Understanding exponential growth and decay models:

• What is exponential growth characterized by?

  Proportional increase to current value
• What is exponential decay characterized by?

  Proportional decrease to current value
• What is the condition for exponential growth in the general formula?

  k > 0
• What happens to the quantity P over time in exponential decay?

  Decreases exponentially
• What is the behavior of exponential decay over time?

  Decreases at a decelerating rate
• Match the decay type with its general form and behavior over time:

  Exponential ↔️ $\frac{dP}{dt} =$$- kP$, Decreases at a decelerating rate

  Linear ↔️ $\frac{dP}{dt} =$$- k$, Decreases at a constant rate
• The rate constant k is negative for exponential decay.


  True
• In exponential decay, the decay rate constant k is always positive.

  False
• What does the variable 'k' represent in exponential decay?

  Decay rate constant
• Exponential decay slows down as the quantity decreases.


  True
• The rate constant in the exponential formula determines how fast a quantity changes over time.
• The growth/decay rate 'k' is found by taking the natural logarithm of the change ratio and dividing by the time interval.


  True
• What type of growth is characterized by an accelerating increase over time?

  Exponential growth
• What is the decay rate constant 'k' called in exponential decay?

  Decay rate constant
• What does the decay rate constant 'k' determine in exponential decay?

  Rate of decrease
• What is the general form of the exponential decay equation?

  $\frac{dP}{dt} =$$- kP$
• In exponential decay, the rate of decrease slows down as the quantity decreases.


  True
• What is the general formula for exponential growth and decay?

  $\frac{dP}{dt} =$$kP$
• Match the characteristic with the type of exponential model:

  Growth rate increases exponentially ↔️ Exponential growth

  Rate constant is negative ↔️ Exponential decay
• What are the initial and final quantities denoted as in the calculation of 'k'?

  $P_{0}$ and $P_{t}$
• The rate constant 'k' is negative in exponential growth models.

  False
• The growth rate constant in exponential growth determines the rate of growth.

  True
• Match the growth type with its general form and behavior over time:

  Exponential ↔️ $\frac{dP}{dt} =$$kP$, Increases at an accelerating rate

  Linear ↔️ $\frac{dP}{dt} =$$k$, Increases at a constant rate
• The decay rate constant in exponential decay determines the rate of decrease.


  True
• Match the growth type with its general form and behavior over time:

  Exponential ↔️ $\frac{dP}{dt} =$$kP$, Increases at an accelerating rate

  Linear ↔️ $\frac{dP}{dt} =$$k$, Increases at a constant rate
• In exponential decay, the rate of decrease decelerates over time.
• What is the general formula for exponential growth and decay?

  $\frac{dP}{dt} =$$kP$
• What does a positive value of 'k' indicate in exponential models?

  Exponential growth
• If a population starts at 100 and grows to 150 in 5 years, what is the growth rate 'k'?

  $k \approx 0.081$
• Exponential decay is a model where a quantity decreases at a rate proportional to its current value.
• In linear decay, the quantity decreases by a fixed amount per unit time.


  True
• Exponential decay is a model where a quantity decreases at a rate proportional to its current value
• In linear decay, the quantity decreases at a constant rate
• Exponential decay occurs when the rate constant 'k' is positive.

  False
• A negative value of the growth/decay rate $k$ indicates exponential decay
• Order the steps involved in calculating the growth/decay rate 'k'.

  1️⃣ Identify the initial and final quantities

  2️⃣ Calculate the change ratio

  3️⃣ Determine the time interval

  4️⃣ Compute the rate 'k' using the formula
• In linear decay, the quantity decreases by a fixed amount per unit time.


  True
• Steps to calculate the growth/decay rate 'k':

  1️⃣ Identify the initial and final quantities (`P_0` and `P_t`).

  2️⃣ Calculate the change ratio: $\frac{P_{t}}{P_{0}}$.

  3️⃣ Determine the time interval (`t`).

  4️⃣ Compute 'k' using the formula: $k =$$\frac{\ln(\frac{P_{t}}{P_{0}})}{t}$.
• What does a negative value of 'k' indicate in exponential models?

  Exponential decay
• Exponential growth is a model where a quantity increases at a rate proportional to its current value.