Contextual Applications of Differentiation

What this unit covers

The topics below follow the published Calculus BC course framework for Unit 4. This unit is worth 5–10% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• Motion Along a Line14 min · 3 objectivesRelate position, velocity, and acceleration through differentiation · Determine when a particle is speeding up or slowing down · Find total distance traveled versus displacement
• Related Rates15 min · 3 objectivesSet up a related-rates problem by relating variables with an equation · Differentiate an equation implicitly with respect to time · Solve for an unknown rate at a specific instant
• Linear Approximation and Its Error14 min · 3 objectivesWrite the tangent-line approximation to f at a point and use it to estimate a nearby value · Decide from concavity whether a linear approximation over- or underestimates · Interpret a derivative in context with correct units
• L’Hôpital’s Rule and Indeterminate Forms15 min · 3 objectivesVerify that a limit has an indeterminate form before applying L’Hôpital’s Rule · Apply the rule repeatedly when the quotient of derivatives is still indeterminate · Convert the forms 0·∞, ∞ − ∞ and 1^∞ into a quotient the rule can handle

Formulas in Unit 4

Every term in Unit 4

All 24 terms we publish for Contextual Applications of Differentiation, 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.

Justification with the Mean Value Theorem

State continuity on the closed interval and differentiability on the open one before invoking the conclusion.

Optimization with a constraint

Reduce to one variable using the constraint, differentiate, and justify the extremum with a sign change or the second derivative.

Related rates with implicit relations

Differentiate the relating equation with respect to time before substituting instantaneous values.

Linearization and error

The tangent line overestimates a concave-down function and underestimates a concave-up one, which is how the direction of the error is justified.

Units of a derivative

Units of y per unit of x, read straight off the notation. Wrong units mean a wrong setup — a five-second diagnostic.

Units of an integral of a rate

Rate units times the units of the variable. Gallons per hour integrated over hours gives gallons.

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 rate

Write "decreasing at a rate of 3.2 units per minute", never "decreasing at −3.2", which states the opposite.

Rate, change, amount

A derivative is a rate; the integral of a rate is a change; an amount is an initial value plus that change.

How fast versus how much

"At what rate" wants a derivative. "By how much did it change" wants an integral of a rate. "How much is there" wants both.

The rate is decreasing

A statement about the second derivative. The quantity may still be increasing while its rate of increase slows.

Increasing at a decreasing rate

f′ > 0 with f″ < 0 — rising while flattening. The most common applied case and the phrasing to recognize instantly.

Decreasing at a decreasing rate

f′ < 0 with f″ > 0 — falling while slowing, the shape of cooling or of decay toward an asymptote.

Differentiate before substituting

Substituting an instantaneous value first turns a variable into a constant, whose derivative is zero — destroying the term the problem is about.

Use a constraint to remove a variable

A cone with two varying lengths gives two unknown rates. Similar triangles supply r = h/2, which reduces it to one.

Rates carry signs

Draining is dV/dt < 0; a shrinking radius is dr/dt < 0. Put the sign in at substitution, not onto the final answer.

A rate need not be constant

From dV/dt = (πh²/4)(dh/dt), a fixed dV/dt makes dh/dt inversely proportional to h² — the level falls faster as the cone empties.

Find the missing length at the instant

Use the relation itself. In a ladder problem y comes from the Pythagorean relation with the given x, never from an earlier part.

Linearization

f(x) ≈ f(a) + f′(a)(x − a). Dropping the (x − a) factor treats a rate as a change.

Over- or under-estimate of a tangent line

Concave up puts the tangent below the curve, so the estimate is low. Justify with the sign of f″.

Speed is not a derivative

Speed = |v|, the magnitude of velocity. A particle with velocity −7 has speed 7.

Changing direction requires a sign change

v = 0 is not enough. For v = (t − 3)² the particle pauses and continues the same way.

Speeding up test

v·a > 0. Same signs means speeding up regardless of direction — including v < 0 with a < 0.

Marginal cost

C′(x) approximates the cost of the next unit, in dollars per unit. A rate, not a total.

What examiners penalize here

• Read motion questions carefully: "how far from start" (displacement, signed) and "total distance traveled" (unsigned) are different. For total distance, always locate where v = 0, split the interval, and sum the absolute values of the pieces.
• Related-rates free-response answers must carry units and be evaluated at the stated instant. Write the differentiated equation, then substitute — showing the un-substituted derivative first is what earns the setup points even if arithmetic slips.
• A complete over/under justification names three things: the sign of f″, the concavity that follows, and the position of the tangent line relative to the curve. Write all three.
• BC uses L’Hôpital far beyond limit questions: it is how you show a series term goes to zero, and how improper integrals are evaluated at a bound. The form-checking habit pays off in Units 6 and 10 as much as here.

Practice Calculus BC

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

All 10 units of AP Calculus BC

1. Unit 1 · Limits & Continuity
2. Unit 2 · Differentiation: Definition & Fundamental Properties
3. Unit 3 · Differentiation: Composite, Implicit & Inverse Functions
4. Unit 4 · Contextual Applications of Differentiation
5. Unit 5 · Analytical Applications of Differentiation
6. Unit 6 · Integration & Accumulation of Change
7. Unit 7 · Differential Equations
8. Unit 8 · Applications of Integration
9. Unit 9 · Parametric, Polar & Vector-Valued Functions
10. Unit 10 · Infinite Sequences & Series