The Natural Base, e Objective: Model exponential growth/decay.
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The Natural Base, e Objective: Model exponential growth/decay.
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Transcript of The Natural Base, e Objective: Model exponential growth/decay.
The Natural Base, e
Objective: Model exponential growth/decay.
110log 1log 101 DNE 10log 01log
110log 1log 101 DNE 10log 01log
24.1480)1(1000 10104.
86.1488)1(1000 40404.
The Natural Exponential Function
The Natural Exponential Function
Example 1
Example 1
Example 2
Example 2
Try This
• Find the value of $500 after 4 years invested at an annual interest rate of 9% compounded monthly and continuously.
Try This
• Find the value of $500 after 4 years invested at an annual interest rate of 9% compounded monthly and continuously.
70.715$1500 48
1209.
Try This
• Find the value of $500 after 4 years invested at an annual interest rate of 9% compounded monthly and continuously.
70.715$1500 48
1209.
66.716$500 )409(. e
Example 3
Example 3
Inverse Functions
• Remember, lnx and ex are inverse functions. This means that:
xex ln
xe x ln
Example 4
Example 4
Try This
• How long does it take for an investment to triple at an annual interest rate of 7.2% compounded continuously?
Try This
• How long does it take for an investment to triple at an annual interest rate of 7.2% compounded continuously?
t
t
t
e
e
PeP
072.
072.
072.
ln3ln
3
3
Try This
• How long does it take for an investment to triple at an annual interest rate of 7.2% compounded continuously?
t
t
t
e
e
PeP
t
t
t
26.15072.
3ln
072.3ln
ln3ln
3
3
072.
072.
072.
Radioactive Decay
• Most of the carbon found in the Earth’s atmosphere is the isotope cardon-12, but a small amount is the radioactive isotope carbon-14. Plants absorb carbon dioxide from the atmosphere, and animals obtain carbon from the plants they consume. When a plant or animal dies, the amount of carbon-14 it contains decays in such a way that exactly half of its initial amount is present after 5730 years. The function below models the decay of carbon-14.
teNtN 00012.0)(
Example 5
• Suppose that archaeologists find scrolls and claim that they are 2000 years old. Tests indicate that the scrolls contain 78% of their original carbon-14.
Example 5
• Suppose that archaeologists find scrolls and claim that they are 2000 years old. Tests indicate that the scrolls contain 78% of their original carbon-14.
• Could the scrolls be 2000 years old?
Example 5
• Suppose that archaeologists find scrolls and claim that they are 2000 years old. Tests indicate that the scrolls contain 78% of their original carbon-14.
• Could the scrolls be 2000 years old?
t
t
t
e
eNN
eNtN
00012.
00012.00
00012.0
78.
78.
)(
Example 5
• Suppose that archaeologists find scrolls and claim that they are 2000 years old. Tests indicate that the scrolls contain 78% of their original carbon-14.
• Could the scrolls be 2000 years old?
t
t
t
e
eNN
eNtN
00012.
00012.00
00012.0
78.
78.
)(
t
t
t
e t
5.207000012.
78.ln
00012.78.ln
ln78.ln 00012.
Homework
• Pages 397-399• 13-59 odd• 75,77,79
