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Section 3.5 Exponential and Logarithmic Models 241
Section 3.5
Exponential and Logarithmic Models
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You should be able to solve compound interest problems.
1.
2.
A P 1 r n nt
A Pe rt
You should be able to solve growth and decay problems.
(a) Exponential growth if b > 0 and y ae bx .
(b) Exponential decay if b > 0 and y ae bx .
You should be able to use the Gaussian model
y ae xb2 c .
You should be able to use the logistics growth model
a
y
1 be .
xcd
You should be able to use the logarithmic models
y lnax b and y log 10 ax b.
Vocabulary Check
1. (a) iv (b) i (c) vi (d) iii (e) vii (f) ii (g) v
2. Normally 3. Sigmoidal 4. Bell-shaped, mean
1. y 2e x4 2. y 6e x4
3.
This is an exponential
growth model.
Matches graph (c).
This is an exponential decay
model.
Matches graph (e).
y 6 log 10 x 2
This is a logarithmic model,
and contains 1, 6.
Matches graph (b).
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4. y 3e x22 5 5. y lnx 1
6.
4
y
1 e 2x
Gaussian model
This is a logarithmic model.
Logistics model
Matches (a).
Matches graph (d).
Matches (f ).
7. Since A 10,000e 0.035t , the time to double is
8. Since A 2000e 0.015t , the time to double is
given by
given by
20,000 10,000e 0.035t
2 e 0.035t
ln 2 0.035t
t ln 2 19.8 years.
0.035
Amount after 10 years:
A 10,000e 0.03510 $14,190.68
4000 2000e 0.015t
2 e 0.015t
ln 2 0.015t
t ln 2 46.2 years.
0.015
Amount after 10 years:
A 2000e 0.01510 $2323.67