# Mathematics

Problem Set 1.

This first problem set will cover material introduced in the math review. Students are

encouraged to work in groups, though any students choosing to do so must submit the names

of each student they worked with. All answers submitted must be the final product of an

individual student’s thought. Sections I and II will be worth 20% each while sections III and

IV will each be worth 30%. Theses percentages will be divided evenly among each question

in the cluster.

1.) Calculate the derivatives of the following single variable functions.

a) f (I) = 656Moo6“

b) flit) = (561%

$17

c.) f

log(xa) Hint: Look at the positioning of the parenthesis.

2.) Calculate the derivatives of the following single variable functions.

a) f (56, y) = Iayl‘a

b.) rim) = + wl-ayfi

(ma+y17cx)1i’y

(3-)

d.) my) = Li?“

3.) Solve for the system of equations which defines the solution to the following un-

constrained optimization problems, and provide sufficient conditions to demonstrate the

existence of a solution. When a function is not specified, provide the necessary and suffi-

cient conditions on the functional form. Lastly, note that in some instances, I employ Greek

variables with Latin subscripts. These are simply parameters I have defined to capture a

certain parameter/variable relationship. Treat them as every other parameter.

a.)

maaxflm) ? logx – C(l‘)

b.)

rgalx h(x, y) ? mag/fl – 693x – wyy

1-

(ml-a) ?? (was

e.)

rig/X f (:6, y; 2) = logmz +1og(y)z – ???????? + 6w)

4.) Solve the following constrained optimization problems. When a function is not spec-

ified, provide the necessary conditions on the functional form. Lastly, note that in some

instances, I employ Greek variables with latin subscripts. These are also parameters which

denote a certain parameter/variable relationship. Treat them as every other parameter.

a.)

mmaxflgr) ? logx – px

subject to

a? ? 0

b.)

max mag/5

33:34

subject to:

ny ? K;

c.)

1_

Inajxflmr, y; 1096, Ty) ? (mag/1“”) 7 – wxx – ryy

subject to

(1 + ???? ?? y ? I

d.)

rig/X f (:6, y; 2) = logmz +1og(y)z – ???????? + 6w)

subject to

ow: + 6y ? 4

e.)

or 1-04 B

Iggxflxww) – ??? ???? 2/ $ng wyy

subject to

a? ? 0

3/ ? 0

and

x+y=6

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