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[answered] Midterm 1540 (Section O) Winter 2012 Introductory Mathemati


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Midterm

 

1540 (Section O) Winter 2012

 

Introductory Mathematics for Economists II

 

16 February 2012 (Thursday)

 

Instructor: Ferdous Jalil

 

75 minutes

 

Last Name: __________________________________

 

First Name: __________________________________

 

ID: _______________________________________ sh is

 

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m Signature: ___________________________________ Instructions: There are two sections, Section A and Section B. Section A has questions from 1 to 5; Section

 

B has questions from 6 to 10. Do 3 out of 5 questions from each section. There is 1 bonus question. You

 

may attempt to do that. Write all your answers in the booklet that you have been provided. Make

 

sure you show your work. Writing down only answers will give you at most 10% (if correct). You may

 

use your non-programmable calculator. You have 75 minutes. So keep your eyes on your own exam.

 

Time is your only constraint. Good luck! Section A (Do any 3 out of 5 questions. Each is worth 10 points.)

 

1. Consider the function f (x; y) below: f (x; y) = x3 + 2xy 5x y2 a) Compute the partial derivatives of the ?rst and second order. b) Compute the stationary points of f (x; y) and classify them into local maximum points, local minimum points, or saddle points. Th 2. Consider the function f (x; y) below: f (x; y) = 2x2 + 2xy y 2 + 18x 14y a) Find the stationary point of f (x; y): b) Prove that the stationary point you ?nd in (a) is the global maximum point of f (x; y): 3. Find all the stationary points of the function f (x; y) below and classify them into local maximum

 

points, local minimum points, or saddle points.

 

f (x; y) = x2 https://www.aceyourstudies.com/file/7978689/1540-old-midterm-exam/ 1 y2 xy x3 4. Find if the function f (x; y) below is concave, convex or neither.

 

f (x; y) = ex+y + ex y 5. Calculate that if

 

x y+z

 

x+y z w= x + 10y h then

 

@w

 

@w

 

@w

 

+y

 

+z

 

=0

 

@x

 

@y

 

@z sh is

 

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m x Section B Do any 3 out of 5 questions. (Each is worth 10 points) 6. The following system of equations de?nes u = u(x; y) and v = v(x; y) as di?erentiable functions of x

 

and y around the point R = (x; y; u; v) = (1; 1; 1; 0) : u + xey + v x+e u+v 2 = y e = e 1 1 a) By taking the di?erential of the system, ?nd the equations du and dv.

 

b) Find the values of du du dv dv

 

dx ; dy ; dx ; dx at the point R: 7 Calculate the di?erential of the following functions Th a) z = ln(x2

 

2 b) z = ew c) z = yex y2 ) 2 +x +y 2 2 8. a) Consider the function f (x; y) below: f (x; y) = xy

 

x2 + y 2 Is the function homogeneous, and, if yes, of which degree? What is xf10 (x; y) + yf20 (x; y) equal to?

 

b) Consider the function f (x; y) below:

 

f (x; y) = xy 2 + x3

 

00

 

00

 

00

 

Is the function homogeneous, and if, yes, of which degree? What is x2 f11

 

+ 2xyf12

 

+ y 2 f22

 

equal to? https://www.aceyourstudies.com/file/7978689/1540-old-midterm-exam/ 2 00

 

00

 

00

 

9. Find zx0 ; zy0 ; zxy

 

; zxx

 

; zyy

 

when x3 + y 3 + z 3 3z = 0: 10. a) Using the chain rule, ?nd @w

 

@t when x = x2 + y 2 + z 2

 

p

 

=

 

t+s y = ets z = s3 w b) Using the chain rule, ?nd @w

 

@t when w = xy 2 z 3

 

t2 y = s z = t sh is

 

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x = Bonus Question (5 points) 1. Find the elasticity of substitution for the CES function

 

F (K; L) = (K Th THE END https://www.aceyourstudies.com/file/7978689/1540-old-midterm-exam/ Powered by TCPDF (www.tcpdf.org) 3 +L ) 1=

 


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