The Mathematical Constant e

Euler's Number ($e$): An irrational number, meaning its decimal expansion is infinite and non-repeating. It is roughly equal to $2.718$, similar to how $\pi$ is roughly $3.142$.

The Natural Exponential Function: Defined as $y = e^x$. Like all exponential functions with a base $a > 0$, it always passes through the point $(0, 1)$ because $e^0 = 1$.

Asymptotic Behavior: The graph of $y = e^x$ has the x-axis ($y = 0$) as a horizontal asymptote. As $x$ becomes increasingly negative, the value of $e^x$ approaches zero but never reaches it.

Self-Derivation: The most significant property of $e^x$ is that it is its own derivative. Mathematically, if $y = e^x$, then $\frac{dy}{dx} = e^x$.

Geometric Interpretation: At any point on the curve $y = e^x$, the gradient (slope) of the tangent is exactly equal to the y-coordinate of that point. For example, at $x = 0$, the y-value is $1$ and the gradient is also $1$.

Rate of Change: This property makes $e$ the natural choice for modeling processes where the rate of growth is proportional to the current amount, such as population growth or radioactive decay.

Linear Power Rule: When the exponent is a linear function of $x$ (e.g., $kx$), the derivative involves multiplying by the coefficient of $x$. For $y = e^{kx}$, the derivative is $\frac{dy}{dx} = ke^{kx}$.

Negative Exponents: For the decay function $y = e^{-kx}$, the derivative is $\frac{dy}{dx} = -ke^{-kx}$. This indicates a negative gradient, consistent with a decreasing function.

Constant Multipliers: If the function is scaled by a constant $A$, such as $y = Ae^{kx}$, the derivative remains proportional: $\frac{dy}{dx} = Ake^{kx}$.

Growth Model: The form $y = Ae^{kx}$ where $k > 0$ represents exponential growth. The value of $A$ represents the 'initial value' (the y-intercept) when $x = 0$.

Decay Model: The form $y = Ae^{-kx}$ where $k > 0$ represents exponential decay. The graph is a reflection of the growth model in the y-axis, showing values that decrease toward the x-axis asymptote.

The Role of k: The constant $k$ determines the 'steepness' of the curve. A larger $|k|$ results in a faster rate of growth or decay.

Check the Coefficient: When differentiating $e^{kx}$, always remember to bring the $k$ down as a multiplier. A common mistake is to subtract 1 from the exponent as if it were a polynomial (e.g., $x^n$).

Identify the Intercept: Unless there is a vertical shift, $y = Ae^{kx}$ will always cross the y-axis at $(0, A)$. Use this as a quick check when sketching graphs.

Exact Form Requirements: If a question asks for an 'exact value', do not use your calculator to find a decimal. Keep the $e$ in your final expression.

Gradient vs. Value: Remember that for $y = e^x$, the gradient and the value are identical. If a question asks for the gradient at $x = 2$, the answer is simply $e^2$.