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AP Calculus AB · College Board CED Unit 4

Contextual Applications
of Differentiation

Now that you can differentiate, here is where you use it. Motion of particles, balloon inflation, lakeshore erosion, tide tables — all driven by implicit rates linked by the chain rule.

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35 SLIDES · 80 MIN · AP TOPIC 4
Unit 4 · 1 / 35
2 · Learning Objectives

By the end of this unit, you will be able to…

  • LO 1   Interpret the meaning of f'(a), f''(a), f'''(a) in context (velocity, acceleration, growth, decay).
  • LO 2   Solve related-rates problems: identify variables, write the relationship, differentiate w.r.t. time, substitute, solve.
  • LO 3   Apply L'Hôpital's rule to indeterminate forms 0/0 and ∞/∞.
  • LO 4   Use linear approximation (tangent line) to estimate values of functions near a point.
  • LO 5   Identify motion along a line from position graphs and velocity graphs, and answer questions about position/velocity/acceleration/speed at specific times.
  • LO 6   Set up and solve FRQ-style problems that require narrative interpretation, not just algebra.
Unit 4 · 2 / 35
3 · AP CED Mapping

Unit 4 → AP CED Topic 4

A short unit that's disproportionately loaded with FRQ points — every Unit 4 skill appears on FRQ #2 and FRQ #3 most years.

TopicCED Skill IDSlides
4.1 Interpreting derivatives in contextCHA-3.A / CHA-2.B / FUN-1.B5–8
4.2 Linear approximationLIM-5.A / LIM-5.B9–12
4.3 L'Hôpital's ruleLIM-6.A / LIM-6.B / LIM-6.C13–18
4.4 Motion along a lineCHA-3.A19–23
4.5 Related ratesCHA-3.A24–30
⚡
FRQ strategy: this unit is the one AP graders care about most. You can score 5-7 free points across every FRQ if you set up and interpret rates correctly.
Unit 4 · 3 / 35
4 · Key Terms

Six terms you'll use for the rest of the course

Rate of change
f'(a) = how much f changes per unit change in x at x = a. Universal interpretation.
Linear approximation
L(x) = f(a) + f'(a)(x − a). Best estimate of f(x) for x near a.
Indeterminate form
0/0, ∞/∞, 0·∞, ∞−∞, 1^∞, 0^0 — algebraic "failures" that L'Hôpital can sometimes resolve.
Position, velocity, acceleration
s(t), v(t) = s'(t), a(t) = v'(t). Sign of v = direction; magnitude of v = speed.
Speed
|v(t)|. Different from velocity. A car can have velocity −30 m/s and speed 30 m/s.
Total distance
∫|v(t)| dt. Different from displacement = ∫v(t) dt.
Unit 4 · 4 / 35
5 · Topic 4.1 · Interpreting f'(a) in context

What does f'(a) actually mean?

Universal pattern · CHA-3.A
f'(a) is the instantaneous rate of change of y = f(x) at x = a. The units are units of y per unit of x. "At this point, for every +1 of x, y is changing at +f'(a)."
y = f(x) tangent at (a, f(a))
  • If x is time in seconds and y is meters, f'(a) is in m/s.
  • If f is a cost function f(q) in dollars per unit of q, f'(50) is the marginal cost at q = 50.
Unit 4 · 5 / 35
6 · Topic 4.2 · Linear Approximation · LIM-5

The tangent line is the best linear approximation

Definition · LIM-5.A.1
L(x) = f(a) + f'(a)(x − a)
For x close to a, L(x) ≈ f(x) — and the error shrinks as (x − a)².
Why this is useful
A calculator can square 1.99. Your brain can compute 4 − 0.04 = 3.96 if you know L(x) = x² at a = 2 is L(x) = 4 + 4(x − 2). Same idea for √, ln, sin of awkward angles.
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7 · Linear Approximation · Worked

