Prova de PNL - IME-USPwalterfm/cursos/mac0427/2008/pnlDiurno1.pdf · Prova de PNL Aluno: No USP RG Prova 2 Aviso: as provas contem quest´ oes parecidas mas com n˜ umeros diferentes.

Prova de PNL

Walter Mascarenhas

30 de abril de 2008

Prova de PNL

Aluno: No USP RG Prova 1

Aviso: as provas contem questoes parecidas mas com numeros diferentes. Parafacilitar a correcao, DEVOLVA ESSA FOLHA e ESCREVA O NUMERO DAPROVA nas suas folhas de resposta.

Questao 1 (2,0 pontos) Desenhe o grafico da funcao f abaixo, destacando seusmaximos e mınimos locais e os pontos nos quais f nao tem derivada.

f (x) = max{−12x2−12x−7, 12x2 +24x−7

}.

Questao 2 (2,0 pontos) Considere as funcoes

f1(x,y) = −27x2−18x−27y2−54y−34, (1)f2(x,y) = 27x2 +36x+27y2 +54y+20. (2)

Determine o conjunto C dos pontos (x,y) para os quais a funcao

f (x,y) = max{ f1(x,y), f2(x,y)}

nao tem gradinte. Desenhe C.

Questao 3 (3,0 pontos) Encontre os maximos e mınimos locais da funcao f daquestao 2. Ha maximos ou mınimos globais? Se houver, quais sao eles?

Questao 4 (3,0 pontos) Desenhe as curvas de nıvel da funcao f da questao 2.

Prova de PNL




f (x) = max{

4x2−12x+5, −4x2 +16x−15}

.


f1(x,y) = 1x2−8x+1y2 +18, (3)f2(x,y) = −1x2 +10x−1y2−18. (4)


f (x,y) = max{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = max{−9x2 +6x+2, 9x2−2

}.


f1(x,y) = 2x2 +12x+2y2−8y+22, (5)f2(x,y) = −2x2−8x−2y2 +8y−10. (6)


f (x,y) = max{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = max{−12x2 +2, 12x2−12x−10

}.


f1(x,y) = −2x2 +16x−2y2−4y−35, (7)f2(x,y) = 2x2−20x+2y2 +4y+41. (8)


f (x,y) = max{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = max{−2x2 +4x−1, 2x2−8x−1

}.


f1(x,y) = 9x2−18x+9y2 +24y+22, (9)f2(x,y) = −9x2 +24x−9y2−24y−30. (10)


f (x,y) = max{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = max{

27x2−54x+24, −27x2 +72x−36}

.


f1(x,y) = −9x2−18x−9y2 +12y−12, (11)f2(x,y) = 9x2 +24x+9y2−12y+16. (12)


f (x,y) = max{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = max{

1x2 +2x−2, −1x2 +2}

.


f1(x,y) = −27x2 +72x−27y2−36y−58, (13)f2(x,y) = 27x2−90x+27y2 +36y+74. (14)


f (x,y) = max{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = max{

18x2 +48x+31, −18x2−60x−41}

.


f1(x,y) = 18x2−24x+18y2 +48y+38, (15)f2(x,y) = −18x2 +36x−18y2−48y−42. (16)


f (x,y) = max{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = max{−9x2 +12x, 9x2−6x

}.


f1(x,y) = 1x2 +2x+1y2−2, (17)f2(x,y) = −1x2−1y2 +2. (18)


f (x,y) = max{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = max{

18x2−24x+12, −18x2 +36x−4}

.


f1(x,y) = 8x2−8x+8y2 +24y+24, (19)f2(x,y) = −8x2 +16x−8y2−24y−12. (20)


f (x,y) = max{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = max{

1x2 +2x−2, −1x2 +2}

.


f1(x,y) = −8x2 +32x−8y2−24y−46, (21)f2(x,y) = 8x2−24x+8y2 +24y+30. (22)


f (x,y) = max{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = max{

18x2−12x+6, −18x2 +24x+6}

.


f1(x,y) = 4x2−8x+4y2 +16y+16, (23)f2(x,y) = −4x2 +12x−4y2−16y−24. (24)


f (x,y) = max{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = max{−18x2−24x−4, 18x2 +12x−4

}.


f1(x,y) = 1x2−4x+1y2 +8y+22, (25)f2(x,y) = −1x2 +6x−1y2−8y−18. (26)


f (x,y) = max{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = max{−12x2−36x−30, 12x2 +48x+30

