Exponential Models with Differential Equations - AP Calculus BC
Card 1 of 30
What is the exponential decay model?
What is the exponential decay model?
Tap to reveal answer
$y = y_0 e^{-kt}$. Initial value $y_0$ times exponential with negative rate $k$.
$y = y_0 e^{-kt}$. Initial value $y_0$ times exponential with negative rate $k$.
← Didn't Know|Knew It →
Identify the integrating factor for the differential equation $\frac{dy}{dt} + Py = Q$.
Identify the integrating factor for the differential equation $\frac{dy}{dt} + Py = Q$.
Tap to reveal answer
$e^{\textstyle \int P , dt}$. Multiplying factor to convert equation to exact form.
$e^{\textstyle \int P , dt}$. Multiplying factor to convert equation to exact form.
← Didn't Know|Knew It →
What is the solution to the differential equation $\frac{dy}{dt} = ky$?
What is the solution to the differential equation $\frac{dy}{dt} = ky$?
Tap to reveal answer
$y = Ce^{kt}$. Exponential function with arbitrary constant $C$ and growth rate $k$.
$y = Ce^{kt}$. Exponential function with arbitrary constant $C$ and growth rate $k$.
← Didn't Know|Knew It →
What is the expression for $y(t)$ if $\frac{dy}{dt} = ky$ and $y(0) = y_0$?
What is the expression for $y(t)$ if $\frac{dy}{dt} = ky$ and $y(0) = y_0$?
Tap to reveal answer
$y(t) = y_0 e^{kt}$. Exponential growth formula with given initial condition.
$y(t) = y_0 e^{kt}$. Exponential growth formula with given initial condition.
← Didn't Know|Knew It →
Solve $\frac{dy}{dt} = -4y$ given $y(0) = 9$.
Solve $\frac{dy}{dt} = -4y$ given $y(0) = 9$.
Tap to reveal answer
$y(t) = 9e^{-4t}$. Exponential decay with rate $k = 4$ and initial value 9.
$y(t) = 9e^{-4t}$. Exponential decay with rate $k = 4$ and initial value 9.
← Didn't Know|Knew It →
Which function represents exponential decay: $y = e^{kt}$ or $y = e^{-kt}$?
Which function represents exponential decay: $y = e^{kt}$ or $y = e^{-kt}$?
Tap to reveal answer
$y = e^{-kt}$. Negative exponent causes exponential decrease over time.
$y = e^{-kt}$. Negative exponent causes exponential decrease over time.
← Didn't Know|Knew It →
What is the equilibrium solution for $\frac{dy}{dt} = ky(1 - \frac{y}{L})$?
What is the equilibrium solution for $\frac{dy}{dt} = ky(1 - \frac{y}{L})$?
Tap to reveal answer
$y = 0$ and $y = L$. Points where $\frac{dy}{dt} = 0$, so population remains constant.
$y = 0$ and $y = L$. Points where $\frac{dy}{dt} = 0$, so population remains constant.
← Didn't Know|Knew It →
Solve $\frac{dy}{dt} = 3y$ if $y(0) = 6$. What is $y(t)$?
Solve $\frac{dy}{dt} = 3y$ if $y(0) = 6$. What is $y(t)$?
Tap to reveal answer
$y(t) = 6e^{3t}$. Growth rate 3 with initial condition $y(0) = 6$.
$y(t) = 6e^{3t}$. Growth rate 3 with initial condition $y(0) = 6$.
← Didn't Know|Knew It →
What is the solution to $\frac{dy}{dt} = 0.5y$ with $y(0) = 15$?
What is the solution to $\frac{dy}{dt} = 0.5y$ with $y(0) = 15$?
Tap to reveal answer
$y(t) = 15e^{0.5t}$. Growth model with rate $k = 0.5$ and initial value 15.
$y(t) = 15e^{0.5t}$. Growth model with rate $k = 0.5$ and initial value 15.
← Didn't Know|Knew It →
Find $y(t)$ for $\frac{dy}{dt} = ky$, $y(0) = y_0$.
Find $y(t)$ for $\frac{dy}{dt} = ky$, $y(0) = y_0$.
Tap to reveal answer
$y(t) = y_0 e^{kt}$. Standard exponential solution with initial value specified.
$y(t) = y_0 e^{kt}$. Standard exponential solution with initial value specified.
