Resolucions d'Exàmens de Matemàtiques: Guia Completa i Detallada

Examen 1

1. 1/5 √2·3^3 - 1/2 √2·3^2 + 1/10 √2^3·3 -> 1/5 ·3√6 - 1/2 ·3√2 + 1/10 ·10√6 -> 8√6/5 -3√2/2

2. √18y - √2y + 5/2 · √8y - 4/3 · √9y -> 3√2y - √2y + 5/2 ·2 √2y - 4/3 · 3√y -> 7√2y - 4√y

3. ^3√3a^5b · √2ab^4 = ^6√3^2a^10b^2 · 2^3a^3b^12 -> ^6√3^2a^13b^142^3 -> a^2b^2 ^6√3^2ab^22^3

^3√a·a^8 -> ^6√a^a -> a^6√a^3 -> a√a

4. 9 ^3√6 - 4 ^3√6 + ^3√6 -> 6 ^3√6 / a^3 b^3 c^4 ^3√c^2

5. log2 0,125=x -> log2(2^-3)=x->-3=x->-3

logx 3=-1-> 1/3

log5 x=3-> x=5^3-> x=125

logx 0,04=-2-> 0,04=x^-2 (1/25) =x^-2-> 1/25 =1/x^2=x^2=+/-√25=+/-5

6. log 18->descomp. factors-> log=log (2·3^2) = log 2+log 3^2->log3^2=2log3->log 18=log2+2log3

log^3√√6->6 1/6-> log(6 1/6)=1/6log6->log6=log(2·3)=log2+log3->1/6log6=1/6(log2+log3)->log^3√√6=1/6log2+1/6log3

7. log(a/b)=loga-logb->log(x^-1-y^3/√z=log(x^-1-y^3) - log(√z)-> log√z=logz 1/2=1/2logz->log z^-1 - y^3 -1/2logz->log x^-1=-logx->logy^3=logy^3=3logy->log x^-1-y^3=log(-logx-3logy)->espandida-> log x^-1 - y^3/√z= log x^-1-y^3-1/2logz

Examen 2

1. f(x)={x^3-2x+1->x>0->(1,1,inf){5x^2->x<=0->(0,inf)

2. f(x)=2x-3/x^3-27->Dom:R-{3}(x^3-27=0->x^3=27)graf:recta

g(x)=x^4-9x^3+3->polinomi, Dom:R,graf:c

(x)=log(x-1)->x-1>0->x>1, Dom:R-{x>1},graf:c,tumbada

3. a) 12+3x b) funció lineal->f(x)=mx+n c) f(23)=12+3·23=12+69=81 d) 39=12+3x->39-12=3x->27=3x->x=27/3->x=9

4. a) (1,1) i (3,3)->m=3-1/3-1=2/2=1->y-1=1(x-1)->y-1=x-1->y=x, interpolació lineal, estima x=2, IL->f(x)=x i f(2)=2

b) (0,2) i (5,5)->m=5-2/5-0=3/5->y-2=3/5(x-0)->y-2=3/5x->y=3/5x+2->IL, estima x=2, f(2)=3/5·2+2=6/5+2=6/5+10/5=16/5=3.2, IL->f(x)=3/5x+2 i f(2)=3.2

5. a) m(15)=-15^2+15·15=-225+225-15=-15 b) -x^2+15x-15=0->x=-15+/-√165/2·(-1)=-15+/-√165/-2

Examen 3

1. a) f(x->0)=5·0+20/0^2+9-20/9->20/9-20/9->f(0)=0

b) f(x)=5x+20/x^2+9-20/9->f '(x)=5·(x^2+9)-(5x+20)·2x/(x^2+9)^2->5x^2+45-10x^2-40x/(x^2+9)^2->-5x^2-40x+45/(x^2+9)^2-> dividir per 5, simpl->x^2+8x-9=0->1 i -9, benefici max->f(1)=f,normal=5·1+20/1^2+9-20/9=25/10-20/9=2.5-20/9

