UNIVERSIDADE FEDERAL DE ITAJUBÁ LÍNGUA INGLESA – PROVA DE TRANSFERÊNCIA INTERNA, EXTERNA E PARA PORTADOR DE DIPLOMA DE CURSO SUPERIOR – 11/06/2017 CANDIDATO: _______________________________________________________ CURSO PRETENDIDO: _______________________________________________ OBSERVAÇÕES:

01 – Prova sem consulta. 02 – Duração: 2 HORAS

Rough cuts Brazil’s fabulous Batista boys The meat-mongers whose testimony could end Michel Temer’s presidency

Print edition | The Americas May 25th 2017| BOA VISTA

JOSÉ BATISTA SOBRINHO helped build Brasília. In 1957 his meat business supplied canteens that fed workers constructing Brazil’s modernist capital. Now his two youngest sons, Wesley and Joesley, are bringing the place down. As the bosses of the company their father founded, renamed JBS in his honour, they are at the centre of a scandal that may force a president out of office for the second time in a year. JBS is the world’s biggest beef exporter. Its revenues rose from 3.9bn reais ($1.8bn) in 2006 to 170bn reais last year, helped by China’s appetite and Brazil’s enthusiasm for national champions. From 2007 to 2015 the development bank, BNDES, injected into Batista enterprises more than 8bn reais in capital and loans. Most of it was to help JBS buy rivals, including American brands like Swift and Pilgrim’s Pride. J&F, the family’s holding company, has diversified into non-meat businesses, including Havaianas, which makes fashionable flip-flops. As JBS was buying up rivals, the Batistas were buying politicians. The company’s declared campaign donations swelled from 20m reais in 2006 to nearly 400m reais in the election in 2014; in that contest it gave more than any other firm. In the past decade the brothers have bankrolled 1,829 candidates; their largesse helped elect a third of the current congress. Little of it was legal. The Batistas have confessed that almost all the declared cash, plus millions paid under the table, was bribes to politicians specifically to further J&F’s interests.

In the past year the Batistas’ firms have faced five criminal investigations. The latest probes J&F’s dealings with BNDES, which provided finance at the behest of paid-off politicians. To save their enterprises, and themselves, the brothers approached prosecutors investigating the metastasising bribery scandal centred on Petrobras, the state-run energy company. The bargain they struck was their niftiest deal yet. In exchange for providing evidence of wrongdoing by major political figures—including, possibly, President Michel Temer—they secured near-total immunity. Unlike Marcelo Odebrecht, boss of a construction firm at the heart of the Petrobras allegations, neither Batista will spend a day in jail or under house arrest. Free to leave Brazil, Joesley has already moved to his posh New York flat with his wife, a former television presenter, and their child. He and Wesley each agreed to pay fines of 110m reais, which leaves them both billionaires. The meat-mongers are not completely off the hook. JBS may face bribery probes and lawsuits from holders of the company’s securities in the United States. Brazil’s markets watchdog is looking into possible insider trading. In the weeks before May 17th, when details of their explosive testimonies leaked to the press, the Batistas sold more than 300m reais’ worth of JBS shares and bought dollars. The shares have lost a third of their value since then; the dollar jumped by 7% on the next day. The brothers and their firms deny allegations of insider trading. Apparently, they are blessed with their father’s foresight. This article appeared in the The Americas section of the print edition under the headline "The fabulous Batista boys" THE ECONOMIST. Brazil’s fabulous Batista boys. Available at: . Access date: May 26th 2017.

After reading the text, choose the correct alternative in questions 1 to 5. 1. In the sentence “Now his two youngest sons, Wesley and Joesley, are bringing the place down.” (1st paragraph), we are informed the Batista brothers are: a) destroying the family business b) expanding the family business c) demolishing JBS office building d) expanding JBS office building 2. In relation to JBS revenues, it is possible to say that they a) have been rising steadily since 2006 b) had a peak in 2017 c) grew more than 40 times in the past decade d) decreased due to China exportation of meet 3. The term largesse in “... their largesse helped elect a third of the current congress.” means: a) big amount of money b) large influence of their company c) large buildings d) generosity 4. The most recent investigations the Batista brothers are facing are related to: a) Finance they obtained from BNDES as a request of bribed politicians. b) Finance they offered to the 2014 elections. c) Money they sent to BNDES d) Money they sent overseas to tax havens

5. Due their involvement with bribery, Joesley and Wesley Batista will a) go under house arrest b) give back all the illegal money the profited in the last 11 years c) have all their assets locked by the federal police d) pay around a hundred million reais Now answer the questions 6 to 10 in ENGLISH. 6. How did J&F expanded their business? By entering the non-meat market, diversifying their business like, for example, Havaianas (Resposta que contenha essa informação).

___________________________________________________ ___________________________________________________ ___________________________________________________ 7. How did the Batista brothers get political support for their company? They bankrolled 1,829 candidates; OU they donated hundreds of millions of reais to help politicians’ elections OU they paid bribe to politicians to favor JBS’s interests (resposta que contenha uma ou outra, ou todas as informações).

___________________________________________________ ___________________________________________________ 8. Why is Marcelo Odebrecht mentioned in the text? To compare his punishment (going to jail) with the Batista brothers who will pay a fine and get free to go wherever they want. (resposta que contenha informação semelhante)

___________________________________________________ ___________________________________________________ ___________________________________________________ 9. Are the Batista brothers completely free of charge? Why/why not? Although they are free in Brazil, they may may face bribery probes and lawsuits from holders of the company’s securities in the United States. (Resposta que mencione essa informação).

___________________________________________________ ___________________________________________________ ___________________________________________________ 10. What’s the relation between the Batista brothers and the dollar value? In the weeks before May 17th, when details of their explosive testimonies leaked to the press, the Batistas sold more than 300m reais’ worth of JBS shares and bought dollars. The shares have lost a third of their value since then; the dollar jumped by 7% on the next day. (Resposta que mencione essa informação ou “insider trading”).

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´ UNIFEI - UNIVERSIDADE FEDERAL DE ITAJUBA ´ MATEMATICA ˆ PROVA DE TRANSFERENCIA INTERNA, EXTERNA E PARA PORTADOR DE DIPLOMA DE CURSO SUPERIOR - 11/06/2017 CANDIDATO: CURSO PRETENDIDO: ˜ OBSERVAC ¸ OES: 1. 2. 3. 4. 5.

Prova SEM consulta; A prova PODE ser feita a l´apis; PROIBIDO o uso de calculadoras e similares; Dura¸c˜ao: 2 HORAS. Nas quest˜oes discursivas EXPLICITAR os c´alculos.

Quest˜ ao 1 (10 pontos). pre¸co

Um produto foi vendido de acordo com a seguinte fun¸ca˜o

100 1 + 0.005x onde x ´e o n´ umero de unidades do produto e P ´e dado em dinheiros. Quantas unidades foram vendidas a 5 dinheiros? P(x) =

a) 150

b) 430 e

c) 1000

d) 380

Resposta: b) Invertendo para x temos, 20(100 − P) P avaliando em P = 5 dinheiros, temos x = 380 unidades. x=

Quest˜ ao 2 (10 pontos).

Encontre o conjunto solu¸ca˜o da inequa¸ca˜o 1 4 < . x−2 x+1

a) x ∈ (−∞, −2)

b) x ∈ (−∞, −2) ∪ (−1, 2)

c) x ∈ (−2, +∞)

d) x ∈ (−∞, −1) ∪ (−1, 2)

Resposta: b) Para x < −1 a inequa¸ca˜o ´e equivalente a 4x + 4 < x − 2 cujo conjunto solu¸c˜ao ´e: x ∈ (−∞, −2). Para x > 2 a inequa¸ca˜o ´e equivalente a 4x + 4 < x − 2 cujo conjunto solu¸c˜ao ´e vazio. E para −1 < x < 2 a inequa¸ca˜o ´e equivalente a 4x + 4 > x − 2 cujo conjunto solu¸ca˜o ´e: x ∈ (−1, 2). Assim, a solu¸c˜ao ´e x ∈ (−∞, −2) ∪ (−1, 2).

Quest˜ ao 3 (10 pontos).

