Exponential Function

The exponential function is an important type of function in calculus with distinctive properties. Understanding the continuity properties of exponential functions is essential for studying more complicated types of function.

Basic Properties

The exponential function has the following basic properties:

• Monotonicity: the exponential function is a monotone function
• Continuity: continuous on its domain
• Differentiability: differentiable on its domain
• Inverse relation: the exponential function and the logarithmic function are inverses of each other

The Natural Exponential Function

Properties:

• Domain: $\mathbb{R}$
• Continuous on $\mathbb{R}$
• The graph is a monotonically increasing exponential curve
• Range: $(0, +\infty)$
• Derivative: $f'(x) = e^x$

The General Exponential Function

Properties:

• Domain: $\mathbb{R}$
• Continuous on $\mathbb{R}$
• Monotonically increasing when $a > 1$, monotonically decreasing when $0 < a < 1$
• Range: $(0, +\infty)$
• Derivative: $f'(x) = a^x \ln a$

Graph Characteristics of Exponential Functions

1. Monotonicity: the exponential function is monotone on its domain
2. Asymptotes: there is a horizontal asymptote $y = 0$
3. Smoothness: the graph is a smooth curve, with no corners or breaks
4. Growth: when $a > 1$, the function values grow rapidly as $x$ increases

Continuity Analysis of Exponential Functions

Determining Continuity

The exponential function is continuous everywhere on $\mathbb{R}$.

Proof of Continuity

The continuity of the exponential function rests on the following facts:

1. Power series expansion: the exponential function can be written as a power series $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$

2. Continuity of power series: a power series is continuous within its interval of convergence

3. Interval of convergence: the power series of the exponential function converges on $\mathbb{R}$

Continuity of Composite Functions

Basic Theorem

If the function $f(x)$ is continuous at $x_0$, then $e^{f(x)}$ is continuous at $x_0$.

Examples

Example 1: $f(x) = e^{x^2 + 1}$

Analysis:

• The inner function $h(x) = x^2 + 1$ is continuous on $\mathbb{R}$
• The outer function $g(x) = e^x$ is continuous on $\mathbb{R}$
• Therefore $f(x) = g(h(x))$ is continuous on $\mathbb{R}$

Example 2: $f(x) = e^{\sin x}$

Analysis:

• The inner function $h(x) = \sin x$ is continuous on $\mathbb{R}$
• The outer function $g(x) = e^x$ is continuous on $\mathbb{R}$
• Therefore $f(x) = g(h(x))$ is continuous on $\mathbb{R}$

Applications of Exponential Functions

1. Scientific Modelling

The exponential function is widely used in science:

• Population growth: describes the population growth model $P(t) = P_0 e^{rt}$
• Radioactive decay: describes the decay of radioactive substances $N(t) = N_0 e^{-\lambda t}$
• Chemical reactions: describe the rate of chemical reactions

2. Economic Applications

The exponential function has important applications in economics:

• Compound interest: describes the growth of capital $A = P e^{rt}$
• Inflation: describes changes in prices
• Economic growth: describes economic development

3. Engineering Applications

The exponential function has important applications in engineering:

• Signal processing: describes signal attenuation
• Circuit analysis: describes the charging and discharging of capacitors $V(t) = V_0 e^{-t/RC}$
• Heat conduction: describes temperature changes

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Exercises

Exercise 1

Determine the continuity of $f(x) = e^{x^2}$ at $x = 0$.

Exercise 2

Determine the continuity of $f(x) = e^{\sin x}$ on $\mathbb{R}$.

Exercise 3

Let $f(x) = \begin{cases} e^x, & x \leq 0 \\ x + 1, & x > 0 \end{cases}$. Determine the continuity of $f(x)$ at $x = 0$.

Exercise 4

Determine the continuity of $f(x) = 2^x$ on $\mathbb{R}$.

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Summary

Symbols Used in This Article