# Lecture 6 - Columbia University ```“I feel like I’m diagonally
parked in a parallel universe”
Math Review
Monday June 7 2003
A) Introduction
a. Symbols
b. Operations
c. Central Tendencies
B) Linear Algebra
C) Correlation/Regression Analysis
D) Applied Calculus
Basic Math Review
B) System of equations
a) 3x - y = -7
5y + 5 = -5x
b) 3x + 4y = 2
2y = 4 - 3/2x
Basic Math Review
D) Applied Calculus
Rate of change (slope): Dy/Dx or (y2-y1)/(x2-x1)
Here
Dy/Dx is constant regardless of the “limit”
160
140
Street Number
120
100
80
60
40
20
0
0
1000
2000
3000
4000
Basic Math Review
D) Applied Calculus
How long does it take to fill one beaker (1L)?
 dV 
t  1000ml /  
 dt 
V(t)
dV 
 1000ml/t


dt 

t
Basic Math Review
D) Differential equations
A differential equation is an equation in which one
or more unknowns depend on its/their rate of
change (or that of other variables included in the
equations)
F  ma
where
dv(t ) d 2 x(t )
a

2
dt
dt
Newton’s principia
Basic Math Review
D) Definition of a derivative
f ( x0 )  lim
Dx 0
f ( x  Dx)  f ( x)
Dx
The derivative of a function (x) at a point “a” is
the slope of the straight line tangent to (x) at
“a”  instantaneous rate of change!
One is pushing to limit to “0”: the slope is close
to real as (x) approaches 0
Basic Math Review
D) Definition of a derivative
f ( x0 )  lim
Dx 0
f ( x  Dx)  f ( x)
Dx
The derivative of a function (x) = f’(x)
Important derivatives:
f(x) = C  f’(x) = 0
f(x) = xn  f’(x) = nxn-1
f(x) = ex  f’(x) = ex
f(x) = lnx  f’(x) = 1/x
Basic Math Review
D) Maxima, Minima
One of the great applications of calculus (particularly
in economics) is to determine the “maxima” and
“minima” of functions.
The derivatives of the maxima and minima = 0
The function neither increase nor decreases!
f(x) = x3 – 3x2 – 24x + 5
f’(x) = 3x2 – 6x – 24
3 (x2 – 2x – 8)
3 (x + 2)(x –4)
f’(x) vanishes (reaches a critical point) only when f’(x) = 0
D) Maxima, Minima
f”(x) &lt; 0  maximum
f”(x) &gt; 0  minimum
250
200
150
100
50
0
-8
-6
-4
-2
0
-50
-100
-150
-200
2
4
6
8
10
D) Application: Derivatives
Ji = -  DS  ([Ci ]/z)
DS = D0  2
J Cz=0 = -3  D0 (8 oC)  ([Ci ]/z)z=0
D) Application: Derivatives
Ji = -  DS  ([Ci ]/z)
DS = D0  2
J Cz=0 = -3  D0 (8 oC)  ([Ci ]/z)z=0
“A River (used to) Run Through It”: Part I
Reservoirs and soil erosion
Fo (%)
0%
2%
4%
6%
8%
10%
0
12%
1200
5
10
Depth (cm)
15
20
25
30
35
40
Fo
45
C/N
50
12
14
16
(C/N)a
18
20
D) Application: Integral
1.2
J-Co (gC/m2.yr)
0.0
1.0
0.2
0.4
0.6
0.8
1.0
0
1200
5
10
15
0.6
Depth (cm)
Jo (gC/m2.yr)
0.8
0.4
0.2
20
25
30
35
40
0.0
0
45
10
50
20
30
40
50
Age (years)
60
70
80
90
D) Application: Integral
(x) = Polynomial (6th degree)
∫ (xn) = [n(n+1)/(n+1)] + C
1.2
1.0
Jo (gC/m2.yr)
0.8
0.6
0.4
0.2
y = -2E-10x6 + 4E-08x5 - 3E-06x4 + 0.0001x3 - 0.0013x2 + 0.0136x + 0.3082
R2 = 0.9795
0.0
0
10
20
30
40
50
60
70
80
90
Age (years)
Integral between two limits (0 and 85 yrs)
∫ (xn)100 - ∫ (xn)0
C-C
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