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

Mathematics

For problems 1 6 evaluate the indicated limits, if they exist.

1. For evaluate,

(a) (b) (c)
[Solution]

2. For evaluate,

(a) (b) (c)
[Solution]

3. For evaluate,

(a) (b) (c)
[Solution]

4. For evaluate,

(a) (b) (c)
[Solution]

5. For evaluate,

(a) (b) (c)
[Solution]

6. For evaluate,

(a) (b) (c)
[Solution]

For problems 7 & 8 find all the vertical asymptotes of the given function.

7. [Solution]

8. [Solution]

Mathematics

For problems 1 6 evaluate the indicated limits, if they exist.

1. For evaluate,

(a) (b) (c)
[Solution]

2. For evaluate,

(a) (b) (c)
[Solution]

3. For evaluate,

(a) (b) (c)
[Solution]

4. For evaluate,

(a) (b) (c)
[Solution]

5. For evaluate,

(a) (b) (c)
[Solution]

6. For evaluate,

(a) (b) (c)
[Solution]

For problems 7 & 8 find all the vertical asymptotes of the given function.

7. [Solution]

8. [Solution]