Exponential Models with Differential Equations, Population Growth, Examples - Calculus

Exponential models describe quantities that change at a rate proportional to their current value, which is captured by the differential equation dy/dt = ky. The solution to this equation is y(t) = y(0)e^(kt), where 'y(t)' is the quantity at time 't', 'y(0)' is the initial quantity, 'k' is the constant of proportionality, and 'e' is the base of the natural logarithm. If 'k' is positive, the model represents exponential growth; if 'k' is negative, it represents exponential decay.

💡How it works
• The Differential Equation: The core idea is that the rate of change of a quantity (dy/dt) is directly proportional to the quantity itself (y), with 'k' as the constant of proportionality.
• The Solution (Exponential Model): By solving this separable differential equation, we arrive at the formula for exponential growth or decay:
⚬ y(t) = y(0)e^(kt)
⚬ y(t): The amount of the quantity at time 't'.
⚬ y(0): The initial amount of the quantity at time 't' = 0.
⚬ k: The growth or decay constant.
⚬ e: The base of the natural logarithm, approximately 2.71828.

💡Applications
This relationship is widely applied in various fields:
• Population Growth: The rate at which a population grows is often proportional to its current size.
• Radioactive Decay: The rate at which a radioactive substance decays is proportional to the amount of the substance remaining.
• Finance: Continuously compounded interest follows an exponential model, where the growth of money is proportional to the amount present.
• Medicine: The elimination of a drug from the bloodstream can often be modeled using exponential decay.

💡Interpreting the constant 'k'
• Positive 'k': Indicates exponential growth, where the quantity increases at an accelerating rate.
• Negative 'k': Indicates exponential decay, where the quantity decreases at a rate proportional to its current value.

💡Worksheets are provided in PDF format to further improve your understanding:
• Questions Worksheet: https://drive.google.com/file/d/1DMK4EA0f8SfF4SgdiOZ39F73He_YWIwe/view?usp=drive_link
• Answers: https://drive.google.com/file/d/1QpzjiCPjfoxydzZqvjyJJsqv9jyRLZ0o/view?usp=drive_link

💡Chapters:
00:00 Exponential models, growth and decay
01:40 Worked examples

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