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

Need solutions to all questions by Dec 18. 9:00pm ET

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

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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