Mathematics practice

VALORE ASSOLUTO

k is a the variable on the other hand of the disequation



If K ≥0  ⋃

¿ (x)∨≥0

g g( x)≤−K g(x) ≥ K

| |

( ) ( )

< −K

g x 0  ≤ g x ≤ K

If K<0  ∀ ∈

¿ (x)∨≥0 x R

g

| |

( ) ∄

< 

g x 0

STUDIO DEL DOMINIO

intersection doesnt allow to the value to be reported in the

the

∙∩∘=∘

graph the intersection allows the value to be reported on the graph



∘∪ ∙=∙

DERIVATE STANDARD

f(x) f’(x)

n 0

X 1

2x

2

x n n−1

x x∗x

x x

e e

x x

∗ln

a a a

Ln(x) 1/ X

log 1/x*ln(a)

(x)

a

√ ( )

√

X 1 / 2 X

|x| |x|/x

Sin(x) Cos(x)

√

Arc sin x) 1/ 2

1−x

Cos(x) -sin(x)

√

Arc cos (x) -1/ 2

1−x

Tg(x) 2 2

1/ (x)+1

(x )¿

cos tg

( )

Arc tg(x) 2

1 / 1+ x

DERIVATE TEOREMI

Linearity '

( ) ' '

( ) ( ) ( ) ( )

=f

f g ± f x g ± f x

Homogeneity (“a” is a constant)

(a∗g( ))'=a∗g (x) ¿

x '

Multiplication/division

'

( ) ' '

( )∗f ( ) ( )∗f ( )+ ( )∗g ( )

=f

f g x g x f g x

( )

' 2

' '

( ) ( )

= ∗g−f ∗g

f / g f / g x

DERIVATE COMPOSTE '

√ √

( ) ( )

1. ( ) ( )

=1 ∗g )

g x / 2∗ g x '( x

x

( )

ln g(¿)

2. ¿

¿

¿

x

( )

(¿)

arc tg g

3. ¿

¿

¿

( )

√

' ( )

( )

( ) 2 '

4. ( ) ( ) ( )

=1 ∗g

arc sin g x / 1−g x x

( )

'

( ) 2 '

5. ( ) ( ) ( )

=−1 ∗g

6 1 / g x / g x x

'

( )

( ) ( ) '

6. ( ) ( ) ( )

=cos ∗g

sin g x g x x

'

( )

( ) ( ) '

7. ( ) ( ) ( )

=−sin ∗g

cos g x g x x

'

( ) '

8. ( ) ( )

g x g x ( )

=e ∗g

3. e x

( )

' '

( ) ( )

( ) ( ) ( ) ( ) ( )∗ ( )

g x ∗g +f

( )

g x f x x x g x

9. ( ) ( )

=e ∗ln −→e

f x f x

ESPONENZIALIF(x)=y IMP

x <0

e IMP

x =0

e verified for every x inside R



x >0

e

FUNZIONI RADICE X<0 impossible X=0 possible

x>0 possible

¿

0 ,+ ∞¿

ID= { }

∈ =¿

x R : x ≥ 0

included ; inf. Excluded (because it

0

is not calculable)

LOGARITMO y=ln(x)

¿

ID: ¿

0 ,+∞

¿

If x=0 ln(x)= IMP

If x=1 ln(x)=0

Ln(x)<0 0<x<1



Ln(x)>0 x>1



Ln(a/b) = ln(a)-ln(b)

Ln(a*b) =ln(a)+ln(b)

Ln(x)=a x=

 a

e

LIMITI ( )=0−¿

f x ¿

lim

x→+∞

( )=0+ ¿

f x ¿

lim

x→−∞ ( )=−∞

x → 1−¿ f x

¿

lim

¿ ( )=+∞

¿

x → 1+ f x

¿

lim

¿ ( )=−∞

x → 0+¿ f x

¿

lim

¿ ( ) =+∞

x → 0−¿ f x

¿

lim

¿

x =0+ ¿

e ¿

lim

x →−∞ x =+∞

lim e

x →+∞ ( )=−π

lim arctg x / 2

x →−∞ ( )=π

lim arctg x / 2

x →+∞

ASYMPTOTE

( ) =±

lim f x ∞ is a vertical asymptote

x=k

x→ k ( ) =k

lim f x is a horizontal asymptote

y=k

x →± ∞ the asymptote is

+∞  ∞

e the asymptote is 0+

−∞ 

e

INDETERMINATE FORM

0 / 0 , ∞ / ∞ , 0−∞ , 0∗∞

Simplify grupping the x with the major exponential and continuing the process

since you can reach a determinate one

ORDER OF INFINITE

The major order will be higher in the evaluation of the indeterminate form

It works only if y>0

x

1. y=e

3

2. y=e

2

3. y=e

1

4. y=e