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- Estudiar la concavidad y convexidad, el crecimiento y el decrecimiento, los extremos y
puntos de inflexi´on de las siguientes funciones:
a) f (x) = 2x^3 9 x^2
b) g(x) =
x
c) h(x) = x^4 12 x^2 + 8
d) t(x) = x · e
2 x
- El beneficio mensual de una compa˜n´ıa depende del n´umero de unidades producidas de
acuerdo a la funci´on:
P (x) = A(x 500)^2 + B
cuando x 0 donde P (x) representa el beneficio en euros y x es el n´umero de unidades
producidas. Sabiendo que el beneficio m´aximo es de 87000 euros (para x=500) y que si se
producen 600 unidades el beneficio es de 86000 euros, se pide determinar las constantes
A y B y representar gr´aficamente P (x).
- Calcula la derivada de las funciones:
1 f (x) =
x^3 +
x^2
x 5
2 f (x) = x(x + 2)
3 f (x) = x^2 (7 2 x)
4 f (x) = (2x 5)(4 3 x)
5 f (x) = (x^2 1)(2 3 x^2 )
6 f (x) = (1 + 5x^3 )(1 + 3x^2 )
7 f (x) = (x^2 3 x + 2)( x + 4x^2 )
8 f (x) = (3x
2 5 x + 4)
3
9 f (x) =
x^3
10 f (x) = (3x^4 3 x^2 + 5)^4
11 f (x) = (1 2 x + 3x^2 4 x^3 )^5
12 f (x) =
2 x 3
3 x
13 f (x) =
x^4 3 x^2 + 7x
2 x + 5
14 f (x) =
2 x^3 3 x + 2
x^2 x + 2
15 f (x) =
x^2 3 x + 4
2 x^2 2 x + 5
16 f (x) =
2 x + 3
(3x^2 2 x + 6)^2
17 f (x) =
5 x^2
(2x 8)^7
18 f (x) =
p 4 3 x
19 f (x) =
5
p 7 x^2
20 f (x) =
p 3 5 + 4x^2
21 f (x) =
p 5 5 x^2 + 2x 7
22 f (x) =
p 1 x
p 1 + x
23 f (x) =
p 2 x + 3
p x
x
24 f (x) = x^2 +
p x^3 2
25 f (x) =
x 3
26 f (x) =
p 3 2 x^4
27 f (x) =
r 1 x
1 + x
28 f (x) =
p 3 ln x
29 f (x) = cos
p 5 x
30 f (x) = 5^2 x
(^2) 4
31 f (x) = log 5 sen x
32 f (x) = 4ln^ x
33 f (x) = tan (2x 3)^4
34 f (x) =
1 + (sen x)
2
cos^2 x
35 f (x) =
x + sen x
x + cos x
36 f (x) = log 4 (x^2 2
p x + 3)
37 f (x) = ln
v u u t
e^3
cos
38 f (x) = ee
ex
39 f (x) = ex
2
40 f (x) = (ex)^2
41 f (x) = log 10
tan
p 2 x 1
42 f (x) = arc cos (1 ln x)
43 f (x) = arctan
2 + x
2 x
44 f (x) =
p 4 + 2x^ + ln
1 + x
45 f (x) = cos^2 (2x + 1)^3
46 f (x) =
x^6
5
3 x^2
2
47 f (x) =
p 3 x^2 +
x^5
48 f (x) =
x
3 2 x
2
x^6
49 f (x) = (x + 2)ex
50 f (x) = (3x 1)cos x
51 f (x) = (x + 2)^2 ln (x + 1)
52 f (x) =
x
2
x + 2
53 f (x) =
x
(x + 1)^2
54 f (x) =
e^3 x
x^2 + 1
55 f (x) = 4x^ + tan (x 1)
56 f (x) = ln (x^2 3)
