AP Calculus BC Unit 7 (BC topics): Euler's Method and Logistic Models

Two BC additions to differential equations: Euler's method, which follows short tangent-line steps to approximate a solution, and the logistic model, whose solutions level off at a carrying capacity.

This page covers only the BC topics of Unit 7. The rest of the unit is the same as AP Calculus AB: AB Unit 7 notes and practice.

BC topics on this page: 7.5 Euler's method; 7.9 Logistic models.

Key formulas and facts

• Euler's method: $x_{n+1}=x_n+\Delta x$ and $y_{n+1}=y_n+\Delta x\cdot\frac{dy}{dx}\Big|_{(x_n,y_n)}$
• Concave up solution: Euler (tangent-line steps) underestimates; concave down: it overestimates
• Logistic: $\frac{dy}{dt}=ky\left(1-\frac{y}{L}\right)$, with $k>0$ and carrying capacity $L$
• For $y(0)>0$: $\lim_{t\to\infty}y(t)=L$; $y$ increases if $y(0)<L$ and decreases if $y(0)>L$
• Fastest growth at $y=\frac L2$, where the graph of $y$ has its point of inflection
• Explicit solution: $y=\frac{L}{1+Ae^{-kt}}$ with $A=\frac{L-y(0)}{y(0)}$

Worked examples

• Euler: $\frac{dy}{dx}=x-y$, $y(0)=2$, two steps of $0.5$: slope $-2$ gives $y(0.5)\approx1$; slope $-0.5$ gives $y(1)\approx0.75$
• Logistic form: $\frac{dP}{dt}=0.6P-0.002P^2=0.6P\left(1-\frac{P}{300}\right)$, so $L=300$ and growth is fastest at $P=150$
• Explicit solution: $\frac{dy}{dt}=2y\left(1-\frac y{10}\right)$ with $y(0)=2$ gives $y=\frac{10}{1+4e^{-2t}}$
• Fastest growth time: $1+4e^{-2t}=2$ gives $t=\frac{\ln4}{2}=\ln2$

Common traps

• Using the slope at the new point before taking the step: each step uses the slope at the point you are leaving.
• Mixing up the step size and the number of steps: from $x=0$ to $x=1$ in steps of $0.25$ is four steps.
• Reading the carrying capacity from $\frac{dy}{dt}=4y-0.01y^2$ as $4$ or $0.01$: factor to $ky\left(1-\frac yL\right)$ first.
• Saying fastest growth is at the carrying capacity: the rate there is $0$; fastest growth is at $\frac L2$.
• Forgetting that a population starting above $L$ decreases toward $L$.

Practice questions

15 multiple-choice questions and 2 free-response questions. Work each one out before you open the answer. In a free-response scoring guide, each line is worth the points it states. A free account opens the worked answers and scoring guides of 20 questions a day; Pro opens every one.

Question 1 · Multiple choice

Let $y=f(x)$ be the solution to $\dfrac{dy}{dx}=x+y$ with $f(0)=1$. What is the approximation of $f(1)$ given by Euler's method with two steps of equal size, starting at $x=0$?

1. $2$
2. $3$
3. $2.5$
4. $2.25$

Question 2 · Multiple choice

The function $y=g(x)$ satisfies $\dfrac{dy}{dx}=3x-y^2$ and $g(1)=2$. Using Euler's method with a step size of $0.1$, starting at $x=1$, what is the approximation of $g(1.2)$?

1. $1.9$
2. $1.869$
3. $1.831$
4. $2.069$

Question 3 · Multiple choice

For the differential equation $\dfrac{dy}{dx}=xy$, the solution through $(1,2)$ is approximated with Euler's method using two steps of size $0.5$. What is the approximate value of $y$ at $x=2$?

1. $3$
2. $4.5$
3. $6.5$
4. $5.25$

Question 4 · Multiple choice

Let $y=h(x)$ be the solution to $\dfrac{dy}{dx}=x^2+y$ with $h(0)=1$. Euler's method with two steps of size $0.5$ gives $h(1)\approx2.375$. Which statement about this approximation is true?

1. It is greater than $h(1)$, because the graph of $h$ is concave up for $0\le x\le1$.
2. It is less than $h(1)$, because the graph of $h$ is concave down for $0\le x\le1$.
3. It is less than $h(1)$, because the graph of $h$ is concave up for $0\le x\le1$.
4. It is greater than $h(1)$, because the graph of $h$ is concave down for $0\le x\le1$.

Question 5 · Multiple choice

A cup of tea cools according to $\dfrac{dT}{dt}=-0.1(T-20)$, where $T$ is its temperature in degrees Celsius $t$ minutes after it is poured, and $T(0)=90$. Using Euler's method with two steps of $5$ minutes each, what is the approximate temperature at $t=10$?

1. $37.5$
2. $55$
3. $41.25$
4. $45$

Question 6 · Multiple choice

The function $y$ satisfies $\dfrac{dy}{dx}=kxy$ for a constant $k$, with $y(0)=4$. Euler's method with two steps of size $0.5$, starting at $x=0$, gives $y(1)\approx5$. What is $k$?

