What is the general solution to the differential equation $\frac{dy}{dt} = ky$?
$y = y_0 e^{kt}$
Formula to find $k$ given $y(t)$, $y_0$, and $t$?
$k = \frac{1}{t} \ln(\frac{y(t)}{y_0})$
What is the integral of $\frac{1}{y} dy$?
$\ln|y| + C$
How to calculate the population at time t?
$y = y_0 cdot e^{kt}$
How do you find the constant $k$ in the exponential growth model?
Using the formula $k = \frac{\ln(\frac{y}{y_0})}{t}$, where $y$ is the population at time $t$, and $y_0$ is the initial population.
What is the formula for the amount of a drug remaining in the bloodstream after time $t$?
$A(t) = A_0 e^{kt}$, where $A_0$ is the initial amount and $k$ is the decay constant.
Formula for the rate of change?
$\frac{dy}{dt}$
What is the formula for the exponential model when the rate is proportional to the current size?
$\frac{dy}{dt} = ky$
What is the formula to solve for $y$ when given $ln|y| = kt + C$?
$y = e^{kt+C} = e^C e^{kt}$
What is the formula to determine the constant C?
$e^C = |y_0|$
How to solve for $k$ when given the initial value and a value at time $t$?
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 $k$.
Steps to model population growth given initial population and growth rate?
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.
Steps to predict future population using exponential model?
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.
How do you solve an exponential decay problem?
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.
Steps to solve for time $t$ when given the initial and final values?
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 $t = \frac{\ln(\frac{y}{y_0})}{k}$.
How to find the decay constant $k$ in a radioactive decay problem?
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 $k$.
How to solve for the initial population when given the population at time $t$?
1. Substitute the given values into $y = y_0 e^{kt}$. 2. Solve for $y_0$ using $y_0 = \frac{y}{e^{kt}}$.
Steps to find the time it takes for a population to double?
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 $t = \frac{\ln(2)}{k}$.
How to determine the equation for the amount of drug remaining?
1. Identify the initial amount $A_0$. 2. Use the given information to find the constant $k$. 3. Write the equation $A(t) = A_0 e^{kt}$.
How to solve a problem where bacteria doubles in five hours?
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 $y = 4y_0$.
Define a differential equation.
An equation that relates a function with its derivatives.
What does $\frac{dy}{dt}$ represent in exponential models?
The rate of change of a quantity $y$ with respect to time $t$.
What does the constant $k$ represent in the differential equation $\frac{dy}{dt} = ky$?
The rate constant, indicating the rate of growth (if $k > 0$) or decay (if $k < 0$).
Define $y_0$ in the context of exponential models.
The initial value of the quantity $y$ at time $t = 0$.
What is an exponential model?
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.
What is the meaning of 'e' in exponential models?
Euler's number, the base of the natural logarithm, approximately equal to 2.71828; important in continuous growth/decay situations.
Define 'rate of change' in calculus.
The measure of how a quantity is changing with respect to another quantity, typically time; represented by a derivative.
What is meant by 'proportional to' in the context of exponential growth?
The rate of growth is a constant multiple of the current value.
Define 'initial condition' in a differential equation.
The value of the function at a specific point, often at $t=0$, used to find a particular solution.
What is meant by 'separating variables' in solving differential equations?
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.
