\"e\ (Cambridge International AS Mathematics)

"e"

What is the number "e" in maths?

• e is an irrational number, sometimes called Euler's number

  • e ≈ 2.718

How do I sketch y = e^x?

• The exponential function is y = e^x

  • As with other exponential graphs y = e^x

    • passes through (0, 1)
    • has the x-axis as an asymptote

What is the special property of "e"?

• y = e^x has the particular property that

$\frac{dy}{dx}=e^{x}$

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

How do I sketch y = e^-x?

• 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