← Didn't Know|Knew It →
Determine $y(t)$ if $\frac{dy}{dt} = -3y$ and $y(0) = 7$.
Determine $y(t)$ if $\frac{dy}{dt} = -3y$ and $y(0) = 7$.
Tap to reveal answer
$y(t) = 7e^{-3t}$. Decay model with initial condition $y(0) = 7$.
$y(t) = 7e^{-3t}$. Decay model with initial condition $y(0) = 7$.
← Didn't Know|Knew It →
Which function represents exponential growth: $y = e^{kt}$ or $y = e^{-kt}$?
Which function represents exponential growth: $y = e^{kt}$ or $y = e^{-kt}$?
Tap to reveal answer
$y = e^{kt}$. Positive exponent causes exponential increase over time.
$y = e^{kt}$. Positive exponent causes exponential increase over time.
← Didn't Know|Knew It →
What is the rate constant $k$ if the population doubles in 3 years?
What is the rate constant $k$ if the population doubles in 3 years?
Tap to reveal answer
$k = \frac{\text{ln}(2)}{3}$. Doubling time formula $t_d = \frac{\ln(2)}{k}$ solved for $k$.
$k = \frac{\text{ln}(2)}{3}$. Doubling time formula $t_d = \frac{\ln(2)}{k}$ solved for $k$.
← Didn't Know|Knew It →
State the formula for the derivative of $e^{kt}$ with respect to $t$.
State the formula for the derivative of $e^{kt}$ with respect to $t$.
Tap to reveal answer
$\frac{d}{dt} e^{kt} = ke^{kt}$. Chain rule applied to exponential function with coefficient $k$.
$\frac{d}{dt} e^{kt} = ke^{kt}$. Chain rule applied to exponential function with coefficient $k$.
← Didn't Know|Knew It →
What is the carrying capacity in the logistic growth model?
What is the carrying capacity in the logistic growth model?
Tap to reveal answer
$L$. Maximum sustainable population in logistic growth model.
$L$. Maximum sustainable population in logistic growth model.
← Didn't Know|Knew It →
What is the solution to $\frac{dy}{dt} = -6y$ for $y(0) = 4$?
What is the solution to $\frac{dy}{dt} = -6y$ for $y(0) = 4$?
Tap to reveal answer
$y(t) = 4e^{-6t}$. Decay with rate 6 and initial condition 4.
$y(t) = 4e^{-6t}$. Decay with rate 6 and initial condition 4.
← Didn't Know|Knew It →
Find $y(t)$ if $\frac{dy}{dt} = 5y$ and $y(0) = 10$.
Find $y(t)$ if $\frac{dy}{dt} = 5y$ and $y(0) = 10$.
Tap to reveal answer
$y(t) = 10e^{5t}$. Initial condition $y(0) = 10$ with growth rate $k = 5$.
$y(t) = 10e^{5t}$. Initial condition $y(0) = 10$ with growth rate $k = 5$.
← Didn't Know|Knew It →
What is the solution to the separable equation $\frac{dy}{dx} = x^2 y$?
What is the solution to the separable equation $\frac{dy}{dx} = x^2 y$?
Tap to reveal answer
$y = Ce^{x^3/3}$. Separation gives $\frac{dy}{y} = x^2 dx$, integrate both sides.
$y = Ce^{x^3/3}$. Separation gives $\frac{dy}{y} = x^2 dx$, integrate both sides.
← Didn't Know|Knew It →
What is the solution to $\frac{dy}{dt} = -0.2y$ with $y(0) = 20$?
What is the solution to $\frac{dy}{dt} = -0.2y$ with $y(0) = 20$?
Tap to reveal answer
$y(t) = 20e^{-0.2t}$. Slow decay with rate 0.2 and initial value 20.
$y(t) = 20e^{-0.2t}$. Slow decay with rate 0.2 and initial value 20.
← Didn't Know|Knew It →
What is the doubling time formula for exponential growth?
What is the doubling time formula for exponential growth?
Tap to reveal answer
$t_d = \frac{\text{ln}(2)}{k}$. Time for quantity to double its original value.
$t_d = \frac{\text{ln}(2)}{k}$. Time for quantity to double its original value.
← Didn't Know|Knew It →
What is the half-life formula for exponential decay?
What is the half-life formula for exponential decay?
