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AP Calculus AB · Unit 4 of 8
Contextual Applications
10–15% of the exam6 lessons · 84 min40 terms
What this unit covers
The topics below follow the published Calculus AB course framework for Unit 4. This unit is worth 10–15% of the exam, so budget your time against that rather than against how long the unit takes to teach.
Related ratesLinearizationL’HôpitalMotion
Lessons in this unit
- Related Rates15 min · 3 objectivesRelate the rates of change of quantities linked by an equation · Differentiate a geometric relationship implicitly with respect to time · Substitute given values only after differentiating
- Linearization & Local Approximation13 min · 3 objectivesWrite the equation of the tangent line as a local linear approximation · Use linearization to estimate function values near a known point · Determine whether an estimate is an overestimate or underestimate from concavity
- L’Hôpital’s Rule13 min · 3 objectivesRecognize the indeterminate forms 0/0 and ∞/∞ · Apply L’Hôpital’s Rule to evaluate qualifying limits · Avoid misapplying the rule to non-indeterminate limits
- Motion Along a Line14 min · 3 objectivesRelate position, velocity, and acceleration through derivatives · Determine when a particle moves right, left, or is at rest, and when it speeds up or slows down · Distinguish displacement from total distance traveled
- Interpreting Derivatives in Context, with Units14 min · 3 objectivesState the units of a derivative from the units of the two quantities · Write a sentence interpreting f′(a) that would earn an interpretation point · Distinguish a rate, a change, and an amount in an applied problem
- Particle Motion: Speed, Acceleration, and Direction15 min · 3 objectivesDistinguish velocity, speed, and acceleration and their sign conditions · Determine when a particle changes direction and when it is speeding up · Compute total distance traveled versus displacement
Formulas in Unit 4
Differentiating a relationship in time
If V = (4/3)πr³, then dV/dt = 4πr² · (dr/dt)
Every variable carries a d/dt. The chain rule links dV/dt to dr/dt through the geometry.
Linear approximation
L(x) = f(a) + f′(a)·(x − a)
The tangent line at x = a. Use it to estimate f(x) for x near a. Concave up ⇒ underestimate; concave down ⇒ overestimate.
L’Hôpital’s Rule
If lim f(x)/g(x) is 0/0 or ∞/∞, then lim f(x)/g(x) = lim f′(x)/g′(x)
Differentiate numerator and denominator separately — not with the quotient rule. Re-check the form after each application.
Motion relationships
v(t) = s′(t) · a(t) = v′(t) · speed = |v(t)|
Speeding up ⇔ v and a share a sign. Direction changes where v = 0 and switches sign.
Units of a derivative
units of dy/dx = (units of y) / (units of x)
Reading the notation literally gives the units. A second derivative has units of y per unit of x squared — for example feet per second per second.
Speeding up and slowing down
speeding up ⟺ v(t)·a(t) > 0 slowing down ⟺ v(t)·a(t) < 0
Same signs means speeding up, regardless of direction of travel. Note this is a product test — it works without deciding which sign each has.
Every term in Unit 4
All 40 terms we publish for Contextual Applications, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.
- L'Hôpital's Rule
- For 0/0 or ∞/∞ only, lim f/g = lim f′/g′. Verify the indeterminate form first, and differentiate numerator and denominator separately — not as a quotient.
- Rectilinear motion
- v(t) = x′(t) and a(t) = v′(t). Speed is |v(t)|, a non-negative quantity distinct from velocity.
- Speeding up or slowing down
- Speeding up when v and a share a sign, slowing down when their signs differ. State both signs in a justification.
- Displacement vs total distance
- Displacement is ∫v dt; total distance is ∫|v| dt, which requires splitting the integral where v changes sign.
- Related rates procedure
- Write an equation relating the quantities, differentiate both sides with respect to time, then substitute the instantaneous values — substituting before differentiating is the classic error.
- Related rates units
- The answer needs units, typically a rate such as cm³ per second. A missing unit is a lost point on the free-response section.
- Linear approximation
- L(x) = f(a) + f′(a)(x − a), the tangent line used as an estimate near a. It overestimates where the function is concave down.
- Interpreting a derivative in context
- "At time t = 3 seconds, the volume is increasing at 5 cubic centimeters per second." Value, units, and increasing or decreasing.
- Units of a derivative
- The units of f divided by the units of x. A volume in liters against time in minutes gives liters per minute.
- Units of a definite integral
- The units of the integrand multiplied by the units of the variable, which is why integrating a rate returns the original quantity.
- Interpreting f′(a) in a sentence
- State the value, the units and whether the quantity is increasing or decreasing at that moment. All three are scored.
- Particle at rest vs changing direction
- At rest means v = 0; changing direction requires v to CHANGE SIGN there, which is a stronger condition.
- Position from velocity
- x(b) = x(a) + ∫ₐᵇ v(t)dt. The initial position is required — an integral alone gives displacement, not position.
- Related rates with a cone or sphere
- Use the constraint to eliminate a variable before differentiating, or you will be left with two unknown rates.
- Common related-rates error
- Substituting the instantaneous values before differentiating. Those values are only true at one instant and differentiating them gives zero.
- Units as a setup check
- Read dy/dx literally as y-units per x-unit. If the units of your answer come out wrong, the setup is wrong — a five-second diagnostic worth running every time.
