Contextual applications of differentiation practice questions (AP Calculus)

Sample questions

All 2 free AP Calculus questions on this unit, from the free AP Calculus AB Practice Exam and 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 · Related rates
   The radius of a sphere is increasing at 2 centimeters per second. In terms of $\pi$, how fast is the volume $V = \frac{4}{3}\pi r^3$ increasing when the radius is 3 centimeters?
   • A.
     $24\pi$ cubic centimeters per second
   • B.
     $36\pi$ cubic centimeters per second
   • C.
     $72\pi$ cubic centimeters per second
   • D.
     $144\pi$ cubic centimeters per second
   Show answerFrom the free AP Calculus AB Practice Exam →
2. Question 22 marks · Limits (L'Hôpital)
   Evaluate $\displaystyle\lim_{x \to 0} \frac{e^{2x} - 1 - 2x}{x^2}$.
   • A.
     $0$
   • B.
     $1$
   • C.
     $2$
   • D.
     $4$
   Show answerFrom the free AP Calculus BC Practice Exam →

Key terms

The definitions and formulas AP Calculus expects for contextual applications of differentiation, typeset as they appear on the paper.

Derivative as a rate of change with units

If $f(t)$ is measured in gallons and $t$ in minutes, $f'(t)$ is in gallons per minute. Interpretation questions want the quantity, the units and the time in one sentence.

Position, velocity and acceleration

Velocity is the derivative of position, acceleration the derivative of velocity, and speed is the absolute value of velocity. Speed is increasing when velocity and acceleration have the same sign.

$$v(t) = s'(t), \qquad a(t) = v'(t), \qquad \text{speed} = |v(t)|$$

Related rates

Write an equation linking the quantities, differentiate implicitly with respect to time, then substitute the known values and rates. Substitute only after differentiating.

$$\frac{dV}{dt} = \frac{dV}{dr}\cdot\frac{dr}{dt}$$

Linearization and tangent-line approximation

Near $x = a$ the function is approximated by its tangent line. The approximation is an underestimate where the graph is concave up and an overestimate where it is concave down.

$$L(x) = f(a) + f'(a)(x - a)$$

L'Hospital's rule

For a limit of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, the limit of the quotient equals the limit of the quotient of the derivatives, if that limit exists. State the indeterminate form before applying it.

$$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$$

Average versus instantaneous rate of change

The average rate over $[a, b]$ is the slope of the secant line; the instantaneous rate at a point is the derivative, the slope of the tangent. Table questions ask you to estimate the second from the first.

$$\text{average rate} = \frac{f(b) - f(a)}{b - a}$$

Frequently asked questions

Other AP Calculus topics

• Limits and continuity
• Differentiation: definition and basic rules
• Composite, implicit and inverse functions
• Analytical applications of differentiation
• Integration and accumulation of change
• Differential equations
• 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