1. Substitute the given values into $y = y_0 e^{kt}$. 2. Divide both sides by $y_0$. 3. Take the natural logarithm of both sides. 4. Solve for $k$.

1. Write the equation $y = y_0 e^{kt}$. 2. Determine $y_0$ (initial population). 3. Find $k$ (growth rate). 4. Substitute $y_0$ and $k$ into the equation.

1. Determine the exponential model $y = y_0 e^{kt}$. 2. Substitute the known values of $y_0$, $k$, and $t$ (future time). 3. Calculate $y$ to find the future population.

1. Identify the initial amount $y_0$. 2. Use the given information to find the decay constant $k$. 3. Substitute $y_0$ and $k$ into the equation $y = y_0 e^{kt}$. 4. Solve for the desired variable.

1. Substitute the given values into $y = y_0 e^{kt}$. 2. Divide both sides by $y_0$. 3. Take the natural logarithm of both sides. 4. Solve for $t$ using $t = \frac{\ln(\frac{y}{y_0})}{k}$.

1. Use the half-life formula $k = \frac{\ln(2)}{T}$, where $T$ is the half-life. 2. Substitute the given half-life value into the formula. 3. Calculate $k$.

1. Substitute the given values into $y = y_0 e^{kt}$. 2. Solve for $y_0$ using $y_0 = \frac{y}{e^{kt}}$.

1. Set $y = 2y_0$ in the equation $y = y_0 e^{kt}$. 2. Simplify to $2 = e^{kt}$. 3. Take the natural logarithm of both sides. 4. Solve for $t$ using $t = \frac{\ln(2)}{k}$.

1. Identify the initial amount $A_0$. 2. Use the given information to find the constant $k$. 3. Write the equation $A(t) = A_0 e^{kt}$.

1. Assign an initial value to the bacteria, $y_0$. 2. Note that after 5 hours, $y = 2y_0$. 3. Use the exponential growth equation to find $k$. 4. Use the exponential growth equation to find $t$ when $y = 4y_0$.

If $k > 0$, the quantity $y$ is growing exponentially. If $k < 0$, the quantity $y$ is decaying exponentially.

It provides a specific value of the function at a particular point, allowing us to determine the constant of integration and find a unique solution.

Exponential growth occurs when the rate of increase of a quantity is proportional to the quantity itself. This leads to rapid increases over time.

Separate the variables, integrate both sides with respect to their respective variables, and solve for the function.

It represents the family of solutions that satisfy the differential equation; it is determined by initial conditions.

They are used to model phenomena such as population growth, radioactive decay, compound interest, and the spread of diseases.

Exponential decay occurs when the rate of decrease of a quantity is proportional to the quantity itself. This leads to rapid decreases over time.

To isolate each variable on one side of the equation, allowing for direct integration and solving.

It represents the function that satisfies the relationship between the function and its derivatives.

The natural logarithm is the inverse function of the exponential function with base $e$, allowing us to isolate variables in exponential expressions.

An equation that relates a function with its derivatives.

The rate of change of a quantity $y$ with respect to time $t$.

The rate constant, indicating the rate of growth (if $k > 0$) or decay (if $k < 0$).

The initial value of the quantity $y$ at time $t = 0$.

A mathematical model that describes the growth or decay of a quantity over time, where the rate of change is proportional to the current amount.

Euler's number, the base of the natural logarithm, approximately equal to 2.71828; important in continuous growth/decay situations.

The measure of how a quantity is changing with respect to another quantity, typically time; represented by a derivative.

The rate of growth is a constant multiple of the current value.

The value of the function at a specific point, often at $t=0$, used to find a particular solution.

A technique where terms involving one variable are isolated on one side of the equation, and terms involving the other variable are on the other side, allowing for integration.