e

Examples of e in the following topics:

• Graphs of Exponential Functions, Base e

  • The function $f(x) = e^x$ is a basic exponential function with some very interesting properties.
  • The basic exponential function, sometimes referred to as the exponential function, is $f(x)=e^{x}$ where $e$ is the number (approximately 2.718281828) described previously.
  • The graph's $y$-intercept is the point $(0,1)$, and it also contains the point $(1,e).$ Sometimes it is written as $y=\exp (x)$.
  • The graph of $y=e^{x}$ is upward-sloping, and increases faster as $x$ increases.
  • $y=e^x$ is the only function with this property.

• The Number e

  • Note that $\ln (e) =1$ and that $\ln (1)=0$.
  • There are a number of different definitions of the number $e$.
  • Another is that $e$ is the unique number so that the area under the curve $y=1/x$ from $x=1$ to $x=e$ is $1$ square unit.
  • Another definition of $e$ involves the infinite series $1+\frac{1}{1!}
  • It can be shown that the sum of this series is $e$.

• Conics in Polar Coordinates

  • With this definition, we may now define a conic in terms of the directrix: $x=±p$, the eccentricity $e$, and the angle $\theta$.
  • For a conic with a focus at the origin, if the directrix is $x=±p$, where $p$ is a positive real number, and the eccentricity is a positive real number $e$, the conic has a polar equation:
  • For a conic with a focus at the origin, if the directrix is $y=±p$, where $p$ is a positive real number, and the eccentricity is a positive real number $e$, the conic has a polar equation:

• Cramer's Rule

• Eccentricity

  • The eccentricity, denoted $e$, is a parameter associated with every conic section.
  • The value of $e$ is constant for any conic section.
  • The value of $e$ can be used to determine the type of conic section as well:

• Natural Logarithms

  • The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828.
  • The natural logarithm is the logarithm with base equal to e.
  • Just as the exponential function with base $e$ arises naturally in many calculus contexts, the natural logarithm, which is the inverse function of the exponential with base $e$, also arises in naturally in many contexts.
  • Its value at $x=1$ is $0$, while its value at $x=e$ is $1$.

• Applications of Hyperbolas

  • However, this is the very special case when the total energy $E$ is exactly the minimum escape energy, so $E$ in this case is considered to be zero.
  • Blue is a hyperbolic trajectory ($e > 1$).
  • Green is a parabolic trajectory ($e = 1$).
  • Red is an elliptical orbit ($e < 1$).
  • Grey is a circular orbit ($e = 0$).

• Scientific Notation

  • Because superscripted exponents like $10^7$ cannot always be conveniently displayed, the letter E or e is often used to represent the phrase "times ten raised to the power of" (which would be written as "$\cdot 10^n$") and is followed by the value of the exponent.
  • Note that in this usage, the character e is not related to the mathematical constant $\mathbf{e}$ or the exponential function $e^x$ (a confusion that is less likely if a capital E is used), and though it stands for exponent, the notation is usually referred to as (scientific) E notation or (scientific) e notation, rather than (scientific) exponential notation.

• Introduction to Exponential and Logarithmic Functions

  • Let us consider instead the natural log (a logarithm of the base $e$). 
  • The natural logarithm is the inverse of the exponential function $f(x)=e^x$.
  • It is defined for $e>0$, and satisfies $f^{-1}(x)=lnx $.
  • That is, $e^{lnx}=lne^x=x$.

• Parts of an Ellipse

  • All conic sections have an eccentricity value, denoted $e$.
  • All ellipses have eccentricities in the range $0 \leq e < 1$.
  • $\displaystyle{ \begin{aligned} e &= \frac{\sqrt{a^2 - b^2}}{a} \\ &= \sqrt{ \frac{a^2 - b^2}{a^2} } \\ &= \sqrt{ 1 - \frac{b^2}{a^2}} \end{aligned} }$