Exponential Growth & Decay

Exponential Growth is defined by the function $y = Ae^{kt}$, where $k > 0$. In this model, the quantity increases at an accelerating rate over time, which is common in population biology and compound interest scenarios.

Exponential Decay follows the form $y = Ae^{-kt}$ (or $y = Ae^{kt}$ where $k < 0$). This describes quantities that decrease rapidly at first and then more slowly as they approach a horizontal asymptote at $y = 0$, typical of radioactive isotopes or drug concentrations in the blood.

The constant $A$ represents the initial value or starting quantity. This is found by evaluating the function at $t = 0$, where $e^0 = 1$, leaving only the coefficient $A$.

The choice of the base $e$ (Euler's number) is foundational because the derivative of $e^x$ is simply $e^x$, making the rate of change of the function directly proportional to the function itself. This mirrors physical laws where the growth rate of a population or the decay rate of an atom depends on the current number present.

Every general exponential function $y = a^x$ can be converted into the natural form $y = e^{kx}$ by identifying that $a = e^{\ln a}$. This allows for a unified mathematical approach using $k = \ln a$ as the continuous growth or decay constant.

The inverse relationship between $e^x$ and $\ln x$ is critical for solving modelling problems. To isolate a time variable $t$ located in an exponent, one must take the natural logarithm of both sides, effectively 'undoing' the exponential operation.

Solving for Constants

• Step 1: Use the initial condition ($t=0$, $y=y_0$) to identify the constant $A$, as $y_0 = Ae^0 = A$.
• Step 2: Substitute a known coordinate $(t_1, y_1)$ into the equation to solve for the rate constant $k$. This usually involves the rearrangement $k = \frac{1}{t_1} \ln(\frac{y_1}{A})$.

Linearization for Data Analysis

• To verify if data follows an exponential model, transform $y = Ae^{kt}$ into the linear form $\ln y = kt + \ln A$. By plotting $\ln y$ on the vertical axis against $t$ on the horizontal axis, an exponential relationship will appear as a straight line with gradient $k$ and intercept $\ln A$.

Finding Rates of Change

• The rate of change at any specific moment is found by differentiating the model. For growth, $\frac{dy}{dt} = Ak e^{kt}$, and for decay, $\frac{dy}{dt} = -Ak e^{-kt}$. This result is useful for determining how fast a population is expanding or a substance is vanishing at a precise time.

• Check the Units: Always verify the units of $t$ (seconds, years, etc.) and $y$ (grams, thousands of people). Exams often require converting units before substituting values into the exponential model.

• Sanity Check the Sign: Before calculating, decide if the scenario implies growth or decay. If you calculate a negative $k$ for a population increase, you likely made an algebraic error with signs or logarithms.

• Intercept Interpretation: When presented with a linear log-graph, remember that the intercept is $\ln A$, not $A$. You must calculate $A = e^{\text{intercept}}$ to find the true initial value.

• Asymptotic Awareness: If a question asks for the long-term value of a decay model, the answer is usually $0$ or the vertical shift constant if the model is $y = Ae^{-kt} + C$.

• Log Law Errors: A frequent mistake in linearization is writing $\ln(Ae^{kt}) = \ln A \cdot kt$ or forgetting to take the log of $A$ entirely. Always use the addition rule: $\ln(AB) = \ln A + \ln B$.

• The Initial Value Trap: In modified models like $y = 100 - 50e^{-kt}$, the initial value is NOT the coefficient of $e$. You must substitute $t=0$ to find $y(0) = 100 - 50 = 50$.

• Half-life confusion: When using half-life to find $k$, remember the relationship $k = \frac{\ln 2}{T_{1/2}}$. A common error is using $0.5$ instead of $\ln 0.5$ or forgetting the reciprocal relationship.