AHL 2.9—HL modelling functions Notes

Exponential Models and Half-Life

Exponential models are widely used in Math AI to describe phenomena that exhibit rapid growth or decay.

The general form of an exponential model is:

$$f(x) = a \cdot b^x$$

where $a$ is the initial value, $b$ is the base (usually $e$ for natural exponential functions), and $x$ is the independent variable.

For half-life calculations, we often use the decay form:

$$N(t) = N_0 \cdot e^{-\lambda t}$$

where:

• $N(t)$ is the quantity at time $t$
• $N_0$ is the initial quantity
• $\lambda$ is the decay constant.

Natural Logarithmic Models

Natural logarithmic models are used when the rate of change of a quantity is inversely proportional to its current value. These models take the form:

$$f(x) = a + b \ln x$$

where:

• $a$ and $b$ are constants
• $\ln$ is the natural logarithm.

Sinusoidal Models

Sinusoidal models are used to describe periodic phenomena, such as sound waves, alternating current, and seasonal temperature variations. The general form is:

$$f(x) = a \sin(b(x-c)) + d$$

where:

• $a$ is the amplitude (half the distance between the maximum and minimum values)
• $b$ affects the period (which is $2\pi/b$ in radians)
• $c$ is the phase shift (horizontal translation)
• $d$ is the vertical shift (midline of the oscillation)

Logistic Models

Logistic models are used to describe growth in situations where there's a limit or carrying capacity. The general form is:

$$f(x) = \frac{L}{1 + Ce^{-kx}}$$

where:

• $L$ is the carrying capacity (upper asymptote)
• $C$ is related to the initial value
• $k$ is the growth rate

Piecewise Models

Piecewise models combine different functions over different intervals. They're useful for describing complex behaviors that change abruptly at certain points. The general form is:

$$f(x) = \begin{cases} f_1(x) & \text{if } x < a \\ f_2(x) & \text{if } a \leq x < b \\ f_3(x) & \text{if } x \geq b \end{cases}$$

where:

• $f_1(x)$, $f_2(x)$, and $f_3(x)$ are different functions
• $a$ and $b$ are the boundary points.

Log-Log and Semi-Log Graphs for Model Identification

When working with different types of models, graphing data using log-log or semi-log scales can help identify the best-fitting function.

• Log-Log Graphs:
  • Used to identify power functions of the form$y = ax^b$.
  • A straight-line pattern in a log-log plot suggests a power-law relationship.
• Semi-Log Graphs:
  • Used for exponential models. If $y = a e^{bx}$, then plotting $\ln y$ vs. $x$ produces a linear graph, making it easier to determine the parameters.

By transforming non-linear data into a straight-line relationship, these graphs simplify model selection and parameter estimation.

Using a GDC for Model Fitting

• Graphing calculators (GDCs) are powerful tools for analyzing real-world data and fitting mathematical models.
• The regression functions allow students to determine the best-fitting exponential, logarithmic, or logistic models for given data.

To use a GDC for model fitting:

1. Enter the data into a statistics list.
2. Choose the regression type (e.g., exponential, logarithmic, logistic).
3. Analyze the equation and correlation coefficient ($R^2$).
4. Graph the model along with the data points to check for accuracy.

Misinterpreting Growth Rate in Logistic vs. Exponential Models

A frequent misconception occurs when interpreting the growth rate ($r$) in logistic and exponential models:

• Exponential growth assumes a constant rate of increase, where the function keeps growing indefinitely: $$N(t) = N_0 e^{rt}$$ Here, $r $ directly represents the per-unit time growth rate.
• Logistic growth accounts for a limiting factor (carrying capacity $L$), meaning growth slows down as $N$ approaches $L$: $$N(t) = \frac{L}{1 + Ce^{-rt}}$$ ​In logistic models, $r$ determines the rate of approach to the carrying capacity, not the initial growth speed alone.