## (solution) Let f(x, y) be a two-variable, fourth-order homogenous function

Let f(x, y) be a two-variable, fourth-order homogenous function satisfying

fy(-1,-4) = -2; fxx(2, 8) = 8; fxy(1, 4) = 4. Then f(1; 4) equals

Question 13 is the one I need help on. I can't seem to answer it without knowing anything about the degree of the function. I know that the degree and the order aren't the same thing and that for a homogeneous function, f(tx,ty)=t^n*f(x,y), where n is the degree.

Exam code: 0

Bar-Ilan University

The Department of Economics

Mathematics for Economists 66-111-18

Final Examination, First Semester, Moed C - 3.8.2015

Duration of Exam: 3 hrs.

Instructor: Dr. Z. Hellman

The Disciplinary Committee warns!

It is forbidden to remove the questionnaire from the exam room or copy it or photocopy it or mark it

with a magic marker. It is absolutely forbidden to go to the bathroom. Once you have received the

questionnaire/notebook, you must take the exam and return it. You may leave the exam room only after

half an hour. It is forbidden to talk during the exam. Please comply with the supervisor?s instructions.

Remove electronic devices, beeper and mobile phone. Holding a mobile phone, even if turned off, will lead to

immediate invalidation of the exam. A student who will be found with forbidden auxiliary material or who

will be caught cheating will be severely punished and may even be expelled from the university. A complaint

will be submitted to the discipline committee against anyone transgressing these instructions.

I herewith declare that I have read and understood the instructions on the questionnaire and that I have no

material in my possession that is forbidden for use.

ID no. Signature Instructions

The exam contains 16 multiple choice questions.

Choose the correct answer and indicate it on the attached answers sheet.

The only material permitted in the examination room is a hand-held calculator.

GOOD LUCK!

Time Duration: 3 Hours Permitted materials: calculator and paper

Question

The limit limx?0 (ln x) 1

1

x?e is 1. e1/e

2. 1/e

3. 1

4. None of the other answers is correct

Question

The limit

1.

2. 2x

?1

limx?0 e2x ?1 2

is 2

ln 2

1

ln 2 3. ?

4. None of the other answers is correct

1 Exam code: 0

Question 3 The solution of the integral

1.

2.

3. ln |x + 4| + ln | x+4

x+1 | + c

1

2

3 ln |x + 4| ? 3 ln |x + 1| +

ln |x + 1| ? 13 ln | x+1

x+4 | + c R x

x2 +5x+4 dx is 1

3 c 4. None of the other answers is correct

Question 4

R2 4?2y

R

f (x, y)dxdy equals

The integral

0 x 1.

2. R4 2?

R2 f (x, y)dydx 0 0

R2 4?2X

R

0 0 f (x, y)dydx 0 3. Changing the order of integration is not permissible here

4. None of the other answers is correct

Question 5 The solution of the integral

1. 1 ? Re

1 2

e ln x

x2 dx is 2. ? 2e

3. ? 1e

4. None of the other answers is correct

Question 6 Suppose that f (x) is continuous over the interval [a,b] and that

Rb

f (x)dx = 0. Which of the following is correct?

a 1. There is a point x in (a,b) such that f (x)=0

2. f (x)=0 for all x in (a,b)

Rb

3. |f (x)|dx = 0

a 4. None of the other answers is correct

Question

Lat f (x) = x + e 1/x 7 . Which of the following is correct? 1. The line y = x + 1 is a right-side asymptote of f

2. The line y = ?x + 1 is a left-side asymptote of f

3. The function f is concave everywhere in its domain of definition

4. The function attains a minimum at the point x = 1

Question

3 8 3 Let f (x, y) = x + y ? 3x ? 12y + 20 .Which of the following is correct?

1. f has a saddle point at (1,-2)

2. f attains a minimum at the point (?1, ?2)

3. f attains a maximum at the point (1, 2)

4. f attains a maximum at the point (?1, 2)

2 Exam code: 0

Question 9 Find an approximation for (1.08)3.96 , using differential approximation:

1. 1.32

2. 1.356

3. 1.382

4. None of the other answers is correct

Question

The limit 10 x2 ?1

lim(x,y)?(0,0) xy+y

2 is 1. Undefined

2. ??

3. 1

4. None of the other answers is correct

Question 11 The function of the tangent line to the implicit function

?

xy + y + x2 y = xy 2 at the point (1,2) is

1. y = 0.2x + 1.8

2. y = 0.5x + 1.5

3. The tangent line cannot be calculated based on the implicit function alone

4. None of the other answers is correct

Question 12 The Taylor series expansion of

2 3 1

1?x around the point x = 0 is 4 1. 1 + x + x + x + x + ...

2. 1 + x + x2 /2! + x3 /3! + x4 /4! + ...

3. 1 ? x + x2 ? x3 + x4 ? ...

4. None of the other answers is correct

Question 13 Let f (x, y) be a two-variable, fourth-order homogenous function satisfying

fy (?1, ?4) = ?2, fxx (2, 8) = 8, fxy (1, 4) = 4. Then f (1, 4) equals

1. 3.5

2. 14

3. -0.5

4. None of the other answers is correct

Question 14 Let the function u(x, y, z) = xy

z ln x + xf (y/x, z/x) be given, where f is a two-variable function. Then xux + yuy + zuz equals

1. u(x, y, z) + xy/z

2. u(x, y, z)

3. 0

4. None of the other answers is correct

3 Exam code: 0

Question 15 Suppose that a consumer receives utility ln(x + 1) + ln(y + 1) from consuming two products, Product 1 and

Product 2, in quantities x and y, respectively. The price of one item of Product 1 is 3 shekels and the price

of one item of Product 2 is 3 shekels. What are the optimal quantities x and y, respectively, of Product 1

and Product 2 that the consumer ought to consume for optimal utility if his budget is 10 shekels?

1. x = 2.75, y = 1.5

2. x = 5, y = 0

3. x = 2.5, y = 1 23

4. None of the other answers is correct 4

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Sep 13, 2020

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