1. $2$
2. $\frac{1}{2}$
3. $4$
4. $1$

Question 7 · Multiple choice

The solution of $\dfrac{dy}{dx}=y$ with $y(0)=1$ has $y(1)=e$. Euler's method with four steps of size $0.25$ is used to approximate $y(1)$. What is $e$ minus this approximation?

1. $0.718$
2. $0.277$
3. $2.441$
4. $0.159$

Question 8 · Multiple choice

A population $P(t)$ satisfies $\dfrac{dP}{dt}=0.3P\left(1-\dfrac{P}{800}\right)$ with $P(0)=100$. What is $\displaystyle\lim_{t\to\infty}P(t)$?

1. $400$
2. $240$
3. $100$
4. $800$

Question 9 · Multiple choice

The number of people $N$ who have heard a rumor satisfies $\dfrac{dN}{dt}=0.02N(500-N)$, with $0<N(0)<250$. For what value of $N$ is the rumor spreading fastest?

1. $500$
2. $250$
3. $125$
4. $10$

Question 10 · Multiple choice

A quantity $y$ satisfies $\dfrac{dy}{dt}=4y-0.01y^2$ with $y(0)=50$. What is the carrying capacity of this logistic model?

1. $400$
2. $4$
3. $200$
4. $100$

Question 11 · Multiple choice

Let $P(t)=\dfrac{1200}{1+5e^{-0.4t}}$ be the number of members of a hiking club $t$ years after it opened. How many years after opening does the club's membership increase at its greatest rate?

1. $\frac{2 \ln{\left(5 \right)}}{5}$
2. $\frac{\ln{\left(5 \right)}}{2}$
3. $\frac{5 \ln{\left(5 \right)}}{2}$
4. $\frac{5 \ln{\left(5 \right)}}{4}$

Question 12 · Multiple choice

For which differential equation does every solution with $y(0)>0$ approach $50$ as $t\to\infty$?

1. $\frac{dy}{dt}=0.1y(y-50)$
2. $\frac{dy}{dt}=0.1y(50-y)$
3. $\frac{dy}{dt}=50y(1-0.1y)$
4. $\frac{dy}{dt}=0.1y(1-50y)$

Question 13 · Multiple choice

A population satisfies $\dfrac{dP}{dt}=0.5P\left(1-\dfrac{P}{60}\right)$ with $P(0)=10$. For which values of $P$ is the graph of $P$ against $t$ concave up?

1. $30<P<60$
2. $0<P<60$
3. $15<P<45$
4. $0<P<30$

Question 14 · Multiple choice

A population satisfies $\dfrac{dP}{dt}=0.2P\left(1-\dfrac{P}{600}\right)$ and $P(0)=900$. Which statement describes $P(t)$ for $t>0$?

1. $P$ increases without bound.
2. $P$ decreases and approaches $0$.
3. $P$ decreases and approaches $600$.
4. $P$ increases and approaches $900$.

Question 15 · Multiple choice

A bacteria colony is modeled by $B(t)=\dfrac{2000}{1+19e^{-0.3t}}$, where $t$ is in hours. To the nearest thousandth of an hour, when does the colony first reach $1500$ bacteria?

1. $13.477$
2. $9.815$
3. $3.662$
4. $15.0$

Question 16 · Free response (9 points)

A fish population $y(t)$, in thousands, satisfies $\dfrac{dy}{dt}=\dfrac{y(6-y)}{8}$, where $t$ is in years, and $y(0)=2$.

1. (a) Find $\displaystyle\lim_{t\to\infty}y(t)$, and the population at which it is growing fastest. [2 points]
2. (b) Use Euler's method, starting at $t=0$ with two steps of equal size, to approximate $y(2)$. [2 points]
3. (c) Find $\dfrac{d^2y}{dt^2}$ in terms of $y$. Is the graph of $y$ concave up or concave down at $t=0$? Give a reason. [2 points]
4. (d) Use separation of variables to find $y(t)$, the particular solution with $y(0)=2$. [3 points]

Question 17 · Free response (9 points)

Let $y=f(x)$ be the solution to the differential equation $\dfrac{dy}{dx}=2x-2y$ with $f(0)=2$.

1. (a) Use Euler's method, starting at $x=0$ with two steps of equal size, to approximate $f(1)$. [2 points]
2. (b) Find $\dfrac{d^2y}{dx^2}$ in terms of $x$ and $y$, and evaluate it at $(0,2)$. [2 points]
3. (c) Write the tangent line to the graph of $f$ at $x=0$, and use it to approximate $f(0.2)$. [1 point]
4. (d) Show that $y=x-\frac12+\frac52e^{-2x}$ satisfies the differential equation and the initial condition, and find the exact value of $f(1)$. [2 points]
5. (e) Is the Euler approximation from part (a) an overestimate or an underestimate of $f(1)$? Explain using part (d) or the concavity of $f$. [2 points]