Lista-C1V-SI1-4

Prévia do material em texto

Ca´lculo a uma Varia´vel
Professora: Maite´ Lista 4
1. Usando a definic¸a˜o, determine a derivada das seguintes func¸o˜es:
(a) f(x) = 1− 4x2
(b) f(x) =
1
x + 2
(c) f(x) =
1√
2x− 1
(e) f(x) = 2x2 − x− 1
(f) f(x) =
1− x
x + 3
(g) f(x) = 3
√
x + 3
2. Usando a definic¸a˜o, determine a derivada das seguintes func¸o˜es e
estabelec¸a os domı´nios da func¸a˜o e da derivada:
(a) f(x) = 5x + 3
(b) f(x) =
√
1 + 2x
(c) f(x) =
x + 1
x− 1
(d) f(x) =
1
x2
(e) f(x) = 5− 4x + 3x2
(f) f(x) = x +
√
x
(g) f(x) = x4
(h) f(x) =
4− 3x
2 + x
3. Calcule f ′(x):
(a) f(x) = 3x2 + 5
(b) f(x) = 5 + 3x−2
(c) f(x) =
2
3
x3 +
1
4
x2
(d) f(x) = 3x +
1
x
(h) f(x) = x3 + x2 + 1
(i) f(x) = 3x +
√
x
(j) f(x) = 3
√
x +
√
x
(k) f(x) =
4
x
+
5
x2
4. Obtenha a derivada das func¸o˜es abaixo:
(a) f(x) = (x− 1)(x + 1)
(b) f(x) = (3x5− 1)(2− x4)
(c) f(t) =
t− 1
t + 1
(d) f(t) =
2− t2
t− 2
(e) f(x) =
3
x4
+
5
x5
(f) f(x) = (2x+ 1)(3x2 + 6)
(g) f(x) = (5x−3)−1(5x+3)
(h) f(t) =
3t2 + 5t− 1
t− 1
(i) f(x) =
x + 1
x + 2
(3x2 + 6x)
(j) f(x) =
1
2
x4 +
2
x6
(k) f(s) = (s2 − 1)(3s− 1)(5s3 + 2s)
5. Calcule F (x) onde F ′(x) e´ igual a:
(a)
√
x
x + 1
(b)
3
√
x + x√
x
(c)
√
x +
3
x3 + 2
(d)
x + 4
√
x
x2 + 3
6. Dada a func¸a˜o f(t) = 3t3 − 4t + 1, encontre f(0)− tf ′(0).
7. Calcule f(x) onde f ′(x) e´ igual a:
(a) 3x2 + 5 cosx
(b) x2tgx
(c)
x + 1
tgx
(d) xsenx
(e)
cosx
x2 + 1
(f)
3
senx + cosx
8. Seja f(x) = x2senx + cosx. Calcule:
(a) f ′(x)
(b) f ′(3a)
(c) f ′(0)
(d) f ′(x2)
9. Calcule f ′(x):
(a) f(x) = x2ex
(b) f(x) = ex cosx
(c) f(x) =
1 + ex
1− ex
(d) f(x) = 4 + 5x2 lnx
(e) f(x) =
ex
x2 + 1
(f) f(x) = 3x + 5 lnx
(g) f(x) = x2 lnx + 2ex
(h) f(x) =
x + 1
x lnx
(i) f(x) = ln xx
(j) f(x) =
ex
x + 1
10. Calcule F ′(x) onde F (x) e´ igual a:
(a) xex cosx
(b) exsenx cosx
(c) x2(cosx)(1 + lnx)
(d) (1 +
√
x)extgx
11. Determine a derivada:
(a) y = sen 4x
(b) f(x) = e3x
(c) y = (sen t)3
(d) y = esen t
(e) y = (senx + cosx)3
(f) y = sen(cosx)
(g) f(x) = cos(x2 + 3)
(h) y = tg 3x
(i) g(t) = ln(2t + 1)
(j) f(x) = cos ex
(k) y =
√
3x + 1
(l) f(x) = etgx
(m) g(t) = (t2 + 3)4
(n) y =
√
x + ex
12. Seja f : R −→ R deriva´vel e seja g(t) = f(t2 + 1). Supondo f”(2) =
5, calcule g′(1).
13. Seja f : R −→ R deriva´vel e seja g dada por g(x) = f(e2x). Supondo
f”(1) = 2, calcule g′(0).
14. Derive:
(a) y = xe3x
(b) y = e−xsenx
(c) y = t3e−3t
(d) y = (sen3x + cos 2x)3
(e) y = ln(x +
√
x2 + 1)
(f) y = ln(secx + tgx)
(g) y = ex cos 2x
(h) y = e−2tsen3t
(i) g(x) = ex
2
ln(1 +
√
x)
(j) y =
√
ex + e−x
(k) y =
√
x2 + e
√
x
(l) y = cos3 x3