the number e
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PreCalculus NYOS Charter School Quarter 4 "Nature's great book is written in mathematics." ~Galileo. The Number e. The Number e. Leonhard Euler found an interesting irrational number named e . The number is the sum of + … - PowerPoint PPT PresentationTRANSCRIPT
PRECALCULUSNYOS CHARTER SCHOOLQUARTER 4
"NATURE'S GREAT BOOK IS WRITTEN IN MATHEMATICS." ~GALILEO
The Number e
The Number e
Leonhard Euler found an interesting irrational number named e.
The number is the sum of + …
We can round this off to two decimal places as an estimation of e.
The Number e
We use e in exponential growth and decay problems.
N is the final amountN0 is the initial amount
k is a constantt is time
The Number e
Example: DDT is effective against insects, but was found to be harmful to humans in 1973. More than 1 * 109 kg of DDT had already been used before the risk was identified. How much will remain in the environment in 2020, assuming k = -0.0211, if we stopped using DDT in 1973?
The Number e
Example: 1 * 109 kg; k = -0.0211, 1973 - 2020?
kg
The Number e
We also use e in problems involving continuously compounding interest.
A is the final amountP is the initial amount
r is the annual ratet is time in years
The Number e
Example: Compare the balance after 25 years of a $10,000 investment earning 6.75% interest compounded continuously to the same investment compounded semiannually.
The Number e
Example: t = 25; P = $10,000; r = 6.75%; semiannually vs. continuously
The Number e
Example: t = 25; P = $10,000; r = 6.75%; semiannually vs. continuously
A = $52,574.62 A = $54,059.49
The Number e
Example: t = 25; P = $10,000; r = 6.75%; semiannually vs. continuously
A = $52,574.62 A = $54,059.49
The Number e
If we were to invest the same amount each month in an account with continuously compounding interest, our formula would be
A is the final amount
M is the monthly payment amount
r is the monthly rate
t is time in months
The Number e
Example: If we invest $1000 per month in an account that has 6% continuous compounding, how much will we have at the end of one year?
The Number e
Example: M = $1000; t = 12; r = .06/12
The Number e
Example: If we invest $500 per month in an account that has 4% continuous compounding, how much will we have at the end of 10 years?
The Number e
Example: M = $500; t = 120; r = .04/12
The Number e
Example: Make a graph of the account balance over the 10 years.
This is part of #5 on the rubric for the project.