Matematyka Ogólna, RMT

2010/2011, sem. zimowy

Funkcje cyklometryczne

Zad.1. Wyznaczyć dziedzinę danej funkcji:

(1) f (x) =

√

− log cos x,

(2) f (x) =

√

1−x

2

2

x

−4

,

(3) f (x) =

log x·

√

tg x

x

2

+3x+5

,

(4) f (x) =

√

log(2x

2

−8)

3−3

x

,

(5) f (x) = 10

2

3 arcsin x

,

(6) f (x) = 2 − arccos 2x − 1,

(7) f (x) = ln

√

1 − sin 2x,

(8) f (x) = ln

q

1
2

− cos 3x,

(9) f (x) =

arcsin(2−x)

x

2

−4x+4

,

(10) f (x) = arcsin

√

3x,

(11) f (x) =

parccos(2x + 1),

(12) f (x) = arcsin

q

x+1

x

,

(13) f (x) =

q

ln

2x

x−1

,

(14) f (x) = ln

q

3x−4
4x+3

,

(15) f (x) = arcsin

3

(ln x + 5),

(16) f (x) =

1

arccos(1−ln x)

,

(17) f (x) =

p√

2x − 3,

(18) f (x) =

p

5 −

√

5 − x,

(19) f (x) = arcsin(x − 1),

(20) f (x) = arccos(

3x−1

2

),

(21) f (x) = arctg(12 −

2x

√

x

2

+1

),

(22) f (x) = arccos(x

2

− 4),

(23) f (x) = arcsin

x−3
x−4

,

(24) f (x) = arccos

2−x
x−5

,

(25) f (x) =

q

arcsin(x +

1
3

),

(26) f (x) =

1

arccos

x

2

,

(27) f (x) =

√

x+4

arcsin

x−2

6

,

(28) f (x) =

2

√

3x+1

+ arctg

1

x

.

(29) f (x) = ln(4x − 3) + arcsin

x+2

4

.

Zad.2. Obliczyć:

(1) 3 arcsin 1 − 2 arccos 0,

(2) 4arctg1 − arcctg(−1),

(3)

1
2

arccos

√

3

2

+ arctg(−

√

3),

(4) arcctg

√

3 − 3 arcsin

√

2

2

,

(5) 2 arccos(−

1
2

) − arctg1,

(6) arctg

√

3

3

− 2 arcsin

1
2

,

(7) cos(3 arcsin

√

3

2

+ arccos(−

1
2

)).

Zad.3. Naszkicować wykresy funkcji:

(1) f (x) = arcsin(x − 3),

(2) f (x) = arccos(x + 2),

(3) f (x) = arctgx +

π

2

,

(4) f (x) = arcctgx − 1,

(5) f (x) = 2arcctgx,

(6) f (x) = −arctgx.

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