Avalie √ x3 + 2x2 lim x→0 |x|

a) @

b)

√

2

c)

√ 2 8

Resposta: c)

b) 4

2 4

√ √ x3 + 2x2 |x| x + 2 √ lim = lim = 2. x→0 x→0 |x| |x|

Quest˜ ao 4 (10 pontos). m´ınimo local de f. a) 23

√

d)

c) 0

Se f(x) = 2x3 − 39x + 216x + 23 determine o ponto de d) 9

Resposta: d) Usando a derivada temos f ′ (x) = x2 − 13x + 36 = 0, donde x = 9 ou x = 4. Calculando a segunda derivada vemos que f ′′ (4) < 0 e f ′′ (9) > 0, logo 9 ´e o ponto de m´ınimo local.

2

Quest˜ ao 5 (10 pontos).

Encontre o conjunto solu¸ca˜o da equa¸ca˜o x3 − 1 = |x3 − 3|

a) x = ∅

b)

√ 3 3

c) x = 0 ou x =

√ 3 3

d) x =

√ 3 2

Resposta:√d) Para√x > 3 3 a equa¸ca˜o ´e equivalente a x3 − 1 = x3 − 3 que n˜ao admite solu¸ ca˜o. Para √ 3 3 3 3 x ≤ 3 a equa¸c˜ao ´e equivalente a x − 1 = 3 − x que tem como solu¸ca˜o x = 2. Quest˜ ao 6 (10 pontos).

Considere o sistema linear   3x + 2y + z = 1 6x + ay + 2z = b   x+y+z=0

Para quais valores de a e b o sistema admite solu¸c˜ao u ´nica?

Resposta: Para que um sistema linear admita solu¸c˜ao u ´nica ´e necess´ario e suficiente que o determinante associado a ele seja n˜ao nulo. Calculando o determinante associado ao sistema temos 2a − 8, logo a ̸= 4 e b ∈ R.

3

Quest˜ ao 7 (10 pontos).

Avalie ( ) x lim x + x→0+ |x|

Resposta: Para x > 0 temos |x| = x, logo

( ) x = lim+ (x + 1) = 1 lim x + x→0 x→0+ |x|

Quest˜ ao 8 (10 pontos). Uma empresa produz uma cadeira ao custo de 20 dinheiros. Analisando as vendas os diretores acreditam que se a cadeira ´e vendida por x dinheiros, com x < 80, ser˜ao vendidas 80 − x cadeiras. Qual deve ser o pre¸co para que o lucro da empresa seja m´aximo?

Resposta: A fun¸c˜ao custo ´e dada por C(x) = 20(80 − x) e a receita ´e dada por R(x) = (80 − x)x, logo o lucro ´e dado por L(x) = (80 − x)x − 20(80 − x). Encontrando os pontos cr´ıticos L ′ (x) = 0, temos x = 50. Avaliando L ′′ (50) = −2, logo o lucro m´aximo ´e quando x = 50 cadeiras.

4

Quest˜ ao 9 (10 pontos).

Para x ∈ [−3, 3] encontre o valor de m´aximo da fun¸c˜ao f(x) = x3 − 3x + 4.

Resposta: Derivando, temos f ′ (x) = 3x2 − 3. portanto no intervalo desejado os pontos cr´ıticos s˜ao x = 1 e x = −1. Avaliando a fun¸ca˜o nos pontos cr´ıticos e extremos do intervalo temos f(−3) = −14,

f(−1) = 6,

f(1) = 0 e f(3) = 22.

Assim, o valor de m´aximo ´e 22 atingido em x = 3.

Quest˜ ao 10 (10 pontos).

Resolva o sistema { x + 4y3 = 40 logy2 (2) = logxy (4).

Resposta: Passando a segunda equa¸c˜ao para a base y temos logy (2) logy (4) = . logy (y2 ) logy (y) + logy (x) logo, logy (2) 2 logy (2) = ⇒ 2 1 + logy (x)

logy (x) = 3 ⇒

voltando na primeira equa¸c˜ao, temos 5y3 = 40 ⇒ Assim, a solu¸c˜ao ´e x = 8 e y = 2.

5

y = 2.

x = y3