Lista - Aplicações de Derivada.pdf
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7/17/2019 Lista - Aplicações de Derivada.pdf
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2012.1
t
f (t) = 16t + t2
0 ≤ t ≤ 8
[b, b + h], 0 ≤ b < 8
[3;3, 1], [3;3, 01] [3;3, 001]
t
t = 3
t
y = b
t + ct y t
t = 2?
x
x = 3t2 − t3
x
t
4
4
4
W (t) =
20 + 1
2(t + 4)2, 0 ≤ t ≤ 60
24, 4t + 604, 60≤
t≤
90,
t
t = 50
51
t = 80
90.000
t
2500t2
[2, 5]
2
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x
x = 3 + 2t − t2, t ≥ 0
t
t
v(t)
x
x = −t3 + 3t2, t ≥ 0
v(t)
a(t)
limt→∞(−t3 + 3t2)
x =
−t3 + 3t2, t
≥0
P
y = 3x2 − 2x
x(t)
y(t)
P
dx
dt = 0 y P
x
P
P
y =
1
x2 + 1
x
5(m/s)
y
x = 10
a)f (x) = 2x − 1 b)f (x) = 3x2 + 6x + 7
c)f (x) = (x − 1)(x − 2)(x + 3) d)f (x) = x
2 + sin x
e)f (x) = e−x f )f (x) = x2
x − 1
g)f (x) = ex sin x, x ∈ [0, 2π].
a)f (x) = 3x2 + 6x + 1 b)f (x) = 4x3 − 8x2
c)f (t) =
t
−1
t + 1 d)f (x) = |2 − 6x|
e)f (x) =
x + 4, x ≤ 2x2 − 2, x > −2
f )f (t) =
3 − 4t, t > 04t + 3, t ≤ 0
g)f (x) =
1 + x, x < −11 − x2, x ≥ −1
h)f (x) =
10 − (x − 3)2, x ≤ −25(x − 1), −2 < x ≤ −1
− 91 + (x − 2)2, x > −1h)f (x) = 7x2 − 6x + 3 b)f (x) = 1
3x3 + 3x2 − 7x + 9
i)f (x) = 1
4x4 − 5
3x3 + 4x2 − 4x + 8 d)f (t) =
t2, t < 03t2, t ≥ 0
j)f (x) = 3 + (2x + 3)43 f )f (x) =
x + 1
x2 − 2x + 2
k)f (x) = x2√
16 − x
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y = logax
x x = e
a > 1
a
b
f (x) = x3
+ ax2
+ b
(−2, 1)
a,b,c d f (x) = 2ax3 + bx2 − cx + d x = 0
x = 1
a > 0
a)f (x) = −x3 + 5x2 − 6x b)f (x) = 3x4 − 10x3 − 12x2 + 10x + 9
c)f (x) = 1
x + 4 d)f (x) = 4
√ x + 1 −
√ 2
2 x2 − 1
e)f (t) = t2 + 9
(t − 3)2 f )f (t) = e−t cos t, t ∈ [0, 2π]
g)f (x) =
x2 − 4, x ≤ 24 − x2, x > 2
a)f (x) = x3 − 3x2 + 1 b)f (x) = x + 1
x c)f (x) = x2 + 1
x
d)f (x) =
x
1 + x2 e)f (x) = e2x
− ex
f )f (x) = e
1x
g)f (x) = x3 − x2 + 1
x h)f (x) =
3x2 + 4x
1 + x2 i)f (x) = −x4 + 4x3 − 4x2 + 2
j)f (x) = x2 − x + 1
2(x − 1)
l
l
a
A B
l
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375
2
3, 5
2 2 2, 5
400 2
a b a
400
2
a b
9
12
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a)limx→−14x3 + x2 + 3
x5 + 1 b)limx→1
x100 − x2 + x − 1
x10 − 1 c)limx→0+ xe
1x
d)limx→+∞e3x
x2 e)limx→+∞
lnx
e3x f )limx→0+ sin x ln x
g)limx→0+(1 − cos x) ln x h)limx→+∞(x2 + 1) 1lnx i)limx→0+
1
x + ln x
j)limx→0−(1 − cos x)1x l)limx→0
tg3x − sin x
sin3 xm)limx→0
sec3 x
1 − cosx
n)limx→+∞ x3e−4x o)limx→+∞
x − 3√
x3 − x
p)limx→1−e
1
x2−1
x − 1
q )limx→+∞
x
x2 + 1
x
f (x) = x2 sin 1x
g(x) = x limx→0 f (x) = limx→0 g(x) = 0 limx→0 f (x)g(x) = 0
limx→0
f
(x)
g
(x) 1