Contextual Applications of Differentiation

Unit Overview

Having learned how to differentiate, this unit is about what derivatives mean in the real world. A derivative is a rate of change, and interpreting it correctly — with units and direction — is a core AP skill. You will study motion (position, velocity, acceleration), related rates (how linked quantities change together in time), linearization (using a tangent line to approximate), and L'Hôpital's Rule (using derivatives to resolve indeterminate limits).

This unit is heavy on interpretation and problem set-up. On the exam it is a frequent source of free-response questions.

Big picture: A derivative carries units of output per unit of input and tells you both how fast and which direction a quantity is changing.

Learning Objectives

• Interpret a derivative's value, sign, and units in context.
• Analyze rectilinear motion: relate position, velocity, acceleration, speed, and displacement.
• Set up and solve related-rates problems.
• Use local linearity / linearization to approximate function values.
• Apply L'Hôpital's Rule to indeterminate forms 0/0 and ∞/∞.

Prerequisite Concepts

• Fluent differentiation, especially the Chain Rule (Unit 3) — essential for related rates.
• Geometry formulas (area, volume, Pythagoras, similar triangles).
• Indeterminate forms from Unit 1.

Key Definitions

• Rate of change in context: f'(a) has units of (units of f) per (unit of x); its sign tells direction (increasing/decreasing).
• Velocity: v(t) = s'(t). Acceleration: a(t) = v'(t) = s''(t). Speed: |v(t)|.
• Displacement: net change in position. At rest: v(t) = 0.
• Related rates: quantities linked by an equation, each changing with time; differentiate the relation with respect to t.
• Linearization of f at a: L(x) = f(a) + f'(a)(x − a) (the tangent-line approximation).
• Indeterminate form: an expression like 0/0 or ∞/∞ whose value is not determined by the parts alone.

Important Theorems / Rules

L'Hôpital's Rule: If lim f/g gives 0/0 or ∞/∞ and the limits of f' and g' behave, then lim f(x)/g(x) = lim f'(x)/g'(x). Check the indeterminate form before applying; a form like 1/0 is not indeterminate.

Speeding up / slowing down: a particle is speeding up when v and a have the same sign, and slowing down when they have opposite signs.

Important Theorems / Rules

L'Hôpital's Rule: If lim f/g gives 0/0 or ∞/∞ and the limits of f' and g' behave, then lim f(x)/g(x) = lim f'(x)/g'(x). Check the indeterminate form before applying; a form like 1/0 is not indeterminate.

Speeding up / slowing down: a particle is speeding up when v and a have the same sign, and slowing down when they have opposite signs.

Important Formulas

• Linearization: L(x) = f(a) + f'(a)(x − a); differential dy = f'(x) dx.
• Common related-rates relations (differentiate w.r.t. t):
  • Circle area A = πr² → dA/dt = 2πr (dr/dt)
  • Sphere volume V = (4/3)πr³ → dV/dt = 4πr² (dr/dt)
  • Cube volume V = s³ → dV/dt = 3s² (ds/dt)
  • Right-triangle / Pythagoras x² + y² = z² → 2x(dx/dt) + 2y(dy/dt) = 2z(dz/dt)

Concept Explanations

Interpreting derivatives (4.1). Always attach units and read the sign. If T(t) is temperature in °C after t minutes, then T'(5) = −2 means "cooling at 2 °C per minute at t = 5." A second derivative describes how the rate itself is changing (e.g., increasing at a decreasing rate).

Motion (4.2). Differentiate position to get velocity, again to get acceleration. Watch the distinction between displacement (integral/net of velocity, can be negative) and total distance (accounts for turnarounds where v = 0). Use the same-sign rule to decide speeding up vs. slowing down.

Related rates (4.4–4.5) — the standard method:

1. Draw and label; identify which rates are given and which is wanted.
2. Write an equation relating the variables (geometry).
3. Differentiate both sides with respect to t (Chain Rule everywhere).
4. Substitute the known instantaneous values last — never before differentiating.
5. Solve for the unknown rate; include units.

Linearization (4.6). Near x = a, a smooth curve looks like its tangent line. Use L(x) to approximate hard values (e.g., √4.1 ≈ 2.025). For a concave-up function the tangent lies below the curve, so the estimate is an underestimate; concave-down gives an overestimate.

L'Hôpital's Rule (4.7). When substitution yields 0/0 or ∞/∞, differentiate top and bottom separately and re-evaluate; repeat if still indeterminate. It elegantly recovers the fundamental limits, e.g. lim_{x→0} sin x / x = lim cos x / 1 = 1.

Worked Examples

Example 1 (Motion — distance). For s(t) = t³ − 6t² + 9t + 1, find total distance on [0, 2]. v(t) = 3(t−1)(t−3), so it turns at t = 1. s(0) = 1, s(1) = 5, s(2) = 3. Distance = |5 − 1| + |3 − 5| = 4 + 2 = 6.

Example 2 (Speeding up?). v(t) = t² − 4, a(t) = 2t. At t = 3: v = 5 > 0 and a = 6 > 0 — same sign — so the particle is speeding up.

Example 3 (Related rates — balloon). Air enters a sphere at 100 cm³/s. How fast is r growing when r = 5? dV/dt = 4πr²(dr/dt) → 100 = 4π(25)(dr/dt) = 100π(dr/dt) → dr/dt = 1/π cm/s.

Example 4 (Related rates — ladder). A 10-ft ladder's base slides out at 2 ft/s. When the base is 6 ft out, the top is 8 ft up. 2x·x' + 2y·y' = 0 → 2(6)(2) + 2(8)(y') = 0 → y' = −1.5 ft/s (sliding down).

Example 5 (Linearization). Approximate √4.1 with f(x) = √x at a = 4: f'(4) = 1/4, so L(4.1) = 2 + (1/4)(0.1) = 2.025.

Example 6 (L'Hôpital). lim_{x→0} (1 − cos x)/x² = (0/0) → lim sin x/(2x) = (0/0) → lim cos x/2 = 1/2.

Graphical Interpretation

• The velocity graph's sign shows direction of motion; its zeros mark turning points; its own slope is acceleration.
• Linearization is literally the tangent line; the approximation error grows with distance from a and with the curve's concavity.
• A quantity increasing at a decreasing rate has f' > 0, f'' < 0 — rising but leveling off.

Common Mistakes

• Omitting units or misreading the sign of a rate in context.
• Confusing displacement with total distance (forgetting to split at v = 0).
• Substituting numbers before differentiating in related rates — this freezes variables that are actually changing.
• Forgetting the Chain Rule (dr/dt, dh/dt, …) when differentiating a related-rates relation.
• Applying L'Hôpital to a non-indeterminate form (e.g., 1/0), or forgetting to re-check the form after one application.

AP Exam Tips

• Related-rates and motion questions are free-response staples — practice clean set-ups and always state units.
• Justify "speeding up/slowing down" by comparing the signs of v and a explicitly.
• For approximations, state the linearization and note whether it's an over- or under-estimate using concavity.
• Confirm the indeterminate form in writing before using L'Hôpital's Rule.

Unit Summary

• A derivative in context has units and a sign — interpret both.
• Motion: v = s', a = v'; speed = |v|; distance splits at v = 0; same-sign v, a ⇒ speeding up.
• Related rates: relate → differentiate in t → substitute last.
• Linearization L(x) = f(a) + f'(a)(x − a) approximates near a; concavity decides over/under.
• L'Hôpital's Rule resolves 0/0 and ∞/∞ by differentiating top and bottom.

Next unit: use derivatives to analyze the shape of graphs — extrema, concavity, and optimization.