Exponential Functions Explained

Exponential functions are essential components in calculus, modelling natural growth and decay. This note delineates important concepts, properties, and applications of exponential functions and their inverses, with a specific emphasis on Euler's number, $e$.

Key Terms: Definitions

• Euler's Number $e$: Approximately 2.71828. It serves as the base for natural logarithms and is pivotal for continuous growth models in mathematics.
• Exponential Function: Represented as $f(x) = ke^x$, where $k$ is a constant.
• Natural Logarithms: These are the inverses of exponential functions, denoted as $\ln(x)$.

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Understanding Differentiation of $e^x$

• Differentiation Formula:
  • The Derivative of $e^x$ is $e^x$: $\frac{d}{dx} (e^x) = e^x$.
  • Uniqueness: This property indicates that the function grows at a rate proportional to its value, a unique aspect in calculus.

Derivation Steps:

• Begin with the limit definition: $\frac{d}{dx}(e^x) = \lim_{h \to 0} \frac{e^{x+h} - e^x}{h}$
• Factor out $e^x$: $= e^x \cdot \lim_{h \to 0} \frac{e^h - 1}{h}$
• Constant Explanation: The limit $\lim_{h \to 0} \frac{e^h - 1}{h} = 1$ is fundamental for validating properties of exponential growth.

Euler's Number, $e$

• Definition and Limit Definition:
  • Euler's Number (e): $e = \lim_{n \to \infty} \left( 1 + \frac{1}{n} \right)^n$

Historical Context:

• Discovery:
  • Initially identified by Jacob Bernoulli through his study of compound interest.
• Development:
  • Leonhard Euler expanded its application in calculus with rigorous proofs.

Significance in Exponential Functions

• Derivative Property:
  • Uniqueness:
    • The distinguishing feature of $e^x$ is that its derivative is $e^x$, highlighting a constant rate of steep ascension.
• Fundamental Role of Logarithms:
  • Being the base of natural logarithms makes it indispensable for resolving growth and decay equations.

Graphical Behaviour of $e^x$

Graph Characteristics:

• Exponential Growth:
  • The graph displays rapid ascent as $x$ increases.
• Constant Slope:
  • The graph maintains consistent growth, with tangent line slopes equaling the function values at any point.
• Asymptotic Behaviour:
  • Approaches the x-axis but never intersects it.

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Applications and Worked Examples

Worked Examples

• Example 1: Differentiate $e^x$.
  • Solution: $\frac{d}{dx}(e^x) = e^x$.
• Example 2: Differentiate $e^{ax+b}$.
  • Let's break this down step by step:
    1. Apply the chain rule: $\frac{d}{dx}(e^{ax+b}) = e^{ax+b} \cdot \frac{d}{dx}(ax+b)$
    2. Differentiate the inner function: $\frac{d}{dx}(ax+b) = a$
    3. Therefore: $\frac{d}{dx}(e^{ax+b}) = ae^{ax+b}$

Continuous Growth Models

• Real-life Applications:
  • Commonly used in calculating Compound Interest and modelling Population Growth.

Common Errors

• Mistakes with Exponents:
  • Avoid altering the base or mishandling exponents when performing differentiation.

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Inverse Relationship: Exponential and Logarithmic Functions

Definitions and Properties

• Inverse Functions:
  • The operations of $y = e^x$ and $y = \ln x$ effectively neutralise each other.
• Key Properties:
  • Domain for $e^x$: $x \in \mathbb{R}$
  • Domain for $\ln x$: $x > 0$

Graphical Representation

• Reflection Across $y = x$:
  • The inverse relationship is evident through their mirrored graphs.

Problem Solving

• Algebraic and Graphical Applications:
  • Confirm transformations by verifying reflections.

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Exam Tips

• Remember to thoroughly verify the domains of exponential and logarithmic functions to prevent errors.
• Employ graphing technology for visualising problems, ensuring accuracy in solutions.