Mathematics
Grade 12
15 min
Describe linear and exponential growth and decay
Describe linear and exponential growth and decay
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1
Introduction & Learning Objectives
Learning Objectives
Differentiate between linear and exponential models by analyzing their respective rates of change.
Formulate linear functions (f(x) = mx + c) and exponential functions (f(t) = A₀e^(kt)) from word problems or data sets.
Interpret the parameters (slope, initial value, growth/decay factor, and continuous rate) within the context of a given model.
Calculate and compare the constant rate of change of a linear function with the variable instantaneous rate of change of an exponential function using derivatives.
Analyze the long-term behavior of linear and exponential functions by evaluating their limits as the independent variable approaches infinity.
Solve multi-step real-world problems involving concepts like compound interest, population dynamics, and radioacti...
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Key Concepts & Vocabulary
TermDefinitionExample
Linear Growth/DecayA quantity experiences linear growth or decay when it changes by a constant *amount* over equal time intervals. Its rate of change is constant.A bank account that has $50 deposited into it every month. The balance increases by a constant $50 each month, regardless of the current balance.
Exponential Growth/DecayA quantity experiences exponential growth or decay when it changes by a constant *percentage* or *factor* over equal time intervals. Its rate of change is proportional to its current value.A bacterial colony that increases by 20% every hour. The number of new bacteria added each hour depends on how many bacteria were present at the start of that hour.
Constant Rate of ChangeThe defining characteristic of a linear function, represented by the...
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Core Formulas
Linear Function Model
f(x) = mx + c
Used to model quantities with a constant rate of change. 'm' is the slope (the constant rate of change), and 'c' is the y-intercept (the initial value at x=0).
Exponential Function Model (Continuous)
A(t) = A_0 e^{kt}
Used to model quantities with a rate of change proportional to the current amount. 'A₀' is the initial amount at t=0, 'k' is the continuous growth (k>0) or decay (k<0) rate, and 't' is time.
Derivatives as Rates of Change
Linear: f'(x) = m \\ Exponential: \frac{d}{dt}(A_0 e^{kt}) = k \cdot (A_0 e^{kt}) = k \cdot A(t)
The derivative gives the instantaneous rate of change. For a linear function, this rate is constant. For an exponential function, this rate is...
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Challenging
City A has a population of 100,000 and grows by a constant 2,500 people per year. City B has a population of 120,000 and grows at a continuous rate of 2% per year. At approximately what time 't' are their instantaneous population growth rates equal?
A.t ≈ 1.5 years
B.t ≈ 2.0 years
C.t ≈ 3.1 years
D.Their rates are never equal.
Challenging
A radioactive sample has a mass of 10 grams after 20 years and a mass of 5 grams after 40 years. Assuming exponential decay A(t) = A₀e^(kt), what was the initial mass A₀ of the sample?
A.15 grams
B.25 grams
C.20 grams
D.50 grams
Challenging
A population P(t) is governed by the differential equation dP/dt = -0.03P. If the initial population P(0) is 8000, which statement accurately describes the population's behavior?
A.The population decays exponentially, approaching a limit of 0 as t → ∞.
B.The population decays linearly until it reaches 0.
C.The population grows exponentially, approaching ∞ as t → ∞.
D.The population remains constant at 8000.
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