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Maths I

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Submitted By jmqd1995
Words 888
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´
SOLUCIO
Soluci´ provisional Examen Setembre 2009-10 o 1. (a) Trobeu l’equaci´ d’una circumfer`ncia que passi pels punts (0, 2), (2, 0) i (4, 2). o e
(b) Trobeu el valors de k que fan que aquesta equaci´ sigui d’una circumfer`ncia, i en aquest o e cas indiqueu-ne el centre i el radi: x2 − 2kx + y 2 − 4y = −20
´
SOLUCIO
(a) (x − 2)2 + (y − 2)2 = 4.
(b) k > 4 o k < −4; centre (k, 2), radi



−16 + k 2 .

2. Digueu a quines funcions corresponen les seg¨ents gr`fiques, sabent que una s’obt´ a partir u a e d’una exponencial de la forma Aax i l’altra ´s un polinomi. Cal donar l’expressi´ de cada e o funci´ en la forma f (x) = ... o 3

2

1

-5

-4

-3

-2

-1

0

1

2

3

4

5

-1

-2

-3

´
SOLUCIO
3x
1
2 ∗ 2 i 2 (x − 2)(x − 3)(x + 1).
3. La capacitat de producci´ el`ctrica d’una central va ser l’any 2000 de 1000 unitats, o e
(a) Quant tardar` la producci´ a ser igual a 2000 unitats, si la producci´ augmenta un 3% a o o anual?
(b) Si la producci´ augment´s en 50 unitats anuals, quant tardar` a ser superior a 2000 o e a unitats? Plantegeu i resoleu aquest apartat usant desigualtats.
´
SOLUCIO
(a) 1000(1.03)t = 2000 → t = 23.44
(b) 1000 + 50t > 2000 → t > 20.
1

1
1−x2

4. Donada la funci´ f (x) = o (a) Calculeu l’aproximaci´ quadr`tica en x = 0. o a
(b) Useu aquesta aproximaci´ per a trobar el valor aproximat de o 100
.
99

´
SOLUCIO
(a) 1 + x2
(b) Si fem
5. (a) Calculeu

100
99

=

￿￿

1
1−x2

x10 +

obtenim x = 0.1 i per tant

1 x2 + 4ex + 1 +

100
99

√ ￿ x3 dx

= 1 + 0.12 ≈ 1.01

(b) Calculeu l’`rea tancada entre la funci´ f (x) = −(x − 2)(x + 2) i l’eix de les x. a o
´
SOLUCIO
(a)

x11
11

+

−1 x + 4ex + x +

(b)


￿

2
5



x5 + C

2

−2

x2 + 4 dx = −[

x3
32
− 4x]2 =
−2
3
3

6. (a) Donats els vectors ￿ = (1, 2), ￿ = (0, −2) i ￿ = (2, x), es demana a b c i. Trobeu un vector k￿ on k ∈ R tal que tingui norma (longitud) igual a 1. a ii. Comproveu que ||￿ + ￿ ≤ ||￿ || + ||￿ a b|| a b||

iii. Trobeu els valors de x que fas que ￿ sigui combinaci´ lineal de ￿ i ￿ c o a b.
(b) Donada la seg¨ent equaci´ matricial, u o
DX + BX + CX = 2X + D,

on totes les matrius s´n quadrades i de la mateixa dimensi´, digueu quines propietats o o s’han de complir perqu` es pugui a¨ e ıllar i a¨ ılleu la matriu X.
´
SOLUCIO
(a)

i.

1
√ (1, 2).
5

ii. ||￿ + ￿ = a b||



iii. Totes les x

1 = 1 ≤ ||￿ || + ||￿ = a b||



5+



4 = 4.23

(b) X = (D + B + C − 2I)−1 D; la matriu (D + B + C − 2I) ha de tenir inversa. 7. PREGUNTA (20 punts) Donada la seg¨ent equaci´: u o y 5 + x2 y 3 + y = 42, la representaci´ de la qual ´s o e

2

-5

-4

-3

-2

-1

0

1

2

3

4

5

(a) Calculeu els valors de y que corresponen a x = 1 i marqueu-los en el gr`fic. a (b) Calculeu usant derivaci´ impl´ o ıcita la derivada y ￿ i calculeu tamb´ y ￿ (1). e (c) Calculeu la recta tangent en el punt que t´ x = 1 i representeu-la al gr`fic. e a
(d) Calculeu el diferencial de y quan passem del punt de la corba que t´ x = 1 al punt que e t´ x = 1.1. Marqueu aproximadament aquest valor al gr`fic. e a
´
SOLUCIO
(a) y = 2
3

3

2xy
2∗2
(b) y ￿ = − 5y4 +3x2 y2 +1 . y ￿ (x = 1, y = 2) = − 5∗24 +3∗22 +1 = − 16
93

(c) y = 2 − − 16 (x − 1)
93
(d) df = − 16 ∗ 0.1 = 0.01720.
93
8. PREGUNTA (20 punts)
Considerem la funci´ f (x) = (x − 1)(x + 1)(x + 2)(x − 2) = x4 − 5x2 + 4, o (a) Trobeu el seu domini.

(b) Calculeu els seus punts estacionaris i digueu raonadament, utilitzant algun dels criteris explicats a classe, si s´n `ptims locals. En cas que ho siguin, indiqueu si s´n m`xims o o o o a m´ ınims locals.
(c) Calculeu raonadament els seus punts d’inflexi´. Heu de demostrar, mitjan¸ant algun o c procediment vist a classe, que s´n efectivament punts d’inflexi´. o o
(d) Representeu gr`ficament la funci´. Feu una gr`fica aproximada, tenint en compte que a o a ´s un polinomi de grau 4. e (e) Trobeu els seus m`xims i m´ a ınims globals i el seu rang.
(f) Calculeu i representeu la funci´ f (x + 1) i resoleu la desigualtat f (x + 1) ≤ f (x). o 3

´
SOLUCIO
(a) R
￿
(b) x = 0 m`xim local x = ± 10 m´ a ınims locals.
4
￿
(c) x = ± 10
12

(d) El gr`fic de l’´ltim apartat ´s a u e 5

2,5

-10

-7,5

-5

-2,5

0

2,5

5

7,5

10

-2,5

-5

￿
￿
10
(e) x = ± 4 m´ ınims globals i el rang ´s [f ( 10 ), ∞) = [−2.25, ∞). e 4
(f) (−∞, −2) ∪ (−1/2, 1)

4

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