Growth
Authors: Bill Davis, Horacio Porta and Jerry Uhl
©19962007
Publisher: Math Everywhere, Inc.
Version 6.0
1.03 Instantaneous Growth Rates
BASICS
B.1) Instantaneous growth rates
Here is a friendly function
f
@
x
D
=
1
+
2x
3

x
4
and a plot:
Clear
@
f, x
D
;
f
@
x
_
D
=
1
+
2x
3

x
4
;
fplot
=
Plot
@
f
@
x
D
,
8
x,

1, 2
<
, PlotStyle
Ø
88
Thickness
@
0.01
D
, Blue
<<
,
AspectRatio
>
1
ê
GoldenRatio, AxesLabel
Ø
8
"x"
,
""
<D

1.0

0.5
0.5
1.0
1.5
2.0
x

2

1
1
2
·
B.1.a.i)
Measure the net growth of
f
@
x
D
=
1
+
2x
3

x
4
over the interval
@

1,2
D
.
Then measure the average growth rate of
f
@
x
D
over the
interval
@

1,2
D
.
·
Answer:
Here you go:
Over the interval
@

1,2
D
, the function starts out at:
f
@

1
D

2
And it ends up at:
f
@
2
D
1
Its net growth is:
fgrowth
=
f
@
2
D

f
@

1
D
3
Its average growth rate in units on the
y
axis per unit on the
x
axis over the interval
@

1,2
D
is:
xgrowth
=
2

H

1
L
3
fgrowth
xgrowth
1
As
x
grows from

1
to
2
, on the average,
f
@
x
D
grows at a rate of
1
unit every time
x
grows by
1
unit.
The average growth rate of
f
@
x
D
over the interval
@

1,2
D
is
1
.
·
B.1.a.ii)
Measure the average growth rate of
f
@
x
D
=
1
+
2x
3

x
4
over the interval
@
x,x
+
0.5
D
.
Interpret the result.
·
Answer:
Clear
@
f, x
D
;
f
@
x
_
D
=
1
+
2x
3

x
4
1
+
2x
3

x
4
Over the interval
@
x,x
+
0.5
D
, the function starts out at:
f
@
x
D
1
+
2x
3

x
4
And it ends up at:
f
@
x
+
0.5
D
1
+
2
H
0.5
+
x
L
3

H
0.5
+
x
L
4
Its net growth over the interval
@
x,x
+
0.5
D
is:
fgrowth
=
Expand
@
f
@
x
+
0.5
D

f
@
x
DD
0.1875
+
1.x
+
1.5x
2

2.x
3
Its average growth rate in units on the
y
axis per unit on the
x
axis over the interval
@
x,x
+
0.5
D
is:
xgrowth
=
0.5;
Clear
@
fAverageGrowthRate, x
D
;
fAverageGrowthRate
@
x
_
D
=
Expand
B
fgrowth
xgrowth
F
0.375
+
2.x
+
3.x
2

4.x
3
On the interval
@
x,x
+
0.5
D
, the average growth rate of
f
@
x
D
is
0.375
+
2x
+
3x
2

4x
3
.
For instance, when you look at:
fAverageGrowthRate
@
0
D
0.375
then you see that, on the average,,
f
@
x
D
grows
0.375
times as fast as
x
grows as
x
advances from
0
to
0
+
0.5
=
0.5
.
But when you look at:
fAverageGrowthRate
@
1.5
D

3.375
Then you see that, on the average,
f
@
x
D
goes down
3.375
times as fast as
x
grows as
x
advances from
1.5
to
1.5
+
0.5
=
2
.
·
B.1.a.iii)
Given a positive number
h
, measure the average growth rate of
f
@
x
D
=
1
+
2x
3

x
4
over the interval
@
x,x
+
h
D
.
Interpret the result.
·
Answer:
Clear
@
f, x
D
;
f
@
x
_
D
=
1
+
2x
3

x
4
1
+
2x
3

x
4
Over the interval
@
x,x
+
h
D
, the function starts out at:
f
@
x
D
1
+
2x
3

x
4
And it ends up at:
Clear
@
h
D
;
Expand
@
f
@
x
+
h
DD
1
+
2h
3

h
4
+
6h
2
x

4h
3
x
+
6hx
2

6h
2
x
2
+
2x
3

4hx
3

x
4
Its net growth over the interval
@
x,x
+
h
D
is:
fgrowth
=
Expand
@
f
@
x
+
h
D

f
@
x
DD
2h
3

h
4
+
6h
2
x

4h
3
x
+
6hx
2

6h
2
x
2

4hx
3
Its average growth rate in units on the
y
axis per unit on the
x
axis over the interval
@
x,x
+
h
D
is:
xgrowth
=
h;
Clear
@
fAverageGrowthRate, x, h
D
;
fAverageGrowthRate
@
x
_
, h
_
D
=
Expand
B
f
@
x
+
h
D

f
@
x
D
h
F
2h
2

h
3
+
6hx

4h
2
x
+
6x
2

6hx
2

4x
3
This gives you the measurement of the average growth rate of
f
@
x
D
on the interval
@
x,x
+
h
D
.
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 Spring '08
 Staff
 Math, Mathematica, Instantaneous Growth Rate

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