ECO 3145: Spring/Summer 2021
Quiz 2
May 25, 2021
I) Consider the equation
G(x, y) = x2 − 3xy + y3 − 7 = 0
1. Are the conditions of the implicit function theorem satisfied at the
point (x0, y0) = (4, 3)? Justify your answer. 3 marks
2. Find the value of ∂y∂x at (x0, y0) = (4, 3). 2 marks
II) Consider the following three equations
xy − w = 0
y − w3 − 3z = 0
w3 + z3 − 2zw = 0
1. Determine the total differential of the system. 2 marks
2. Represent the total differential of the system in matrix form
JV = Udz,
where J is the Jacobian matrix , V = (dx dy dw)
′
and U a vector. 2
marks
3. Are the conditions of the implicit function theorem satisfied at the
point (x, y, w; z) = (14 , 4, 1, 1)? Justify your answer. 3 marks
4. Using the Cramer’s rule, find the expressions of ∂x∂z ,
∂y
∂z and
∂w
∂z at
(x, y, w; z) = (14 , 4, 1, 1). 3 marks
1

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