Exponential Models and Half-Life
Exponential models are widely used in Math AI to describe phenomena that exhibit rapid growth or decay.
ExampleOne of the most common applications is in calculating half-life, which is particularly useful in fields like nuclear physics and pharmacology.
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.
In a radioactive decay problem, if we have 100 grams of a substance with a half-life of 5 hours, we can model this as:
$N(t) = 100 \cdot e^{-(\ln 2 / 5)t}$
To find the amount left after 12 hours:
$N(12) = 100 \cdot e^{-(\ln 2 / 5) \cdot 12} \approx 29.3$ grams
TipRemember that the half-life $T_{1/2}$ is related to the decay constant $\lambda$ by the equation:
$T_{1/2} = \frac{\ln 2}{\lambda}$
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.
These models are particularly useful in scenarios involving:
- Sound intensity and perceived loudness
- pH scale in chemistry
- Earthquake magnitude on the Richter scale
The perceived loudness $L$ of a sound in relation to its intensity $I$ can be modeled as:
$L = 10 \log_{10}(\frac{I}{I_0})$
where $I_0$ is the reference intensity. This can be rewritten in the natural logarithmic form:
$L = 10 \cdot \frac{\ln(I/I_0)}{\ln(10)}$
NoteNatural logarithmic models often appear in situations where there's a diminishing return effect, such as the relationship between additional study time and improvement in test scores.
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)
In the IB curriculum, radian measure is assumed for sinusoidal models unless otherwise stated.
ExampleTo model daily temperature fluctuations in a city where the temperature varies between 15°C and 25°C, with the highest temperature at 3 PM (15:00), we could use:
$T(t) = 5 \sin(\frac{\pi}{12}(t-15)) + 20$
Here, $t$ is the time in hours (0-24), 5 is the amplitude, $\pi/12$ gives a period of 24 hours, -15 shifts the peak to 3 PM, and 20 is the average temperature.
Common MistakeStudents often confuse the period and frequency. Remember, frequency is the reciprocal of the period. In radians, period = $2\pi/b$, while frequency = $b/2\pi$.
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:
