Growth and Decay in Exponential Functions
Exponential functions are paramount in modelling scenarios where quantities grow or decay at a rate proportional to their current value. Let's dissect these concepts further. For a more detailed exploration of exponential functions, see exponential functions.
Exponential Growth
Exponential growth is a phenomenon where a quantity proliferates at a rate proportional to its current value, leading to an increasingly steep curve on a graph.
- Definition: Exponential growth occurs when a quantity increases by the same percentage over equal intervals of time.
- Mathematical Form: The general form of an exponential growth function is y = abx, where:
- a is the initial amount,
- b is the growth factor (where b > 1),
- x is the exponent, often representing time.
Example Question 1
Suppose a population of bacteria doubles every hour. If the initial population is 100, find the population after 5 hours.
Solution:
- Given: a = 100, b = 2 (since it doubles), and x = 5.
- Using the formula: y = abx, we find y = 100 * 25 = 3200.
Exponential Decay
Conversely, exponential decay describes scenarios where a quantity diminishes at a rate proportional to its current value, resulting in a gradually flattening curve on a graph. To understand how exponential functions relate to logarithmic trends, refer to logarithmic functions.
- Definition: Exponential decay transpires when a quantity decreases by the same percentage over equal intervals of time.
- Mathematical Form: The general form of an exponential decay function is y = abx, where:
- a is the initial amount,
- b is the decay factor (where 0 < b < 1),
- x is the exponent, often representing time.
Example Question 2
A radioactive substance decays to half its mass every 3 years. If the initial mass is 100g, find the mass after 9 years.
Solution:
- Given: a = 100, b = 0.5 (since it halves), and x = 9/3 = 3 (since it halves every 3 years).
- Using the formula: y = abx, we find y = 100 * 0.53 = 12.5.
IB Maths Tutor Tip: Understanding exponential functions is crucial for modelling real-life phenomena, from rapid population growth to financial investments, highlighting their versatility across biology, finance, and physics.
The Base e
The number e, approximately equal to 2.71828, is a mathematical constant that emerges naturally in various domains, such as calculus, complex analysis, and number theory, due to its unique properties.
Definition and Properties
- Definition: e is the limit of (1 + 1/n)n as n approaches infinity.
- Properties:
- The function f(x) = ex is its own derivative, i.e., f'(x) = ex.
- The area under the curve y = e-x from x = 0 to infinity is equal to 1.
Exponential Functions with Base e
Exponential functions with base e are especially prevalent in mathematical modelling due to their unique properties and are often represented in the form y = ae^(kx). This specific form is crucial in understanding differentiation and integration of exponential functions.
- Mathematical Form: y = ae(kx), where:
- a is the initial amount,
- k is the rate of growth (if k > 0) or decay (if k < 0),
- x is the variable, often representing time.
Example Question 3
If a bank account is continuously compounded at an annual interest rate of 5%, and the initial deposit is £1000, find the amount after 10 years. This example illustrates the principle of compound interest.
Solution:
- Given: a = 1000, k = 0.05, and x = 10.
- Using the formula: y = ae(kx), we find y = 1000 * e(0.05 * 10) = 1000 * e0.5 = £1648.72
Applications of Base e
- Continuous Compounding: A = Pe(rt), where P is the principal amount, r is the rate of interest, and t is the time.
- Natural Growth/Decay: N(t) = N0e(kt), where N0 is the initial quantity, k is the rate of growth/decay, and t is the time.
Applications in Various Domains
Exponential functions permeate through various fields, including biology, finance, and physics, providing a robust mathematical framework to model diverse phenomena, such as population growth, financial compounding, and radioactive decay, respectively.
Population Growth
In biology, exponential growth models can describe population growth under ideal conditions, where resources are unlimited. However, in real-world scenarios, other models like logistic growth, which considers carrying capacity, might be more applicable. These concepts are key in understanding complex real-world scenarios.
