Economics 214
Lecture 6
Polynomial Functions
. to1
from integers are polynomial in the exponents theand
numbers real are ,,2,1,0, parameters the
)(
form theakesfunction t polynomial univariateA 2
210
n
nia
xaxaxaaxfy
i
nn
Polynomial Functions The degree of the polynomial is
the value taken by the highest exponent.
A linear function is polynomial of degree 1.
A polynomial of degree 2 is called a quadratic function.
A polynomial of degree 3 is called a cubic function.
Roots of Polynomial Function
a
acbbxx
cbxaxy
bax
bxay
2
4, :roots
:function Quadratic
/ :root
:functionLinear
zero. equalfunction themakethat
argument its of values theare polynomial a of roots The
2
21
2
3 cases for roots of quadratic function
b2-4ac>0, two distinct roots. b2-4ac=0, two equal roots b2-4ac<0, two complex roots
Quadratic example
34
12
4
57
4
257
4
24497
2*2
2*3*477
2
4
2
1
4
2
4
57
4
257
4
24497
2*2
2*3*477
2
4
372
22
2
22
1
2
a
acbbx
a
acbbx
xxy
Plot of our Quadratic function
Roots of Quadratic Equation
-4
-2
0
2
4
6
8
10
12
14
-2 -1 0 1 2 3 4 5
x
y y
Exponential Functions
The argument of an exponential function appears as an exponent.
Y=f(x)=kbx
k is a constant and b, called the base, is a positive number.
f(0)=kb0=k
Exponential Functions
0
and with decreasesly montonical 0,b1When
axis.-y theacross function theof
reflection a is 1
ofgraph theThus1
b
1
exponents of rulesour Using
0
case In this . with increases then ,1When
x.of any valuefor parameter the
ofsign theas same theisfunction thisofsign The
lim
lim
x
x
x
x
x
xx
x
x
x
x
kb
xkb
b
bb
b
kb
xbb
k
Figure 2.16 Some Exponential Functions
Exponential functions with k<0
Exponential Functions with k<0, b=3/2 and 2/3
-8000
-7000
-6000
-5000
-4000
-3000
-2000
-1000
0
1000
-25 -20 -15 -10 -5 0 5 10 15 20 25
x
y
y2
y1
Regional Growth in the U.S.
Region 1990.4 1995.4 2000.4 1991-1995 1996-2000 New England Region 303846 370457 510649 4.0% 6.6% Mideast Region 1000386 1209580 1600680 3.9% 5.8% Great Lakes Region 818083 1048335 1337536 5.1% 5.0% Plains Region 327328 417824 553255 5.0% 5.8% Southeast Region 1051354 1392657 1857465 5.8% 5.9% Southwest Region 443615 598893 850857 6.2% 7.3% Rocky Mountain Region 130333 184108 264158 7.2% 7.5% Far West Region 888273 1073627 1515873 3.9% 7.1%
Annual Grow RateYear and Quarter
New England Income
1264012154
Change
328640303846)(1.0816303846)()04.1(
)1()1)(1(
)1()1()1(
4:1992
316000303846)(04.1
)04.1()1(
4:1991
Income England New
.
)1(
relation thehave wegeneral,In
4:19914:19924:19904:1991
2
4:19902
4:1990
4:19904:19914:1992
4:19904:19904:1991
1
XXXX
XrXrr
XrrXrX
XXrX
formdecimalinrategrowthannualr
tyearinincomeXwhereXrX ttt
Growth Formula
form. decimalin expressed is
)1(
is period,each of end at the
rate by the growsit when periodin level its and ,
period,in variablea of valueebetween th iprelationsh The
gCompoundin
Period-of-End Discrete with FormulaGrowth
r
whereXrX
r
ntt, X
tn
nt
t
New England Personal Income Annual Growth
Rate = 4% Our bar Chart is
approximately a step function.
We assume growth doesn’t occur until end of the year.
050000
100000150000200000250000300000350000400000
1990:4
1991:4
1992:4
1993:4
1994:4
1995:4
PersonalIncome
Step Function
1990
19
90
19
91
19
92
19
93
19
94
19
95
Income
Year
Income Growth
We have depicted the growth of income over time as a step function.
It is usually more natural to think of continuous growth, which would be reflected in a smooth evolution over time of variables like income and population.
Multiple Compounding in one Period
t
k
t
1t
t
Xk
rX
kr
X
t, X
1
is period, theduring times compounded rate the
by growsit when , period,next theof beginning
at the valueits and , period of beginning at the
variablea of valueebetween th iprelationsh The
1
Effect of Number of compounds at 4%
Compounds 1,000.00$ 1 1,040.00$ 2 1,040.40$ 4 1,040.60$
12 1,040.74$ 365 1,040.81$
Continuous 1,040.81$
Effect of Number of Compounds at 4%
$1,000 Compounded over 1 year
$1,039.40
$1,039.60
$1,039.80
$1,040.00
$1,040.20
$1,040.40
$1,040.60
$1,040.80
$1,041.00
1 2 4 12 365 Continuous
Number of Compunds
$1,000
Figure 3.2 Compounding at Different Frequencies