2.1.3 "e"

What is e, the exponential function?

• The exponential function is y = e^x
  • e is an irrational number
  • e ≈ 2.718
• As with other exponential graphs y = e^x
  • passes through (0, 1)
  • has the x-axis as an asymptote

  

What is the big deal with e?

• y = e^x has the particular property

• ie for every real number x, the gradient of y = e^x is also equal to e^x (see Derivatives of Exponential Functions)

The negative exponential graph

• y = e^-x is a reflection in the y-axis of y = e^x
• They are of the form y = f(x) and y = f(-x) (see Transformations of Functions - Reflections)

  

 

What is exponential growth and decay?

 

• y = Ae^kx (k > 0) is exponential growth
• y = Ae^-kx (k > 0) is exponential decay
• A is the initial value
• k is a (usually positive) constant
• “-“ is used in the equation making clear whether it is growth or decay

Worked Example