IB Tutor Advice: Practise applying the exponential function formulas to diverse scenarios, ensuring you're comfortable transitioning between growth and decay contexts for problem-solving in exams.
Financial Compounding
In finance, exponential functions model compound interest, where the amount of interest is calculated on the initial principal, which also includes all of the accumulated interest from previous periods on a deposit or loan. Continuous compounding, utilising the base e, is a specific case where the interest is compounded instantly.
Radioactive Decay
In physics, exponential decay models describe the process of radioactive decay, where the decay rate is proportional to the remaining quantity of the substance. This model is pivotal in various applications, including carbon dating and medical imaging.
FAQ
The exponent in an exponential function plays a crucial role in determining the nature and rate of growth or decay. Specifically, if the base of the exponential function is greater than 1, the function represents exponential growth, and the quantity will increase as the exponent (often representing time) increases. Conversely, if the base is between 0 and 1, the function represents exponential decay, and the quantity decreases as the exponent increases. The actual value of the exponent influences the steepness of the curve, with larger exponents resulting in steeper growth or decay in the graphical representation of the function.
The concept of 'half-life' in radioactive decay is intrinsically related to exponential decay functions. Half-life is defined as the time required for a quantity to reduce to half its initial value. In the context of exponential decay, if we consider a decay model of the form y = a * bx, where 0 < b < 1, the half-life can be determined by setting y = a/2 and solving for x. This provides a quantifiable measure of the rate of decay and is particularly useful in various scientific fields, such as physics, chemistry, and environmental science, to understand and predict the decay of radioactive substances over time.
Exponential functions are adept at modelling real-world phenomena like population growth and financial investments due to their inherent property of representing quantities that change at a rate proportional to their current value. For instance, in population growth, if a species doubles its population every certain time period, this growth can be modelled with an exponential function, where the rate of growth is a constant proportion. Similarly, in finance, when interest is compounded, the amount of interest earned grows exponentially because it is calculated as a percentage of the continually growing amount. Thus, exponential functions provide a mathematical framework to describe and predict such phenomena.
The number e, approximately equal to 2.71828, is of paramount importance in mathematics due to its unique properties and applications, especially in exponential functions. It is the only number for which the derivative of ex is ex itself, which makes it extremely useful in calculus, particularly in problems related to growth and decay. Moreover, e is the base for the natural logarithm and is prevalent in mathematical models describing real-world phenomena, such as population growth, radioactive decay, and compound interest with continuous compounding. Its properties make calculations and manipulations in various mathematical and scientific applications more straightforward and elegant.
While exponential functions are widely used to model growth and decay, they are not universally applicable to all kinds of real-world scenarios. Exponential models are ideal for situations where a quantity grows or decays at a rate proportional to its current value. However, in scenarios where growth rate changes due to external factors, like limited resources or carrying capacity (as seen in logistic growth models), or where growth is influenced by other variables, exponential functions may not be the most accurate or appropriate model to use.
Practice Questions
The population of the bacteria can be modelled by the exponential growth function, which is given by y = abx. Here, a is the initial population, b is the growth factor, and x is the time. Given that the initial population a is 200, the bacteria doubles every 3 hours, so b = 2, and we are finding the population after 12 hours, so x = 12/3 = 4 (since the population doubles every 3 hours). Substituting these values into the formula, we get y = 200 * 24 = 200 * 16 = 3200. Therefore, the population of the bacteria after 12 hours will be 3200.
To find the amount in the account after 6 years with continuous compounding, we can use the formula A = Pe(rt), where P is the principal amount, r is the rate of interest, t is the time, and e is approximately 2.71828. Given that P = £5000, r = 0.04 (converting the percentage to a decimal), and t = 6, substituting these values into the formula, we get A = 5000 * e(0.04 * 6) = 5000 * e0.24. Calculating the value, we get A = 5000 * 1.27125 = £6356.25. Therefore, the amount in the account after 6 years will be £6356.25.
