1. Lista 12 de C´lculo I
a 2010-2 22
UFES - Universidade Federal do Esp´
ırito Santo Fun¸˜o logar´
ca ıtmica
DMAT - Departamento de Matem´tica
a
Fun¸˜o exponencial
ca
LISTA 12 - 2010-2 Fun¸˜es hiperb´licas
co o
ln x2 − 3
1. Seja f (x) = . Determine o dom´ ınio de f , os valores de x onde a f se anula e
(x − 1)(x + 3)
os intervalos onde a f ´ positiva e onde a f ´ negativa.
e e
Nos exerc´
ıcios 2. a 5. esboce o gr´fico da fun¸˜o.
a ca
2. f (x) = ln |x − 4| 4. F (x) = e|x+2|
3. y = | ln |x + 1| | 5. g(t) = 1
2 − e−t
Derive as fun¸˜es dos exerc´
co ıcos 6. a 16. (se for conveniente, use deriva¸˜o logar´
ca ıtmica)
√
e sen 2x x 10. f (x) = ex
x
14. f (x) = xπ + π x
6. f (x) =
ecos 3x
√ √ 11. f (x) = (xx )x
7. f (x) = e x ln x 15. f (x) = (ln x)x xln x
√
8. f (x) = ln x x2 + 1 12. f (x) = log2 x2 √
x+1
9. f (x) = (ex )x 13. f (x) = ( sen x) arcsen x 16. ln
(x − 1)3
Calcule y ′ nos exerc´
ıcios 17. a 19.
x y
17. ln + =5
y x
18. sen exy = x
y 2 cos x
19. = 2ln y , para x = 0 e y = 1
ex
1
20. Se senh x = − , encontre: (a) cosh x (b) tanh x (c) senh 2x
4
1
21. Determine x tal que tanh x = − .
4
Nos exerc´
ıcios 22. a 24. mostre que as igualdades se verificam.
x2 − 1
22. tanh(ln x) =
x2 + 1
23. cosh x + senh x = ex
24. (cosh x + senh x)n = cosh nx + senh nx (sugestˆo: use o exerc´ 23.)
a ıcio
Derive as fun¸˜es dos exerc´
co ıcios 25. a 27.
25. f (x) = tanh( sen x)
26. f (x) = senh (ln 2x) + cosh(ln 2x)
27. f (x) = xcosh x
√
π 3
28. Mostre que cot (π cosh (ln 3)) − arcsen (tan (π senh (ln 2))) = − .
2 3

2. Lista 12 de C´lculo I
a 2010-2 23
RESPOSTAS
√
1. Dom´
ınio = (−∞, −3) ∪ 3, ∞ ; 2x2 + 1
8. f ′ (x) =
f = 0 em x = 2 x (x2 + 1)
2
f > 0 para x < −3 ou x > 2; 9. f ′ (x) = 2xex
√ x
f < 0 para 3 < x < 2 10. f ′ (x) = xx ex (1 + ln x)
y x
2
11. f ′ (x) = (xx ) (x + 2x ln x)
1 2
12. f ′ (x) =
x ln 2
2. 0 x
ln ( sen x)
–1
13. f ′ (x) = ( sen x) arcsen x cot x arcsen x + √
–2
1 − x2
–3 14. f ′ (x) = πxπ−1 + (ln π)π x
y
1 2 ln x
15. f ′ (x) = (ln x)x x ln x + ln (ln x) +
2
ln x x
−(5x + 7)
3. 1 16. f ′ (x) =
2 (x2 − 1)
y
–1 0 x
17. y ′ =
–2 1 2 3 4
x
y
1 − yexy cos exy
10 18. y′ =
xexy cos exy
8
1
6 19. y′ =
4. 2 − ln 2
4
√ √ √
2 17 17 17
20. (a) (b) − (c) −
0 x 4 17 8
–5 –4 –3 –2 –1 1
–2 1 3
21. ln
y 2 5
1
1 1
2 eln x − e− ln x x− x2 − 1
x 22. tanh(ln x) = 1 = x
1 =
–1 2 (eln x + e− ln x ) x+ x
x2 + 1
5. –2
ex − e−x ex + e−x 2ex
–3 23. cosh x + senh x = + = = ex
2 2 2
–4
24. cosh nx+ sen nx = enx = (ex )n = (cosh x+ senh x)n
–5
(1 + 4x cos 2x + 6x sen 3x)e sen 2x 25. f ′ (x) = cos x sech2 ( sen x)
6. f ′ (x) = √
2ecos 3x x 26. f ′ (x) = 2
√ √ √ cosh x
′ e x (1 + x ln x) 27. xcosh x senh x ln x +
7. f (x) == x
2x