Show all supporting work on scratch paper. On the AP Exam, Section I is split into a no-calculator and a calculator-allowed part — each question below is labeled accordingly.
Q1EASY7.1 ModelingNo Calculator
A population grows at a rate proportional to the current population. Which differential equation models this situation?
(A) $\dfrac{dP}{dt}=k$
(B) $\dfrac{dP}{dt}=kP$
(C) $\dfrac{dP}{dt}=\dfrac{k}{P}$
(D) $\dfrac{dP}{dt}=kt$
Q2EASY7.2 Verifying SolutionsNo Calculator
Which of the following functions is a solution to $\dfrac{dy}{dx}=2y$?
(A) $y=2x$
(B) $y=x^{2}$
(C) $y=e^{2x}$
(D) $y=\sin(2x)$
Q3MEDIUM7.1 Modeling (Newton)No Calculator
A cup of coffee cools at a rate proportional to the difference between its temperature $T$ and a room temperature of $70$°F. Which equation models this?
(A) $\dfrac{dT}{dt}=k(T+70)$
(B) $\dfrac{dT}{dt}=k(T-70)$
(C) $\dfrac{dT}{dt}=k\,T$
(D) $\dfrac{dT}{dt}=70-k\,T$
Q4MEDIUM7.3 / 7.4 Slope FieldsNo Calculator
The slope field shown could represent which differential equation?
(A) $\dfrac{dy}{dx}=x$
(B) $\dfrac{dy}{dx}=y$
(C) $\dfrac{dy}{dx}=x+y$
(D) $\dfrac{dy}{dx}=xy$
Q5MEDIUM7.4 ReasoningNo Calculator
For which differential equation are all line segments in the same horizontal row of a slope field identical?
(A) $\dfrac{dy}{dx}=x$
(B) $\dfrac{dy}{dx}=y$
(C) $\dfrac{dy}{dx}=x+y$
(D) $\dfrac{dy}{dx}=\sin x$
Q6EASY7.6 SeparationNo Calculator
If $\dfrac{dy}{dx}=\dfrac{x}{y}$, which is a general solution?
(A) $y^{2}=x^{2}+C$
(B) $y=x+C$
(C) $\ln|y|=\ln|x|+C$
(D) $y^{2}=2x+C$
Q7MEDIUM7.7 Particular SolutionNo Calculator
Let $y$ be the solution to $\dfrac{dy}{dx}=2xy$ with $y(0)=3$. Then $y(1)=$
(A) $3$
(B) $3e$
(C) $3e^{2}$
(D) $e^{3}$
Q8HARD7.7 DomainNo Calculator
Let $y$ be the solution to $\dfrac{dy}{dx}=y^{2}$ with $y(0)=1$. The largest open interval containing $0$ on which the solution is defined is
(A) $(-1,1)$
(B) $(-\infty,1)$
(C) $(-\infty,\infty)$
(D) $(0,1)$
Q9EASY7.8 Exponential GrowthNo Calculator
A quantity $Q$ satisfies $\dfrac{dQ}{dt}=0.05\,Q$ and $Q(0)=200$. Then $Q(t)=$
(A) $200+0.05t$
(B) $200\,e^{0.05t}$
(C) $200\cdot 1.05^{t}$
(D) $\dfrac{200}{e^{0.05t}}$
Q10MEDIUM7.8 Decay (Half-Life)Calculator
A radioactive substance decays so that $\dfrac{dN}{dt}=-0.04\,N$, where $t$ is in years. To the nearest year, the half-life is
(A) $14$ years
(B) $17$ years
(C) $20$ years
(D) $25$ years
Q11MEDIUM7.2 VerifyingNo Calculator
Which of the following is NOT a solution to $\dfrac{dy}{dx}=y$?
(A) $y=e^{x}$
(B) $y=2e^{x}$
(C) $y=-e^{x}$
(D) $y=e^{2x}$
Q12HARD7.4 DE + TangentNo Calculator
Let $y=f(x)$ be the solution to $\dfrac{dy}{dx}=x+y$ with $f(1)=2$. The tangent line to $y=f(x)$ at $x=1$ is used to approximate $f(1.2)$. The approximation is
(A) $2.2$
(B) $2.4$
(C) $2.6$
(D) $3.2$
Q13HARD7.6 SeparationNo Calculator
Which of the following is a general solution to $\dfrac{dy}{dx}=\dfrac{y}{x}$?
(A) $y=Cx$
(B) $\ln|y|=\ln|x|+C$
(C) Both (A) and (B)
(D) $y=x+C$
Q14MEDIUM7.8 ExponentialNo Calculator
A bacterial culture has $1000$ bacteria. After $4$ hours, the culture has $4000$ bacteria. If the growth rate is proportional to the current population, how many bacteria are present after $6$ hours?
(A) $5000$
(B) $6000$
(C) $8000$
(D) $16000$
Q15EASY7.3 Slope FieldsNo Calculator
Consider $\dfrac{dy}{dx}=x-y$. The slope of the line segment in the slope field at the point $(2,1)$ is
(A) $-1$
(B) $0$
(C) $1$
(D) $3$
Q16MEDIUM7.6 Separable IVPNo Calculator
If $\dfrac{dy}{dx}=\dfrac{x}{y^{2}}$ with $y(0)=1$, then $y$ when $x=2$ equals
(A) $\sqrt[3]{4}$
(B) $\sqrt[3]{7}$
(C) $\sqrt{5}$
(D) $2$
Q17MEDIUM7.7 Initial ConditionsNo Calculator
If $y$ satisfies $\dfrac{dy}{dx}=ye^{x}$ with $y(0)=1$, then $y(\ln 2)=$
(A) $e$
(B) $\sqrt{e}$
(C) $e^{2}$
(D) $2e$
Q18HARD7.4 EquilibriumNo Calculator
Consider $\dfrac{dy}{dt}=y(2-y)$. The equilibrium solutions are
(A) $y=0$ only
(B) $y=2$ only
(C) $y=0$ and $y=2$
(D) none exist
PART IIShow All Work
Free-Response Questions
Free-response answers require complete algebraic work: separation of variables, antiderivatives, $+C$, use of initial conditions, and correct solving for $y$. On the AP Exam, skipping any of these steps will cost points.
