Describe linear and exponential growth and decay

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...

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...

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...