57 f (x) =
x 2
2 x + 1
58 f (x) =
p x
sen x
59 f (x) =
p 7 7 x + 15
60 f (x) = 3^5 x^ · cos ( x 4)
61 f (x) = ln x^2
62 f (x) = (ln x)
2
63 f (x) =
p 4 21 x
64 f (x) =
ex^ + e x
ex^ e x
65 f (x) = log 2
3 x + 1
x^2 + 1
66 f (x) =
p x +
p 2 x + 3
67 f (x) = log
x^2
x 1
68 f (x) = cos ln 3 ⇡+x
5
69 f (x) =
xe
2 x
x^2 x 3
70 f (x) = 3^2
4 x
71 f (x) = (x^2 3)ex
72 f (x) = 7tan 3 x
73 f (x) =
p sen x
74 f (x) = 2x +
p x + 1
75 f (x) = tan (x^3 + 1)
76 f (x) =
3
p x^2 + 1
77 f (x) =
2 x
x 1
78 f (x) = 5x cos x
79 f (x) =
tan x
x
80 f (x) = x · sen 2 x
81 f (x) = (2x + 3)^2
82 f (x) = ex
(^2) 4
83 f (x) = ln (3x 2)
84 f (x) = 2^7 x
85 f (x) = cos (5x
2 )
86 f (x) = ln (sen x)
87 f (x) = sen x + cos x
88 f (x) = (x^2 + 1)6x
89 f (x) =
x^2 1
90 f (x) = x · cos
x
91 f (x) =
p 1 x^2
92 f (x) =
1 + sen x
1 sen x
93 f (x) =
p x + sen x
153 f (x) = ln [cos (x^2 1)]
154 f (x) =
1 + tan x
1 tan x
155 f (x) = ln (e
cos x )
156 f (x) =
p 32 x ^1
157 f (x) = ln (sen x · cos x)
158 f (x) = sen (
p x) + cos^2 (
p x)
159 f (x) = cos (2x^3 2
p 3 3 x)
160 f (x) = tan
3 x + 1
2
161 f (x) = sen
p x + 1
162 f (x) = cos (x^3 2)
163 f (x) =
2 x
1 x
164 f (x) =
2 x
(x 2)^4
165 f (x) =
4
p x
cos x
166 f (x) = arc sen
p x + 1
167 f (x) = x · arc cos 3 x
168 f (x) = arc tan (ln (5x))
169 f (x) = 2arc tan x
2
170 f (x) = e
arc sen x
171 f (x) =
arc tan x
x^2 + 1
172 f (x) = (6x 3)^6 x ^3
173 f (x) = (3x^2 2)cos^ x
174 f (x) = (ln 2x)ln 2x
175 f (x) =
✓r x
1 x
s x
1 x
176 f (x) = x^2 arc sen x^2
177 f (x) = sen
3 5
r
tan^2
2 x^2 + 5
3
178 f (x) = sen^3 x ·
5
r
tan^2
2 x^2 + 5
3
179 f (x) = ln (sen 2 x)
1 / 2
180 f (x) = tan
x
2
x^4 + 3
181 f (x) =
1 + 2tan x
x^3
182 f (x) = ln
103 x+
2 x + 8
◆x
183 f (x) =
p cos^3 (5x 3)^2
184 f (x) =
p x
p 3 3 x
2
p 5 3 x^2
e^4
185 f (x) = ln sen
1 x^2
1 + x^2
186 f (x) = log 2
p tan x
187 f (x) = x^2 sen x
188 f (x) =
p x^2 + 1ln x
189 f (x) = x
3 e
x+
190 f (x) =
p excos x
191 f (x) = arc sen (5x^2 )
192 f (x) = e
p x
193 f (x) = ln (sen x) + sen (ln x)
194 f (x) = arc tan (4x^3 )
195 f (x) = sen^2 7 x
196 f (x) =
p x
sen x
197 f (x) =
p 5 x^2
198 f (x) = esen^5 x
3
199 f (x) = arc cos 52 x
200 f (x) = x^5 +5x+
p x+
p 5 x+
p 5
p 5 5 x
