Nanyang Technological University
HE1003 Basic Mathematics for Economists
PROBLEM SET 3
1. Compute all the partial derivatives of the following functions:
(a) 4:323; 7 33:93 + 6x
2. Compute the partial derivatives of the Cobanouglas production function Q :

Nanyang Technological University
HE1003 Basic Mathematics for Economists
PROBLEM SET 6A
1. Suppose that the function y1 (t) and y2 (t) satisfy y1 (0) = 5 and y2 (0) =
dy1
dt
dy2
dt
=
2y1 + 4y2
=
2 and the system
3y1 + 3y2
Find the solutions for y1 (t) and

Nanyang Technological University
HE1003 Basic Mathematics for Economists
PROBLEM SET 6
1. Suppose that the function y(t) satises y(0) =
2 and
dy
ty 2
=p
dt
1 + t2
Find y(t).
2. Use an integrating factor to solve the equation:
p
dy
x
y = 1 + x2
+
dt
1 + x2

Nanyang Technological University
HE1003 Basic Mathematics for Economists
PROBLEM SET 5
1. Use integration by substitution to determine the indenite integral for the following function involving a
product. Please check the answer on your own.
(a)
(b)
(c)
(

Nanyang Technological University
HE1003 Basic Mathematics for Economists
PROBLEM SET 4
1. Solve the following equation for x
(a) 2e6x = 18
2
(b) ex = 1
(c) 2x = e5
(d) 2x
2
=5
(e) ln x2 = 5
(f) ln x5=2
0:5 ln x = ln 25
[(a) x = (ln 9)=6; (b) x = 0; (c) x

Nanyang Technological University
HE1003 Basic Mathematics for Economists
PROBLEM SET 3B
1. Find the partial derivatives of
f (x; y) = x2 y + 5xy 3 + y 2
2. Suppose
f (x; y) = x3 y
x
y
3. Find the second partial derivatives of (1) and (2).
4. Find all the

Nanyang Technological University
HE1003 Basic Mathematics for Economists
PROBLEM SET 1A
1. [Equation of the tangent line] Find the equation of the line that is tangent to the graph of the given
function at the specied point.
x3
(a) y =
5x2 + 3x
(b) y = x5

Nanyang Technological University
HE1003 Basic Mathematics for Economists
PROBLEM SET 1
1. For each of the following functions, plot enough points to sketch to complete graph. Then answer the
following questions:
(a) Where is the function increasing and wh

Nanyang Technological University
HE1003 Basic Mathematics for Economists
LECTURE 8
Matrix Algebra
Matrices and matrix operations
1. A matrix is a rectangular arrays of numbers. The numbers in the array are called the entries in the matrix.
2. For examples

Nanyang Technological University
HE1003 Basic Mathematics for Economists
LECTURE 9
Determinants
Determinants by cofactor expansion
1. Recall that the 2
2 matrix
A=
is invertible if and only if ad
2. The expression ad
"
a b
c d
#
bc 6= 0:
bc is called the

Nanyang Technological University
HE1003 Basic Mathematics for Economists
LECTURE 7
System of Linear Equations
Introduction
1. In two dimensions a line in a rectangular xy-coordinate system can be represented by an equation of the
form
ax + by = c
(a; b 6=

Nanyang Technological University
HE1003 Basic Mathematics for Economists
LECTURE 3
Calculus of Several Variables
Functions of several variables
1. Production functions, cost functions, prot functions, utility functions and demand functions are common in e

Nanyang Technological University
HE1003 Basic Mathematics for Economists
LECTURE 6
Dierential Equations
Introduction
1. A dierential equation (DE) for a function y = f (x) is an equation involving the derivatives of y. For
example,
dy
+y =0
dx
is a dieren

Nanyang Technological University
HE1003 Basic Mathematics for Economists
LECTURE 5
Integration and Its Applications
Indenite integration
1. Previously we devoted to dierential calculus which measures the rate of change of functions. Dierentiation, we lear

Nanyang Technological University
HE1003 Basic Mathematics for Economists
LECTURE 1
One Variable Calculus I: Foundations
Basic notations
1. Some useful notations in mathematics are those for sums, products and factorials.
2. For sum we denote the sum
x1 +

Nanyang Technological University
HE1003 Basic Mathematics for Economists
LECTURE 2
One-Variable Calculus II: Applications of Dierentiation
Increasing and decreasing function
1. The derivative of a function carries much information about the important prop