Differential equations practice questions (AP Calculus)

Sample questions

All 2 free AP Calculus questions on this unit, from the free AP Calculus BC Practice Exam. Each answer is behind its own toggle; the whole paper, its PDF and the full answer key are on the exam page.

1. Question 12 marks · Differential equations
   Let $y = f(x)$ be the solution of $\dfrac{dy}{dx} = x + y$ with $f(0) = 1$. Using Euler's method with two steps of size $0.5$, the approximation to $f(1)$ is
   • A.
     $1.5$
   • B.
     $2$
   • C.
     $2.5$
   • D.
     $3$
   Show answerFrom the free AP Calculus BC Practice Exam →
2. Question 22 marks · Differential equations
   A population $P(t)$ satisfies the logistic differential equation $\dfrac{dP}{dt} = 3P - 0.006P^2$ with $P(0) = 50$. The limiting value of $P$ as $t \to \infty$ and the value of $P$ at which $P$ is growing fastest are, respectively,
   • A.
     500$ and $250

     LaTeX syntax issue: KaTeX parse error: Can't use function '$' in math mode at position 4: 500$̲ and $250
   • B.
     500$ and $500

     LaTeX syntax issue: KaTeX parse error: Can't use function '$' in math mode at position 4: 500$̲ and $500
   • C.
     1000$ and $500

     LaTeX syntax issue: KaTeX parse error: Can't use function '$' in math mode at position 5: 1000$̲ and $500
   • D.
     250$ and $125

     LaTeX syntax issue: KaTeX parse error: Can't use function '$' in math mode at position 4: 250$̲ and $125
   Show answerFrom the free AP Calculus BC Practice Exam →

Key terms

The definitions and formulas AP Calculus expects for differential equations, typeset as they appear on the paper.

Differential equation and its solutions

An equation involving a function and its derivatives. The general solution has an arbitrary constant; an initial condition picks out the particular solution. Verify a proposed solution by substituting it into both sides.

Slope field

Short line segments drawn at grid points with slope given by the differential equation. A solution curve follows the segments; matching slope fields to equations is a standard multiple-choice task.

Separation of variables

Rewrite $\frac{dy}{dx} = f(x)g(y)$ with all $y$ terms on one side and all $x$ terms on the other, integrate both sides, then solve for $y$ and use the initial condition to find the constant.

$$\int \frac{dy}{g(y)} = \int f(x)\,dx$$

Exponential growth and decay

When the rate of change is proportional to the amount present, the solution is exponential. Positive $k$ is growth, negative $k$ is decay; half-life and doubling time follow from solving for $t$.

$$\frac{dy}{dt} = ky \;\Rightarrow\; y = C e^{kt}$$

Euler's method (BC)

A tangent-line stepping approximation to a solution curve from an initial point with step size $h$. Each step uses the slope at the current point from the differential equation.

$$y_{n+1} = y_n + h \cdot f(x_n, y_n)$$

Logistic model (BC)

Growth that slows as the population approaches a carrying capacity $L$. The population grows fastest when it is half the carrying capacity, and tends to $L$ as $t$ increases.

$$\frac{dP}{dt} = kP\left(1 - \frac{P}{L}\right), \qquad \lim_{t \to \infty} P(t) = L$$

Frequently asked questions

Other AP Calculus topics

• Limits and continuity
• Differentiation: definition and basic rules
• Composite, implicit and inverse functions
• Contextual applications of differentiation
• Analytical applications of differentiation
• Integration and accumulation of change
• Applications of integration
• Parametric, polar and vector functions (BC)
• Infinite sequences and series (BC)

All AP Calculus topics · AP Calculus exam guide · All free practice exams