analisi 1, 2, algebra e geometria

Question 1

log_a (a^x)

Answer 1

= x

Question 2

log_a (xy)

Answer 2

= log_a(x) + log_a(y)

Question 3

log_a (x/y)

Answer 3

= log_a(x) - log_a(y)

Question 4

log_a (x^b)

Answer 4

= b * log_a (x)

Question 5

log_b (x) (portarlo in base a)

Answer 5

= log_a (x) / log_a (b)

Question 6

sommatoria da k=1 a n (k)

Answer 6

n*(n+1)
————–
2

Question 7

sommatoria da k=0 a n (q^k)

Answer 7

1-q^(n+1)
————
(1-q)

Question 8

sin (-x)

Answer 8

• sin (x)

Question 9

cos (-x)

Answer 9

cos(x)

Question 10

tan(-x)

Answer 10

-tan(x)

Question 11

cot(-x)

Answer 11

-cot(x)

Question 12

sin (a+b)

Answer 12

sin(a)cos(b) + cos(a)sin(b)

Question 13

sin (a-b)

Answer 13

sin(a)cos(b) - cos(a)sin(b)

Question 14

cos (a+b)

Answer 14

cos(a)cos(b) - sin(a)sin(b)

Question 15

tan (a+b)

Answer 15

1 - tan(a)tan(b)

Question 16

tan (a-b)

Answer 16

1 + tan(a)tan(b)

Question 17

sin (2a)

Answer 17

2*sin(a)cos(a)

Question 18

cos (2a)

Answer 18

cos^2(a) - sin^2(a)

Question 19

tan (2a)

Answer 19

1 - tan^2(a)

Question 20

sin (a/2)

Answer 20

+ o - sqrt ( 1 - cos(a) ) / 2

Question 21

cos (a/2)

Answer 21

+ o - sqrt [ 1 + cos(a) ] / 2

Question 22

tan (a/2)

Answer 22

+ o - sqrt { [ 1 - cos(a) ] / [ 1 + cos(a) ] }

Question 23

sin h (x)

Answer 23

      2

Question 24

cos h (x)

Answer 24

     2

Question 25

disequazione irrazionale:

sqrt(a) > b

Answer 25

sistema 1:—————— sistema 2:
a >= 0 ———————– a >= 0
b >= 0 ———————– b < 0
a > b^2

Question 26

z = x + iy

scrivere coniugato e modulo

Answer 26

coniugato: z’ = x - iy
modulo: | z | = sqrt ( x^2 + y^2 )

Question 27

z = x + iy

scrivere forma trigonometrica

Answer 27

z’ = p ( cos(a) + i*sen(a) )
p = sqrt ( x^2 + y^2)
cos(a) = x / sqrt ( x^2 + y^2 )
sin (a) = y / sqrt (x^2 + y^2)

Question 28

z = x + iy

scrivere forma esponenziale

Answer 28

z’ = p*e^(ia)
p = sqrt ( x^2 + y^2)
cos(a) = x / sqrt ( x^2 + y^2 )
sin (a) = y / sqrt (x^2 + y^2)

Question 29

lim x—–> inf [ 1 + 1/x ] ^(x)

Answer 29

e

Question 30

lim x——>inf [ x^(1/x) ]

Answer 30

1

Question 31

lim x——-> 0 [ 1 + x ]^(1/x)

Answer 31

e

Question 32

derivata di x^a

Answer 32

a* x^( a- 1 )

Question 33

derivata di sqrt(x)

Answer 33

2*sqrt(x)

Question 34

derivata di a^(x)

Answer 34

a^(x)*ln(a)

Question 35

derivata di tan(x)

Answer 35

cos^2(x)

Question 36

TEOREMA DI ROLLE

Answer 36

se una funzione [a,b]—>R è continua in [a,b] e derivabile in ]a,b[, e:
f(a) = f(b)
allora:
esiste c tale che f’(c) = 0

Question 37

TEOREMA DI CAUCHY

Answer 37

siano f,g definite da [a,b] ---> R,
continue in [a,b] e derivabili in ]a,b[
tali che g(a) diverso da g(b)
e non esiste una x tale che g'(x) = f'(x) = 0
allora:
esiste c tale che: 
f(b) - f(a) ......... f'(c)
---------------  = -------
g(b) - g(a) ....... g'(c)

Question 38

TEOREMA DI LAGRANGE

Answer 38

sia f: [a,b]—->R continua in [a,b] e derivabile in ]a,b[
allora:
esiste un c appertenente a ]a,b[ tale che:
…………… f(b) - f(a)
f’(c) = —————-
…………….. b - a

Question 39

TEOREMA DI DE L’HOPITAL

Answer 39

siano f,g derivabili
sia x’ punto di accumulazione tale che
lim x—>x’ f(x) = 0 e lim —>x’ g(x) = 0
(oppure entrambi infinito)
allora, se esiste, il limite:
lim x—>x’ [ f(x) / g(x) ] = lim x—>x’ [ f’(x) / g’(x) ]

Question 40

TEOREMA DI FERMAT

Answer 40

se esiste un massimo o un minimo relativo in una funzione, allora f’(c) = 0
con c punto di max o di min relativo

Question 41

∫ x^α dx

Answer 41

x^(α+1)
———- + c
α+1

Question 42

∫ a^x dx

Answer 42

a^x
——- + c
ln a

Question 43

METODO DI INTEGRAZIONE PER PARTI:

Answer 43

∫ f ′(x) ⋅ g(x) dx = f(x) ⋅ g(x) − ∫ f(x) ⋅ g′(x) dx