# 程序代写案例-ECON6003

University of Sydney

ECON6003 – MATHEMATICAL METHODS OF ECONOMIC ANALYSIS

Mid-semester Exam: Semester 2, 2021
INSTRUCTOR: VL
1) For function f(x)=2x+2 find its inverse. Correct answer: (y-2)/2.
2) Let f ( x ) = − (x2 + 1)/ x where x ∈ R ∖ { 0 }. Find if f(x) concave or convex?
Correct answer: Convex for only negative values of x.
3) Find the value at which the following function f ( x ) = x ( x − 1 )3 is
4) Find the present value of \$10 000 which is paid after 5 years and \$10 000
which is paid after 10 years if the interest rate is compounded annually at 2% per
annum (rounded to the nearest whole number). Correct answer: \$17 261.
5) Find the following definite integral (integral from 0 to 2 of x*exp(x2)).
6) What values of x and y satisfy the following system of two equations?
3 x − y = 1 , 2 x + 7 y = 16. Correct answer: x=1, y=2.
7) If f(x,y)=ln(2x+y) and g(x,y) is the second derivative with respect to x and y,
what is g(1,1)? Correct answer: -2/9.
8) Let f ( x , y , z ) = x 2 y z . Find the partial derivative with respect to y and
evaluate it at ( x , y , z ) = ( 1 , 2 , 3 ). Correct answer: 3.
9) What is the 5th derivative of the following function
y = 2 x 5 + 3 x 4 + 2 x 3 + x 2. Correct answer: 240.
10) Find the derivative of the following function y(x)=(x-1)(x+1)/(x2+1) using
the quotient rule and evaluate it at x=1. Correct answer: 1.

Question 1 : Let p 1 ( x 1 , x 2 ) = x 1
3 + x 2
3 . What is the degree of
homogeneity of p 1 ( x 1 , x 2 ) = x 1
3 + x 2
Let p 2 ( x 1 , x 2 ) = 3 p 1 ( x 1 , x 2 ) . What is the degree of homogeneity of
p 2 ( x 1 , x 2 ) ? Correct answer: 3.
Find the slope, d x2/d x1, of p 2 ( x 1 , x 2 ) at ( x 1 , x 2) = ( 1 , 1 ). Correct
Find the slope, d x2/d x1, of p 2 ( x 1 , x 2 ) at ( x 1 , x 2) = ( 3 , 1). Correct
Find the curvature, d2 x2/(d x1)
2 of p 2 ( x 1 , x 2 ) at ( x 1 , x 2) = ( 1 , 1 ). Correct

Question 2: Dr. Jones is a billionaire and an amateur mathematician. Today, she
wants to invest 40 million dollars at a fixed annual interest rate of 10% and use
this fund to set up a “Nobel prize” for mathematicians: in each year, she awards
a sum of money to the most outstanding mathematician. To combat inflation, the
size of the prize is x for the first year, 1.045x in the second year, and \$1.045t-1x in
year t. Suppose that the first prize is scheduled to be given out immediately and
Dr. Jones wants this to become a legacy that lasts forever, what is the highest
possible value of x in millions of dollars? (For example, if your answer is 5

Question 3: At a given point in time, the marginal product of labor is 7 and the
marginal product of capital is 10, the amount of labor is increasing by 15 per
each unit of time and the rate of change of capital is 5. What is the rate of change

Question 4: The number of patients, n, that seek treatment from Doctor Michael
is a function of the quality of his service, q, and is given by n(q), where
n′(q) > 0, n′′(q) = 0, and q ≥ 0. The government pays Doctor Michael p for every
patient he treats. The cost to Doctor Michael of treating n patients is c(n),
where c′(n) > 0 and c′′(n) > 0, and the cost of delivering quality q is αq. Thus,
the total cost is c(n)+ αq.
(i) Express Doctor Michael's profit as a function of q. (1 mark)
(ii) Doctor Michael chooses q to maximise profit. What is the first order
condition for an interior maximum? (2 marks)
(iii) Check that the solution to (ii) yields a maximum. (2 marks)
(iv) Write down the condition for the solution to Doctor Michael's profit
maximisation problem to be interior. Show on a diagram. (3 marks)
(v) Take the differential of the first order condition in (ii) or use implicit
differentiation to find an expression for dq/dp and sign it. (2 marks)  Email:51zuoyejun

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