Math Review

ECN601 – Section 003
Syracuse University
Fall 2010

Professor Lourenço Paz

Math Review

1. Basic algebra and geometry
Check out the handouts at : http://www1.maxwell.syr.edu/pa.aspx?id=36507223186 2. Calculus of one variable
Let y = axb, where a and b are numbers, with b different from zero. The derivative of y with respect to x is dy/dx = abxb-1
Examples:
y = 2x7, dy/dx = 2*7x7-1 = 14x6 y = 5x0.3, dy/dx = 5*0.3*x0.3-1 = 1.5x-0.7
Exercises: Calculate dy/dx
a) y= 3x2
b) y = 12x0.8

c) y = 6

e) y = 5/x4

g) y = 3

f) y = 4x2 + 2/x3

d) y =
+4/

h) y = 8x

Another useful derivative rule is for the natural logarithm. Let y = ln x, for x>0. Then, dy/dx = 1/x
Example: y = 5 lnx, dy/dx = 5/x
Exercises:
i) y = 3lnx

j) y = -2lnx

k) y = 2x7 – (lnx)/2

1

l) y = 5ln x - 2/x3

ECN601 – Section 003
Syracuse University
Fall 2010

Professor Lourenço Paz

3. Calculus of more than one variable
When the function y depends on more than one variable, e.g. x and z, we use the partial derivative. So, the partial derivative of y with respect to x
( ) provides the change in y given an infinitesimal change in x, keeping z constant (or if you prefer, treating z as a number)!
Examples:
y = x4 +3z;

= 4x3

y = 3x2z8 + z2 +6;

= 3*z8*2*x2-1 = 6z8x

=3

= 3x2*8z8-1 +2z = 24x2z7 +2z y=3lnx + z3;
Exercises: Calculate
a) y = 3lnx + 4lnz
d) y = 3x + 5z0.5

= 3z2

= 3/x and b) y = 10x0.5z0.5
e) y = 4x3z5

c) y = 2x0.2z0.6

Answers
1.
a) dy/dx = 3*2*x2-1 = 6*x
b) dy/dx = 12*0.8*x0.8-1 = 9.6*x-0.2
c) dy/dx =6*(1/2)*x1/2 – 1 = 3*x-1/2 = 3/x1/2
d) dy/dx = ¼ * x¼ -1 = ¼ * x-¾ = 1/(4 x¾)
e) dy/dx = 5*(-4)x-4-1 = -20x-5 = -20/x5
f) dy/dx = 4*2x2-1 + 2*(-3)x-3-1 = 8x – 6/x4
g) dy/dx = 3*1/2*x0.5-1 + 4*(-0.5)x-0.5-1 = 3/(2
2

) -2/x1.5

ECN601 – Section 003
Syracuse University
Fall 2010

Professor Lourenço Paz

h) dy/dx = 8x1-1 = 8x0 = 8
i) dy/dx = 3*1/x = 3/x
j) dy/dx = -2*1/x = -2/x
k) dy/dx =