- Three-part interpretation sentence
- When, what is changing and in which direction, and the rate with units. Readers check for all three.
- Interpreting a negative derivative
- Say decreasing at a rate of 3.2 units per minute, not increasing at −3.2 and never decreasing at −3.2, which reverses the meaning.
- Rate versus change versus amount
- A derivative is a rate, the integral of a rate is a change, and an amount needs an initial value plus that change.
- How fast versus how much
- How fast or at what rate wants a derivative; how much or by how much changed wants an integral of a rate.
- The rate is decreasing
- A statement about the second derivative. The quantity can still be increasing while its rate of increase slows.
- Increasing at a decreasing rate
- f′ > 0 and f″ < 0 — rising while flattening, the most common applied case and the phrasing to recognize instantly.
- Decreasing at a decreasing rate
- f′ < 0 and f″ > 0 — falling while slowing, the shape of cooling coffee or of decay toward an asymptote.
- Speed is not a derivative
- Speed = |v(t)| is the magnitude of velocity. A particle with velocity −7 has speed 7.
- Changing direction test
- v(t) must CHANGE SIGN. v = 0 alone is not enough — for v = (t − 3)² the particle pauses and continues the same way.
- Speeding up test
- v and a have the same sign, equivalently v·a > 0. Works without deciding which sign each has individually.
- Negative acceleration does not mean slowing
- With v < 0 and a < 0 the particle is speeding up while moving backward. The most-missed sign question in the unit.
- Displacement
- ∫v dt, signed. Can be zero for a particle that has moved a great deal.
- Total distance
- ∫|v| dt, which requires splitting at every zero of v and adding the magnitudes.
- Position needs an initial condition
- x(b) = x(a) + ∫v dt. The integral alone gives the change in position, not the position.
- Related rates: differentiate the relation, then substitute
- Substituting the instantaneous values before differentiating turns variables into constants and destroys the problem.
- Related rates: one equation, one unknown rate
- Every quantity that varies contributes a rate by the chain rule. Identify what is given and what is asked before differentiating.
- Cone volume relation
- V = (1/3)πr²h. If the problem gives a fixed ratio between r and h, substitute it BEFORE differentiating to avoid two unknown rates.
- Sphere volume and surface area
- V = (4/3)πr³ gives dV/dt = 4πr²(dr/dt) — note the coefficient is the surface area, which is a useful check.
- Linearization formula
- f(x) ≈ f(a) + f′(a)(x − a). Dropping the (x − a) factor treats a rate as a change.
- Tangent-line estimate and concavity
- Concave up makes the tangent line an underestimate; concave down makes it an overestimate. Justify with the sign of f″.
- L’Hôpital requires 0/0 or ∞/∞
- Check the indeterminate form first and say you did. Applying it to a determinate form gives a wrong answer confidently.
- L’Hôpital differentiates separately
- Differentiate numerator and denominator individually — it is not the quotient rule, and using the quotient rule here is a standard error.
- Marginal cost as a derivative
- C′(x) is the approximate cost of the next unit, in dollars per unit. A rate, not a total.
- Average velocity versus instantaneous
- Average velocity is displacement over elapsed time; instantaneous velocity is v(t). The MVT is what connects them.
What examiners penalize here
- AP related-rates responses must show the differentiated equation with the d/dt factors before any numbers appear. Readers look for that general relationship; a correct final number with no setup earns little credit.
- To justify over- versus underestimate, cite concavity explicitly: "f″ < 0 on the interval, so the graph is concave down and the tangent line overestimates." A bare "overestimate" with no reason to concavity does not earn the justification point.
- Always write the phrase "form is 0/0" (or ∞/∞) before applying L’Hôpital on free response. Stating the indeterminate form is part of a valid justification; jumping straight to differentiating the pieces can cost the point even with a correct answer.
- For "speeding up or slowing down", always compare the signs of v and a at that instant. Same sign ⇒ speeding up; opposite signs ⇒ slowing down. The sign of acceleration by itself tells you nothing about speed.
- When you interpret a **negative** derivative, say "decreasing at a rate of 3.2 units per minute" rather than "increasing at −3.2." Both are mathematically true; only the first reads as an interpretation. And never write "decreasing at a rate of −3.2," which says the opposite of what you mean.
Practice Calculus AB
Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.
Questions about this unit
How much of the AP Calculus AB exam is Unit 4?
Unit 4, Contextual Applications, is worth 10–15% of the Calculus AB multiple-choice section according to the published course framework. Across all 8 units that makes it a substantial share — heavier than an even split would give it.
What topics are covered in Calculus AB Unit 4?
Contextual Applications covers Related rates, Linearization, L’Hôpital and Motion. We publish 40 terms with definitions for this unit, all of them on this page.
How should I study Calculus AB Unit 4?
Read the 6 lessons below first — about 85 minutes — then drill the 40 terms in cram mode until you can produce each definition from memory rather than just recognize it. Recognition is what makes a unit feel finished when it is not. Finish with practice questions and read the explanation for every one you get right by elimination as well as the ones you miss.
All 8 units of AP Calculus AB
Unit names, topics and exam weights follow the published College Board course framework for AP Calculus AB. AP® is a trademark registered by the College Board, which does not endorse this site.