Understanding the Exponential Function

Exponential Function

Definition and Fundamental Property

• The exponential function, denoted as exp, is uniquely defined as a differentiable function on $\mathbb{R}$.
• It satisfies $\exp'(x) = \exp(x)$ and $\exp(0) = 1$.
• Fundamental Property: For all real numbers $a$ and $b$, $\exp(a + b) = \exp(a) \times \exp(b)$.

Notation $e^x$

• $\exp(x)$ is commonly written as $e^x$.
• $e$ is defined as $\exp(1)$, approximately equal to 2.718.
• Properties: For real numbers $x$ and $y$:
  • $e^{x+y} = e^x \times e^y$
  • $e^{x-y} = \frac{e^x}{e^y}$
  • $e^{-x} = \frac{1}{e^x}$
  • $(e^x)^y = e^{xy}$

Studying the Exponential Function

Sign and Variations

• The exponential function is strictly positive over $\mathbb{R}$, hence $e^x > 0$ for all real $x$.
• It is strictly increasing on $\mathbb{R}$.

Limits

• $\lim\limits_{x \to +\infty} e^x = +\infty$
• $\lim\limits_{x \to -\infty} e^x = 0$

Derivative

• The exponential function is differentiable on $\mathbb{R}$, with its derivative being itself: $(e^x)' = e^x$.
• If $u$ is a differentiable function on an interval $I$, then $(e^{u(x)})' = u'(x)e^{u(x)}$.

Table of Variations

Graphical Representation

• The graph of $y = e^x$ is always above zero and increases.

Equations with Exponentials

Fundamental Theorem

• The exponential function is a bijection from $\mathbb{R}$ to $]0; +\infty[$.
• For real numbers $a$ and $b$: $e^a = e^b \Leftrightarrow a = b$.

Inequalities

• Given the exponential function is strictly increasing on $\mathbb{R}$, for real numbers $a$ and $b$:
  • $e^a < e^b \Leftrightarrow a < b$
  • $e^a > e^b \Leftrightarrow a > b$

Examples

• Solving exponential equations and inequalities:
  1. $e^{3x-5} = e^{x+1} \Leftrightarrow 3x - 5 = x + 1 \Leftrightarrow x = 3$
  2. $e^{x^2-3} = \frac{1}{e} \Leftrightarrow e^{x^2-3} = e^{-1} \Leftrightarrow x = \sqrt{2}$ or $x = -\sqrt{2}$
  3. $e^{2x+1} > e^{-x+4} \Leftrightarrow 2x + 1 > -x + 4 \Leftrightarrow x > 1$

Chapter 2: The Topology of $\mathbb{R}^n$

2.1 Open and Closed Sets

• Open Ball Definition: For $\mathbf{x} \in \mathbb{R}^n$ and $r > 0$, $B(\mathbf{x}; r) = {\mathbf{y} \in \mathbb{R}^n : |\mathbf{y} - \mathbf{x}| < r}$.
• Open Set Definition: $E \subset \mathbb{R}^n$ is open if for every $\mathbf{x} \in E$ there exists $r > 0$ such that $B(\mathbf{x}; r) \subset E$.
• Theorem 2.1.3:
  • $\mathbb{R}^n$ and $\emptyset$ are open sets.
  • The intersection of a finite number of open sets in $\mathbb{R}^n$ is open.
  • The union of an arbitrary collection of open sets in $\mathbb{R}^n$ is open.
• Closed Set Definition: $F \subset \mathbb{R}^n$ is closed if $\mathbb{R}^n \setminus F$ is open.
• Theorem 2.1.5:
  • $\mathbb{R}^n$ and $\emptyset$ are closed sets.
  • The union of a finite number of closed sets in $\mathbb{R}^n$ is closed.
  • The intersection of an arbitrary collection of closed sets in $\mathbb{R}^n$ is closed.
• Limit Point Definition: $\mathbf{x} \in \mathbb{R}^n$ is a limit point of $E$ if for every $r > 0$, $B(\mathbf{x}; r) \cap (E \setminus {\mathbf{x}}) \neq \emptyset$.
• Closure Definition: The closure of $E$, $\overline{E}$, includes all limit points of $E$ and the points of $E$.
• Theorem 2.1.7:
  • $E$ is closed if and only if $E = \overline{E}$.
  • $\overline{E}$ is closed.
  • $\overline{E}$ is the smallest closed set containing $E$.
• Interior Definition: The interior of $E$, $E^\circ$, contains all $\mathbf{x} \in E$ for which there exists $r > 0$ such that $B(\mathbf{x}; r) \subset E$.
• Theorem 2.1.9:
  • $E^\circ$ is open.
  • $E^\circ$ is the largest open set contained in $E$.
• Boundary Definition: The boundary of $E$, $\partial E$, contains all $\mathbf{x} \in \mathbb{R}^n$ such that for every $r > 0$, $B(\mathbf{x}; r) \cap E \neq \emptyset$ and $B(\mathbf{x}; r) \cap (\mathbb{R}^n \setminus E) \neq \emptyset$.

2.2 Compact Sets

• Open Cover Definition: A collection of open sets ${G_\lambda}{\lambda \in \Lambda}$ that covers $E \subset \mathbb{R}^n$ is an open cover if $E \subset \bigcup{\lambda \in \Lambda} G_\lambda$.
• Compact Set Definition: $K \subset \mathbb{R}^n$ is compact if every open cover of $K$ has a finite subcover.
• Heine-Borel Theorem (2.2.3): $K \subset \mathbb{R}^n$ is compact if and only if it is closed and bounded.