}.


f1(x,y) = 4x2−12x+4y2 +4y+12, (27)f2(x,y) = −4x2 +16x−4y2−4y−10. (28)


f (x,y) = max{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = max{−2x2−16x−30, 2x2 +12x+10

}.


f1(x,y) = 27x2 +72x+27y2 +18y+53, (29)f2(x,y) = −27x2−90x−27y2−18y−61. (30)


f (x,y) = max{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = max{

1x2 +4x+8, −1x2−2x+8}

.


f1(x,y) = 27x2−36x+27y2 +8, (31)f2(x,y) = −27x2 +54x−27y2−16. (32)


f (x,y) = max{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = max{

4x2−12x+11, −4x2 +16x−9}

.


f1(x,y) = 18x2−48x+18y2 +36y+48, (33)f2(x,y) = −18x2 +36x−18y2−36y−28. (34)


f (x,y) = max{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = max{

27x2 +72x+50, −27x2−90x−58}

.


f1(x,y) = 2x2−4x+2y2−4y+6, (35)f2(x,y) = −2x2−2y2 +4y+10. (36)


f (x,y) = max{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = max{

27x2−36x+8, −27x2 +54x−16}

.


f1(x,y) = −27x2−36x−27y2−10, (37)f2(x,y) = 27x2 +54x+27y2 +14. (38)


f (x,y) = max{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = max{−12x2 +48x−50, 12x2−36x+10

}.


f1(x,y) = −27x2−36x−27y2 +54y−37, (39)f2(x,y) = 27x2 +18x+27y2−54y+17. (40)


f (x,y) = max{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = max{

1x2 +6x+10, −1x2−4x+2}

.


f1(x,y) = 2x2−12x+2y2−8y+30, (41)f2(x,y) = −2x2 +16x−2y2 +8y−26. (42)


f (x,y) = max{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = max{

8x2 +8x+5, −8x2−16x+5}

.


f1(x,y) = 1x2−8x+1y2−6y+27, (43)f2(x,y) = −1x2 +10x−1y2 +6y−27. (44)


f (x,y) = max{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = max{

9x2−4, −9x2−6x}

.


f1(x,y) = 9x2 +18x+9y2−18y+20, (45)f2(x,y) = −9x2−24x−9y2 +18y−18. (46)


f (x,y) = max{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = max{−27x2−36x−15, 27x2 +18x−15

}.


f1(x,y) = 27x2−18x+27y2−72y+51, (47)f2(x,y) = −27x2 +36x−27y2 +72y−45. (48)


f (x,y) = max{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = max{

4x2 +12x+13, −4x2−16x−7}

.


f1(x,y) = −18x2 +36x−18y2−12y−16, (49)f2(x,y) = 18x2−48x+18y2 +12y+28. (50)


f (x,y) = max{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = max{−2x2−8x−12, 2x2 +4x−12

}.


f1(x,y) = 4x2 +8x+4y2 +5, (51)f2(x,y) = −4x2−4x−4y2 +5. (52)


f (x,y) = max{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = max{

4x2−4x+3, −4x2 +7}

.


f1(x,y) = 18x2−12x+18y2 +24y+8, (53)f2(x,y) = −18x2 +24x−18y2−24y−8. (54)


f (x,y) = max{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = max{−1x2−6x−13, 1x2 +4x−5

}.


f1(x,y) = 1x2 +8x+1y2 +20, (55)f2(x,y) = −1x2−10x−1y2−16. (56)


f (x,y) = max{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = max{

27x2−4, −27x2 +18x+8}

.


f1(x,y) = 1x2 +8x+1y2 +8y+32, (57)f2(x,y) = −1x2−6x−1y2−8y−20. (58)


f (x,y) = max{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = max{

3x2−18x+31, −3x2 +24x−29}

.


f1(x,y) = −2x2−4x−2y2 +4y, (59)f2(x,y) = 2x2 +2y2−4y−4. (60)


f (x,y) = max{ f1(x,y), f2(x,y)}




Prova de PNL - IME-USPwalterfm/cursos/mac0427/2008/pnlDiurno1.pdf · Prova de PNL Aluno: No USP RG Prova 2 Aviso: as provas contem quest´ oes parecidas mas com n˜ umeros diferentes.

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