7.8 Exponential Models with Differential Equations

Cards (54)

The general form of exponential models in differential equations is P(t).
Exponential models describe growth or decay proportional to the current amount of a quantity. 
True
In the exponential model P(t) = P₀e^(kt), the term k represents the growth or decay constant.
What is the value of k if a bacterial population doubles every hour?
ln(2)
Match the parameter with its description:
P(t) ↔️ Quantity at time t
P₀ ↔️ Initial quantity at t=0
k ↔️ Growth/decay constant
The standard form of exponential differential equations is dP/dt = kP.
In the standard form dP/dt = kP, k is the growth constant if k > 0. 
True
What is the differential equation for a population starting with 500 individuals and a growth rate of 0.05?
dP/dt = 0.05P
Exponential models in differential equations describe the growth or decay of a quantity proportional to its current amount.
To find the growth constant k for a doubling time, use the formula k = ln(2). 
True
After integrating both sides of dP/P = k dt, we obtain **ln|P| = kt + C**. 
True
Solve the differential equation dP/dt = 0.05P with P(0) = 500.
P(t) = 500e^(0.05t)
Match the variable with its description in exponential differential equations:
P(t) ↔️ Quantity at time t
P₀ ↔️ Initial quantity at t=0
k ↔️ Growth/decay constant
t ↔️ Time elapsed
What is the general form of an exponential model in differential equations?
P(t) = P₀e^(kt)
Suppose a bacterial population starts with 500 cells and doubles every hour. The model is P(t) = 500e^(ln(2)t)
The exponential growth model is given by P(t) = 500e^(kt)
The growth constant k in exponential models is always positive.
False
Exponential models in differential equations describe growth or decay proportional to the current amount
What does P(t) represent in an exponential model?
Quantity at time t
Match the parameter with its description:
P(t) ↔️ Quantity at time t
P₀ ↔️ Initial quantity at t=0
k ↔️ Growth/decay constant
The standard form of an exponential differential equation is dP/dt = kP
The first step in solving exponential differential equations using separation of variables is to separate the variables
Steps to solve exponential differential equations using separation of variables:
1️⃣ Separate variables
2️⃣ Integrate both sides
3️⃣ Solve for P(t)
When integrating dP/P, the result is ln|P|.
True
After solving for P(t), the general form is P(t) = A * e^(kt), where A equals e^C
What is the purpose of interpreting solutions to exponential differential equations in real-world applications?
Understanding the variables
Match the variable with its description:
P(t) ↔️ Quantity at time t
P₀ ↔️ Initial quantity at t=0
k ↔️ Growth/decay constant
t ↔️ Time elapsed
The half-life of a substance modeled by P(t) = 1000e^(-0.02t) is approximately 34.66 years
What type of problems are exponential models most commonly used for?
Growth and decay
Match the exponential model with its description:
Growth ↔️ P(t) = P₀e^(kt)
Decay ↔️ P(t) = P₀e^(-kt)
If a population starts at 5000 and grows at 4% per year, what is the exponential growth model?
P(t) = 5000e^(0.04t)
The decay constant in the decay model is positive.
False
What is the formula for the exponential growth model?
$P(t) =$ $P₀e^(kt)$
In Example 1, the population starts at 5000 and grows at 4% per year.
What is the initial quantity (P₀) in Example 2?
100 grams
Radioactive decay is an example of a process that can be modeled using the decay model.
True
Match the feature with the correct growth model:
Growth rate decreases as population approaches carrying capacity ↔️ Logistic Growth
Carrying capacity is unlimited ↔️ Exponential Growth
Steps to solve the logistic differential equation:
1️⃣ Separate variables
2️⃣ Integrate both sides
3️⃣ Solve for P(t)
What is the general formula for the exponential growth model?
$P(t) =$ $P₀e^(kt)$
In the exponential decay model, the rate constant k is positive.
False