9.3 - The Natural Base, e

9.3 - The Natural Base, e

Leonhard Euler (1707-1783)

Swiss mathematician responsible for the development of a new number "e"

This number would be called the "natural base" or "Euler's number"

Applications include compound interest in finance.

Multiple Choice

How do you pronounce "Euler?"

• You-ler

• Oil-er

• Ee-ler

Multiple Select

Select all of the alternative names for the number "e"

• Natural Number

• Euler's Number

• Natural Base

• Euler's Base

Pi vs. Natural Base

Both irrational numbers.

Pi is the ratio between a circle's circumference and its diameter.

e is base rate of growth shared by all continually growing processes

Multiple Choice

Applications of "e" include financial calculations with continuously compounded interest.

• True

• False

One way of approximating the value of "e"

Another way of approximating the value of e.

Simplifying Natural Base Expressions

 $e^3\cdot e^6$  

 $\frac{16e^5}{4e^4}$   

 $\left(3e^{4x}\right)^2$  

Multiple Choice

Simplify.

 $e^{-4}\cdot e^{-5}$  

•  $e^9$  

•  $e^{-9}$  

•  $e^{-45}$  

•  $e^{45}$  

Multiple Choice

Simplify.

 $\frac{30e^5}{4e^2}$  

•  $\frac{30e^3}{4}$  

•  $\frac{15e^7}{2}$  

•  $\frac{15e^{10}}{2}$  

•  $\frac{15e^3}{2}$  

Multiple Choice

Simplify.

 $\left(2e^{-3}\right)^3$  

•  $6e^{-9}$  

•  $8e^{-9}$  
  

•  $6e^{-6}$  

•  $8e^9$  

Multiple Choice

If k is positive in e^kx, what happens as x increases?

• exponential growth

• decay

• grows in the negative direction

• stays constant

Multiple Choice

If k is negative in e^kx, what happens as x increases?

• Growth

• Slow growth

• decays towards zero

• stays constant

• Gets large and negative

Multiple Choice

If k is fractional in e^kx, what happens as x increases?

• grows rapidly

• grows slowly

• decays towards zero

• stays constant

What is the rate of increase or decrease?

 $y=2e^{-2x}$  

Fill in the Blank

What is the rate of increase or decrease? Round to the nearest tenth (one decimal place) and write as a percent. 

 $y=e^{-0.5x}$