The Exponential Function

• The exponential function, denoted as exp, is the solution to the differential equation $y' = y$ with the initial condition $\exp(0) = 1$

Definition

• $exp : \mathbb{R} \rightarrow \mathbb{R}$ maps x to $\exp(x)$ or $e^x$

Properties

• $\exp(0) = 1$
• $\exp(1) = e \approx 2.718$
• $\exp(a + b) = \exp(a) \times \exp(b)$
• $\exp(a - b) = \frac{\exp(a)}{\exp(b)}$
• $\exp(na) = (\exp(a))^n$
• $\exp(-a) = \frac{1}{\exp(a)}$

Derivative

• The exponential function is differentiable over $\mathbb{R}$ and has itself as its derivative.
• If $f(x) = e^x$, then $f'(x) = e^x$

Variations

• The exponential function is strictly increasing on $\mathbb{R}$.
• As x approaches $-\infty$, $e^x$ approaches 0. As x approaches $+\infty$, $e^x$ approaches $+\infty$

Limits

• $\lim_{x \to -\infty} e^x = 0$
• $\lim_{x \to +\infty} e^x = +\infty$
• $\lim_{x \to +\infty} \frac{e^x}{x} = +\infty$
• $\lim_{x \to 0} \frac{e^x - 1}{x} = 1$

Compared Growth

• The exponential function grows faster than any polynomial function.
• $\lim_{x \to +\infty} \frac{e^x}{x^n} = +\infty$

Applications

• Solving differential equations
• Modeling exponential phenomena, such as population growth and radioactive decay
• Calculating compound interest
• Statistics

Exercises

• Solve the equation $e^x = 5$.
• Calculate the derivative of the function $f(x) = e^{x^2}$.
• Study the variations of the function $f(x) = xe^x$.
• Calculate the limit of the function $f(x) = \frac{e^x}{x^2}$ as x approaches $+\infty$.
• Show that the function $f(x) = e^x$ is convex on $\mathbb{R}$.

Graphical Representation

• The exponential function $f(x) = e^x$ is shown on a Cartesian plane
• The graph starts close to the x-axis on the left, gradually increasing before steeply rising to the right, illustrating exponential growth