Problem Set 1
Econ 302 Intermediate Microeconomic Analysis, Spring 2026
Dr. Hyun Woong Park
Name:
1. Consider the following function:
y = (xa + xb )1/a with x > 0
dy
where a > 0 and b > 0 are constants. Calculate the first-order derivative, dx
and determine its the
sign value, i.e. whether it is positive > 0 or negative < 0.
2. For each of the following single variable function, y = f (x), where a > 0 and b > 0 are constants,
dy
d2 y
calculate the first-order derivative, dx
, and the second-order derivative, dx
2 , and determine the sign
value of each.
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(1) y = (x−1)
b with x > 1.
(2) y = xa with x > 0.
(3) y = ln xa with x > 0.
3. For each of the following multi-variable functions, y = f (x, z), where a > 0 and b > 0 are
constants, calculate the first-order partial derivative with respect to x and that with respect to z.
(1) y = ln xa + ln z b
(2) y = xa z b
(3) y = (xa + z a )1/a
4. Suppose your utility, which is a measure of your satisfaction, denoted by U , is a function of two
goods you consume; the number of each is denoted by q1 and q2 , respectively. Formally, U = f (q1 , q2 ).
Find the total differential.
5. Suppose y = 10x2 − 40x + 3. Find the optimal level of x and the corresponding minimum y. Use
both the first order condition and the second order condition. Illustrate your answer in a diagram.
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