2. f(x)=x^5+2x^2-5/x^4-1->AV:x^4-1=0->(x^2-1)(x^2+1)=0->(x-1)(x+1)(x^2+1)=0->x^2+10x+/-√1=+/-1, AO->f(x)/x=x^5+2x^2-5/x^4-1/x/1=x^5+2x^2-5/x^5-1=x·(x^4-1)=x^5-x->(x^5+2x^2-5)-(x^5-x)=x+2x^2-5->result->x

3. f(x)={x^2-1 si x<= 2 }x+2 si x>2

1. lim x ->1->(x^2-2)->f(1)=1^2-1=1-1=0
2. lim x->2^- ->(x^2-1)->f(x)=2^2-1=3
3. lim x->2^+->(x+2)->f(x)=2+2=4
4. lim x->4->(x+2)->f(x)=4+2=6

b) no són iguals, no existeix

4. t^2+28/(t+2)^2,(t=0)

a) 0^2+28/(0+2)^2=28/4=7 milions, llarg termini->(t->inf)->inf/inf->grau max->f(t)=t^2(1+28/t^2)/t^2(1+4t/t^2+4/t^2)->1/t^2 i 4/t^2->petits,1/1=1 milió

b) derivada->f '(t)=2t·(t+2)^2-(t^2+28)·2(t+2)/(t+2)^4->2·(t^2+28)(t+2)->2(t+2)(t^2+2t-t^2-28)/(t+2)^4->2(t+2)(2t-28)/(t+2)^2->2(2t-28)=0->28/2=14

5. a) y '=(2x^4+3x)^1/5->1/5·(2x^4+3x)^-4/5·(8x^3+3)->8x^3+3/5·(2x^4+3x)^4/5->regla cadena->3(^5√2x^4+3x)^2·8x^3+3/5·(2x^4+3x)^4/5->24x^3+9/5·^5√2x^4+3x

b) y '= 1/3x^2+x-5·(6x+1)->6x+1/3x^2+x-5

c) y '=x^2+2x-1->y=x^-3->du/dx=2x+2->y=-3·1/(x^2+2x-1)^4·(2x+2)->6x+6/(x^2+2x-1)^4

6. x->3->inf

x->0->x^2/x(2x-1)=x/2x-1=0

x->inf->x^2(1+1/x^2)/x^2(1-1/x)->x(x+1/x)/1-1/x->x^2+x/x/1-1/x=x^2/1=inf

Examen 4

1. Dom->R-{-1}, f '(x)=2·(x+1)-2x·1/(x+1)^2=0->2x+2-2x=0->2=0->creixent(-2 i 0)->AV:x=-1 / AH:2x/x=2

2. Dom:R->A->no hi ha, AO->f(x)/x=x·e^2·inf=inf->mt->1·e^2x+x·2e^2x=[e^2x+2xe^2x]=0->e^2x(1+2x)=0->1+2x=0->2x=-1->x=-1/2->imatge->(-1 i 0)->-1/e^2 i 1+0->2deriv->f ''(x)=2e^2x+2e^2x+2e^2x·2x ->2e^2x(2+2x)=0->e^2x=0->punt inflexió=2+2x=0->x=-1

3. f '(x)=x^2+4x-5=0->(1 i -5)->Dom: (-inf,-5)u(1,inf)->intervals f '(x)=1/x^2+4x-5·(2x-4)->2x-4/x^2+4x-5->(-6 i 2)decreix i creix

4. f(x)=2(x+3)(x^2+1-2x)->2(x^3+x-2x^2+3x^2+3-6x)->2(x^3+x^2-5x+3)->2x^3+2x^2-10x+6-> f '(x)=6x^2+4x-10=0->(1 i -5/3)-> f ''(x)=12x+4->imatge(1->mínim/-5/3->màxim)->intervals(-2 i 2)