Johns Hopkins University Math Methods for Economists & Indefinite Integral HW I will upload the original file below.

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Johns Hopkins University Math Methods for Economists & Indefinite Integral HW I will upload the original file below.
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1. Evaluate the following indefinite integral: � 2 6 2. Evaluate the following definite integral: � 12 5 + 8 3 − 16 + 4 6 + 4 − 4 2 + 2 − 6 10 5 3. For the following system of equations find solutions to the values of x, y, and z using the matrix inversion technique shown in this course. 1 23 2 3 2 xyz x yz xyz ++= + −= −−= 4. Find the partial derivative of the following equation with respect to x and y: = � 2 + 2 5. Without substituting u or v into Z, please find the total derivative of Z with respect to x: = ( 2 + 2) − ln � 2 2 − 6 + 3 � = = 2 − 3 6. Determine the critical points of functions for the following function: ( , ) = 4( − 2) − 2 − 3 Determine whether or not they correspond to local maxima, minima, or saddle-points. Clearly distinguish the first-order conditions and the second-order conditions. 2 7. Given the constraint, find the stationary points for the following function and evaluate the second order conditions. Use the Lagrange technique. f xy x y ( , ) ln 2ln = + Subject to 2 2 x y + = 6 Johns Hopkins University

Math Methods for Economists

Fall 2019 Final

Please show all your work. Failure to show all intermediate steps will result in

zero credit. Also, please remember to write your name on your answer sheet.

Finally, please use white unlined paper if possible and submit your file as a PDF.

1. Evaluate the following indefinite integral:

�

2

6

2. Evaluate the following definite integral:

�

10

5

12 5 + 8 3 − 16 + 4

6 + 4 − 4 2 + 2 − 6

3. For the following system of equations find solutions to the values of x, y, and z using

the matrix inversion technique shown in this course.

x+ y+z =

1

2x + 3y − z =

2

3x − y − z =

2

4. Find the partial derivative of the following equation with respect to x and y:

= � 2 + 2

5. Without substituting u or v into Z, please find the total derivative of Z with respect

to x:

= ( 2 + 2 ) − ln �

2

�

2 − 6 + 3

= 2 − 3

=

6. Determine the critical points of functions for the following function:

( , ) = 4( − 2 ) − 2 − 3

Determine whether or not they correspond to local maxima, minima, or saddle-points.

Clearly distinguish the first-order conditions and the second-order conditions.

7. Given the constraint, find the stationary points for the following function and

evaluate the second order conditions. Use the Lagrange technique.

6

f ( x,=

y ) ln x + 2 ln y Subject to x 2 + y 2 =

2

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