Mathematics for Economics Mathematics for Economics

Mathematics for Economics

Table of Contents
Part A...............................................................................................................................................3
Question 1....................................................................................................................................3
Question 2....................................................................................................................................3
Question 3....................................................................................................................................3
Question 4....................................................................................................................................3
Part -B..............................................................................................................................................3
Question 5....................................................................................................................................3
Question 6....................................................................................................................................3
Question 7....................................................................................................................................3
Question 8....................................................................................................................................3
2

Part A
Question 1
(a). Different the following function F (x) = -4x2+3x-1
F(x) = -4 x2+3x-1
F(2x)=2 F(x)
=-4(2 x)2+3(2x)-1=2(-4 x2+3x-1)
=-16x2+6x-1=-8x2+3x-1
=-16 x2+ 8 x2+6x-6x-1+2
=-8 x2+1
=-(8 x2−1 ¿
8 x2−1=0
8 x2=1
x2= 1
8
X=± √ 1
8
The value of x is √ 1
8 or -
√ 1
8
F(2x)=f(x)
=-4(2 x)2+3(2x)-1=-4 x2+3x-1
=-16x2+3x-1=4x2+3x-1
=-16 x2+4 x2+3x-3x-1+1
=-12 x2+4x2=0
=-4x(3x-1)=0
-4x=0 3x-4=0
3

X=0 3x=4 x=4/3
The find the final value of the different function is x=0 and x=1
(b) g(x)=x2 ln x
The function g(x)= x2 ln x is the from f(x)=g(x).h(x) which makes it suitable for appliance of
equation is
F’(x)=g’(x)h(x)+g(x)h’(x)
Following value of each function,
G(x)=x2
H(x)=ln x
G’(x)= 2 x2
2
h’(x)= 1
x
When we substitute each of these into equation is
F’(x)= 2 x2
2 ln x + x2 ln x
= x2 ln x+ x2. 1
x
= x2 ln x + x
=x3 ln x
(c) h(x)= 3ex √ x3 +1
d
dx (h(x)=3ex √x3+1
The different function equation that can follows that,
d
dx (h(x))= dh(u)
du . du
dx ' =u=x
d
du (h(u)=h’(u)
4

d
dx (h’(x)) = d
dx (3ex √x3+1)
= d
dx (1+3ex √x3)
=3( d
dx (x
3
2).ex+x
3
2 . d
dx (ex)¿+0
=3( 3
2 x
3
2-1 .ex +e x . x
3
2 ¿
=3(x
3
2 e x+ 3 √x ex
3 )
=3x
3
2 e x+ 9 √ x ex
3
= 3 √ x (2 x+3) ex
2
2. For function f(x) in 1a, write down the difference ratio ∆ f
∆ x (equivalently indicated with ∆ f
h )
and show that the limit for the increment going to zero is equal to the derivative function.
Formally, show that lim
h→ 0
∆ f
h ( x )=f ' (x ) briefly discuss the meaning of the difference ratio.
3. In a carefully drawn and labelled two-dimensional Euclidean space, draw function f(x) and its
derivative, clearly indicating the coordinates of the maximisers, minimisers, maxima, minima,
and the intersections with both axes, and the intersections between the two functions. Discuss the
relationship between the graphs of f(x) and the graph of f’(x).
Let us start by thinking about the multidimensional functions we can graph:
5

A point (a, b) is a fixed point known as the maximum, minimum, or saddle point. The true value
at a fixed point is called a fixed value. We need a mathematical method to find the fixed points
of the F (X, Y) function and classify them as the maximum, minimum, or saddle point. This
method resembles a method that generates the first and second derivatives of the function, but is
more complex.
Question 2
1.
We can consider the value of lottery is L={1/2,£ 2,000 ; 1
2 ;10,000}
Using the method of utility function is u: R0
+¿ ¿R and consider the function form is u(x)=ln(x)
Expected values on the Lottery E[L]=∑ L . p ( L )
win loss
Gain(L) 10000 2000
Probability (L) 1/2 1/2
E[L]=∑ 10000. p( 1
2 )=5,000
E[L]=∑ 2000. p ( 1
2 )=1,000
6