Tap to reveal answer
$t_{1/2} = \frac{\text{ln}(2)}{k}$. Time for quantity to reduce to half its original value.
$t_{1/2} = \frac{\text{ln}(2)}{k}$. Time for quantity to reduce to half its original value.
← Didn't Know|Knew It →
What is the form of a separable differential equation?
What is the form of a separable differential equation?
Tap to reveal answer
$\frac{dy}{dx} = g(y)h(x)$. Variables can be separated: $\frac{dy}{g(y)} = h(x)dx$.
$\frac{dy}{dx} = g(y)h(x)$. Variables can be separated: $\frac{dy}{g(y)} = h(x)dx$.
← Didn't Know|Knew It →
Find $y(t)$ if $\frac{dy}{dt} = 7y$ and $y(0) = 2$.
Find $y(t)$ if $\frac{dy}{dt} = 7y$ and $y(0) = 2$.
Tap to reveal answer
$y(t) = 2e^{7t}$. Exponential growth with rate 7 and initial condition 2.
$y(t) = 2e^{7t}$. Exponential growth with rate 7 and initial condition 2.
← Didn't Know|Knew It →
Find the particular solution for $\frac{dy}{dt} = 9y$, $y(0) = 1$.
Find the particular solution for $\frac{dy}{dt} = 9y$, $y(0) = 1$.
Tap to reveal answer
$y(t) = e^{9t}$. Fast growth rate 9 with unit initial condition.
$y(t) = e^{9t}$. Fast growth rate 9 with unit initial condition.
← Didn't Know|Knew It →
Determine $y(t)$ if $\frac{dy}{dt} = -3y$ and $y(0) = 7$.
Determine $y(t)$ if $\frac{dy}{dt} = -3y$ and $y(0) = 7$.
Tap to reveal answer
$y(t) = 7e^{-3t}$. Decay model with initial condition $y(0) = 7$.
$y(t) = 7e^{-3t}$. Decay model with initial condition $y(0) = 7$.
← Didn't Know|Knew It →
What is the expression for $k$ if the population triples in 5 years?
What is the expression for $k$ if the population triples in 5 years?
Tap to reveal answer
$k = \frac{\text{ln}(3)}{5}$. Tripling time formula $t = \frac{\ln(3)}{k}$ solved for $k$.
$k = \frac{\text{ln}(3)}{5}$. Tripling time formula $t = \frac{\ln(3)}{k}$ solved for $k$.
← Didn't Know|Knew It →
Which function represents exponential growth: $y = e^{kt}$ or $y = e^{-kt}$?
Which function represents exponential growth: $y = e^{kt}$ or $y = e^{-kt}$?
Tap to reveal answer
$y = e^{kt}$. Positive exponent causes exponential increase over time.
$y = e^{kt}$. Positive exponent causes exponential increase over time.
← Didn't Know|Knew It →
What is the solution to $\frac{dy}{dt} = 0.5y$ with $y(0) = 15$?
What is the solution to $\frac{dy}{dt} = 0.5y$ with $y(0) = 15$?
Tap to reveal answer
$y(t) = 15e^{0.5t}$. Growth model with rate $k = 0.5$ and initial value 15.
$y(t) = 15e^{0.5t}$. Growth model with rate $k = 0.5$ and initial value 15.
← Didn't Know|Knew It →
If $\frac{dy}{dt} = ky$ and $y(0) = y_0$, express $y(t)$ in terms of $y_0$.
If $\frac{dy}{dt} = ky$ and $y(0) = y_0$, express $y(t)$ in terms of $y_0$.
Tap to reveal answer
$y(t) = y_0 e^{kt}$. Standard exponential growth solution with initial value $y_0$.
$y(t) = y_0 e^{kt}$. Standard exponential growth solution with initial value $y_0$.
← Didn't Know|Knew It →
Solve $\frac{dy}{dt} = 2y$ for $y(0) = 3$. What is $y(t)$?
Solve $\frac{dy}{dt} = 2y$ for $y(0) = 3$. What is $y(t)$?
Tap to reveal answer
$y(t) = 3e^{2t}$. Initial condition $y(0) = 3$ gives $C = 3$.
$y(t) = 3e^{2t}$. Initial condition $y(0) = 3$ gives $C = 3$.
← Didn't Know|Knew It →