FRQ 1EASY7.6 / 7.7 SeparationNo Calculator
Consider the differential equation $\dfrac{dy}{dx}=\dfrac{x}{y}$, where $y>0$.
(a) Find the general solution of the differential equation.
(b) Find the particular solution $y=f(x)$ satisfying the initial condition $f(0)=2$.
(c) State the domain of the particular solution from part (b).
Consider the differential equation $\dfrac{dy}{dx}=2xy$.
(a) On the axes provided, sketch a slope field for the given DE at the nine points indicated.
(b) Find the particular solution $y=f(x)$ to the DE with $f(0)=1$.
(c) Use your solution from part (b) to find $f(1)$.
FRQ 3MEDIUM7.1 / 7.7 / 7.8 Tank LeakCalculator
A tank initially contains $500$ gallons of water. Water leaks out at a rate proportional to the amount remaining: $\dfrac{dW}{dt}=k\,W$, where $t$ is in minutes. After $20$ minutes, the tank contains $400$ gallons.
(a) Write an expression for $W(t)$, the amount of water at time $t$, in terms of $k$.
(b) Find the value of $k$. Round to four decimal places.
(c) How much water, to the nearest gallon, remains in the tank after $60$ minutes?
(d) At what rate, in gallons per minute, is water leaking from the tank at $t=60$? Indicate units.
A biologist studies a fish population $P(t)$ in a lake, where $t$ is in years. The population is modeled by the differential equation $\dfrac{dP}{dt}=0.1\,P\left(1-\dfrac{P}{1000}\right)$. (You are not required to solve this differential equation.)
(a) Find $\dfrac{dP}{dt}$ when $P=400$. Interpret the value in context, with correct units.
(b) For what value(s) of $P$ is $\dfrac{dP}{dt}=0$? What do these values represent?
(c) Suppose $P(0)=400$. Use the tangent line to the graph of $P$ at $t=0$ to approximate $P(2)$.
(d) Use implicit differentiation to find $\dfrac{d^{2}P}{dt^{2}}$ in terms of $P$. Use this to determine whether the tangent-line approximation in part (c) is an over- or underestimate. Justify.
PART III — (BC) EXTENSIONSTopics 7.5, 7.9 — BC ONLY
BC-Only Practice
BC ONLY. The following items cover Euler's method (Topic 7.5) and logistic models (Topic 7.9). AB students may skip this section. BC students should be fluent with: (i) Euler iteration $y_{n+1}=y_n+h\cdot f(x_n,y_n)$; (ii) the logistic equation $\dfrac{dy}{dt}=ky\!\left(1-\dfrac{y}{K}\right)$ with carrying capacity $K$ and inflection at $y=K/2$.
Q BC1MEDIUM7.5 Euler's MethodBC ONLYNo Calculator
Let $y=f(x)$ be the solution to $\dfrac{dy}{dx}=x+y$ with $f(0)=1$. Use Euler's method with two steps of equal size $h=0.5$ starting at $x=0$ to approximate $f(1)$.
Suppose the solution $y=f(x)$ to a differential equation is concave up on the interval $[a,b]$. An Euler's-method approximation of $f(b)$ starting from $f(a)$ with positive step size $h$ will be
(A) always greater than the true value $f(b)$.
(B) always less than the true value $f(b)$.
(C) equal to the true value $f(b)$ regardless of $h$.
A population $P(t)$ satisfies $\dfrac{dP}{dt}=0.04\,P\!\left(1-\dfrac{P}{500}\right)$. The carrying capacity and the population value at which $P$ is increasing fastest are, respectively,
Consider the differential equation $\dfrac{dy}{dx}=x-y$ with initial condition $y(0)=2$.
(a) Use Euler's method with two steps of size $h=0.5$, starting at $x=0$, to approximate $y(1)$. Show all computations.
(b) Find $\dfrac{d^{2}y}{dx^{2}}$ in terms of $x$ and $y$. Use this to determine whether the Euler approximation in part (a) is an over- or underestimate of $y(1)$. Justify.
(c) A second student uses Euler's method with four steps of size $h=0.25$ and obtains a different value. Without computing it, predict whether this second approximation will be closer to or farther from the true value of $y(1)$, and explain why.
FRQ BC2HARD7.9 Logistic ModelBC ONLYNo Calculator
A wildlife biologist models a deer population $P(t)$, where $t$ is in years, by $\dfrac{dP}{dt}=0.2\,P\!\left(1-\dfrac{P}{800}\right)$, with $P(0)=100$.
(a) State the carrying capacity. Find $\displaystyle\lim_{t\to\infty}P(t)$ and justify.
(b) At what value of $P$ is the population growing fastest? Find $\dfrac{dP}{dt}$ at that value, with units.
(c) Find $\dfrac{d^{2}P}{dt^{2}}$ in terms of $P$ alone. Use it to identify the value of $P$ at which the graph of $P(t)$ has an inflection point. Justify your answer using a sign analysis of $\dfrac{d^{2}P}{dt^{2}}$.
(d) Sketch a qualitative graph of $P(t)$ for $t\ge 0$. Label the inflection value and the horizontal asymptote.