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8.3 The Number e

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Standards 11, 14

- Goal 1: Use the number e as the base of exponential functions

Goal 2: Use the number e in real-life situations

- 0: zero, null, nill. Invented about 825 by Al-Kwarizmi
- π ≈ 3.141592654... The ratio of the circumference of a circle to the diameter.
- ɸ ≈ 1.618...
- 9,201,976: Swenson’s birthday

- Natural base e
- Discovered by Leonhard Euler (1707 - 1783)
- e ≈ 2.718281828459...

Compound Interest

Compound Interest

Goal 1: Use the number e as the base of exponential functions

asymptote is where?

Graph:

- Goal 1: Use the number e as the base of exponential functions

2.718...

- Goal 1: Use the number e as the base of exponential functions

Simplify the expression.

- Goal 1: Use the number e as the base of exponential functions

Simplify the expression.

- Goal 1: Use the number e as the base of exponential functions

Use a calculator to evaluate.

- Goal 1: Use the number e as the base of exponential functions

1

2

- Goal 1: Use the number e as the base of exponential functions

Growth or decay?

y-intercept?

asymptote?

- Goal 1: Use the number e as the base of exponential functions

We learned that amount after compound interest can be calculated by:

as n →∞

But we know that

So...

for compounding continuously

Goal 2: Use the number e in real-life situations

You deposit $1000 in an account that pays 8% annual interest. Find the balance after one year if the interest is compounded with the given frequency.

Daily

Continuously

Goal 2: Use the number e in real-life situations