Department of Economics
Trent University
ECON3250H – Mathematical Economics
Fall Semester, 2022
Pete
orough Campus
Assignment #1
Due in Class October 3, 2022
General Information and Instructions:
Worth 10% of the course grade, this assignment has 4 questions for a total of 50 marks. Marks for
each question are as indicated and are evenly divided among the parts of the question. Answers to
the questions must be word-processed or hand-written and scanned, and be legible, orderly and
concise; written explanations, where requested, should have no more than a very
ief paragraph of
content. While students may work together on the assignment, they must develop, write up and
submit answers independently. On the due date noted above, answers to the assignment must be
submitted in a single hard copy package at the beginning of class.
Questions:
1. (15 marks) Consider the concept of limit:
a. Provide an ?, ? proof of lim
?→3
√? + 1 = 2.
. Evaluate lim
?→0
? sin
1
?
.
c. State the Pinching Theorem.
2. (10 marks) Consider the property of continuity:
a. State the definition of continuity of the function ?(?) at ? = ?.
. Assuming ?(?) is continuous at ? = ?, prove that ?(?) is continuous at ? = ?(?) if
and only if
? (lim
?→?
?(?)) = lim
?→?
?(?(?))
3. (15 marks) Consider the property of differentiability:
a. State the definition of differentiability of the function ?(?) at ? = ?.
. Use first principles to find the derivative of ?(?) = √? − 1.
c. Use the definition of differentiability to prove that ?′(?) = − sin? if ?(?) = cos?.
{Hint: Use the cosine of a sum of angles formula and the Pinching Theorem}
4. (10 marks) Consider the property of functional independence:
a. State the Functional Independence Theorem.
. Show that the functions ? = ?1 − ?2 and ? = 2?1
2 − 4?1?2 + ?2
2 are functionally
independent.