(x)-(CH)

②

n (x)

=

R'(x)-CCH

('(X)

=

hi(x)

③

n'(x1-((x)

=

0

⑪ Ge+

X!!

⑤

To get

$S:

plug-in

xinto

i(x),

normal

NO

serivate

Infinity

limits

smalle-y

x=-

D 3

small

Krue

curve

• O

smuler=

y

line

motion:

· velocity & acceleration:

-> save sian:Objectspeeding up

-> opposite

siam:object skwing sun.

·

Particle chances

direction when VCH = 0

·

acceleration

(I):

Velocity

increasing

·

acceleration :rerucity

recreasing

Mutt/sel

e

ft

viCt) S'Ct

· averace velocim:AD1+(yet area

under craph)

fHs

·

amacceleration:

IHOC?

Internetutius

men

he

thing

is at

Ives,

itwill

(

(75)

=

0,

tune x-vaue for

he

thing

to

increase

by

1

more

context.

Inevization

Eurer->lefrect

arox(

· If

concue

up: unterestimate f(x)

=

((x)

=

M(x

x,)

y

f

"(N>

· Ifcance bun:

derestimatio

mer

f"(X)(8 similar trinces

irecated Rules

LampOSt:

Trwah:

20

I

=E

(Total

leases

=

53

Cheicht

Of

SIPS:

-mini

trinques

①

Determin. I,l,

k

TCy

=

1x

M

23

4

nu

②

If necessary,

chert

204

=

2x

I

mil. 53

③ Fins

ecuation a

returnie

2

Missing

elements

(sim. As

y

=

Earative!

⑭

plug-in

races

a come

cylinder:

fuctur

I

=

r(qilen)

⑤ Find

Serirune

ucgiens

Fins / in

terms Of

mili

Area->ft/ses

n

volure- Ite/ses

heint/man->fH/sec

radius

->It/sec

L'hospital

Role:

A

COS

(0)

=

1

sin

(0)=

0

EXAMPLE:

lim

+x)

=

B

X

• >

I

*is

ideterminate

I

o

Plythusitem

exe.

+1)

beenre

I

Appy

hospital

=

3

im

e-again

the

e

Quiz:

①

f(x)

=

3 x

2e

3x/2.

• 3e

f(x)

=

3

be

• 3x

g((x)

= 4

4

3

be

3x

lim-male

since

to

x

• yu

i

I

②Yet_init-

wis

indeterminate

liM

to

o

tet-

te=

=

Mit

&

mech value

teorem :If f

is conting on

he

Closes

interal (a,b) and

sidentiable

here

exits

a nveler

is

such much:

f((c)

=

f(x)

• (a)

6-

4

Molle's

Neuvem:

If I

is continues out

sitentiale

on

he

interval

(a,e)

here exists

at leastare number

were

fi(l)

=

f(x)

=

f(b)

my

if

Anuc

.......

· relative or

local extrema:

y-races

on

he

gru

where

he function chances from

increasing

to

secensing

8 vice-versa.

·Alsolue Exrema:

The

kinestor last

y-rac

on

he crap

of

a function or

on

a specific

semain

ofa

function.

Exhere

race never:If I

is continues on

a

Cases

interal,

new

I has both a max

valed

a

min race

on

he interval [a,b].

·

critical

races: on

x-val in

he some off such hat eiler

f()

=

0

a

f(CC)

DNE.

I fuctor equation, get

X-values

Endpoint

a13G

k

Use

constitutes test

t

find

max

8

Min.

optimization

·use ist derivative

test

to

find

Maximize

or

minimize.

Use to justiny.

mit

hieren Sums

al

·

lefthand:

all races exect

he

cast

(0-(n-1)

ja · Richt

hunt:all races except

he first

(1-n)

8

mis

point:create

interals,

use

as a he

misre

rall

·

Travezoid

Sun:trunzis

Fermuln:

Elle,+bz)h,

leut ....

creat

he arrane

of

richt

a left

A

IS

summation

Notation

1x

=

=>

Witt

Unver

limit

Stuna

Pant

sum

E Xnexressin

Xu

=

a

1x an

3

insets

u

=

1

Iver

limit

1Xef(Xn)=)

Find a

sumam string

printhe

arecruace,

we

f(x)

is

he revent

Areu

=

SYX)6X=

y

,

(f(x).x]

Xu

=

a

1X.

1x

=

6

A

A

=

a

+sn=

1

E

Fatima Michel

b.

1x=

=

a

↑ ↑

565

1x

=

1

2

hi(*)

(2n+

i

(H(

=

1(

3

6

LHC1= 18

yaxu

=

a

xXn

6

a

=

1

a

=

1 6

• 1

=

1

b

=

2 b

=

2

00 xu

=

a

xnnn

(

Mp

=

3(7)

4(5)

4(b)

W

=

21

• 20

Si-xx

up

=

15

a

Es

3 A

=

I(z)(2)

2

·

) =

·use critical

⑧

numbers

27

①

Max,

min

2

=

z(2)(2)

=

=

2

P.

I:

F"(X)

=

f(X)

X

=

3,X

=

2

so nee

it

chuces

from

inceasing

in

serana.

increasing:

were

f(x)>C,

FCX)

is

f(x)>

I

increasing,

here fue:

secensing:

(

5,

-3)U(

3,2)

f(x)

cucce

uP:

->

F(X)

=

f(x)

·S3-

F"(X)

f((X)

so wer

candidates

test

f(X)>

f

=

3

=

3

shall san:

X

5

324

F"(x)

=

f(t)

F(X)

-9Y-

3

1 ·S

=

3

so wel

f(X)CO

min:

at

x=-5,

-S

13

z

=

1

MUX:

a

x

=

1,

3