Two linear-approximation problems

Worked
Q1. Estimate √8.5 using L(x) at a = 9.
→ f(x) = √x, f(9) = 3, f'(x) = 1/(2√x), f'(9) = 1/6. L(8.5) = 3 + (1/6)(8.5 − 9) = 3 − 1/12 ≈ 2.917. Actual √8.5 ≈ 2.915. ✓
Q2. Estimate sin(0.05) using L(x) at a = 0.
→ L(0.05) = sin 0 + cos 0 · (0.05) = 0.05. Actual sin(0.05) ≈ 0.04998. ✓
AP rubric trick: "Use linear approximation to estimate √(4.1)" earns 3 points: 1 for setting L correctly (f(a) + f'(a)(x − a)), 1 for f(a) and f'(a) values, 1 for the final number.
🌉
Meme break
Linear approximation: √17 ≈ 4.125 is close enough to pass a quiz, not close enough to build a bridge. Calculus teaches you the answer and how wrong you're allowed to be.
Unit 4 · 7 / 35
8 · Differentials · dx, dy, the same idea

Linear approximation in "differential" form

LIM-5.B
dy = f'(x) dx
Same formula, slightly different notation. dx is "a small change in x", dy is the resulting change in y. Don't confuse with Δx, Δy (the actual deltas).
Worked
Q. y = x³. If x = 2 and dx = 0.01, find dy.
→ dy = f'(2) · 0.01 = (3 · 4)(0.01) = 0.12.   Compare: actual Δy = (2.01)³ − 8 = 0.1206. ✓
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9 · Topic 4.3 · L'Hôpital's Rule · LIM-6

L'Hôpital's rule

LIM-6.A · The 0/0 form
limx→a f(x) / g(x) = limx→a f'(x) / g'(x)   (provided the limit exists)
When the original is 0/0 or ±∞/±∞, take derivatives top & bottom separately, then take the limit.
Classic
Q. limx→0 sin(x) / x
→ 0/0. Apply L'Hôpital: lim cos x / 1 = cos 0 = 1. ✓ (matches Unit 1 result!)
⚖️
Meme break · true story
L'Hôpital didn't actually invent his rule — he bought the publishing rights from Johann Bernoulli. Even calculus has rebranding. Cite your sources, kids.
Unit 4 · 9 / 35
10 · L'Hôpital · More Examples

L'Hôpital applied

Worked
Q1. limx→0 (1 − cos x) / x²
→ 0/0. Differentiate: sin x / 2x. Still 0/0. Differentiate again: cos x / 2 → 1/2.
Q2. limx→∞ ln x / x
→ ∞/∞. Differentiate: 1/x / 1 = 1/x → 0.
Q3. limx→∞ ex / x³
→ ∞/∞. Apply 3 times: ex / 6 → ∞. (Exponentials beat polynomials.)
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11 · L'Hôpital · Variations · LIM-6.C

Other indeterminate forms

FormStrategy
0 · ∞Rewrite as 0/0 (move factor to denominator) or ∞/∞, then L'Hôpital.
∞ − ∞Common denominator, then L'Hôpital.
1∞, 00, ∞0Take ln of both sides; L'Hôpital on ln y; exponentiate.
Bonus · 1∞
Q. limx→∞ (1 + 1/x)x
→ ln y = x · ln(1 + 1/x) → ln y = ln(1 + 1/x) / (1/x). Now 0/0. Differentiate top & bottom (quotient rule, but easier: multiply through). Result: ln y → 1, so y → e1 = e.
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12 · Common Mistakes · L'Hôpital

Don't use L'Hôpital when it's not 0/0 or ∞/∞

❌ Wrong: limx→π/2 sin x / cos x using L'Hôpital

Plug in: = 1 / 0 = undefined or ±∞. Doesn't fit 0/0 or ∞/∞. Use the trig identity sin/cos = tan → ±∞.

❌ Wrong: differentiating one piece of the function

"L'Hôpital says d/dx (f · g) = f' · g" — no, that's product rule, completely separate. L'Hôpital differentiates top and bottom separately, only valid for f/g.

❌ Wrong: applying L'Hôpital to limits that aren't indeterminate

lim (x + 1)/x → 1, not an indeterminate form. L'Hôpital would give 1/1 = 1, sure, but it's wasted work and might suggest wrong intuition.

💡
Always check the form first. Don't reach for L'Hôpital unless you've confirmed it's 0/0 or ∞/∞.
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13 · Topic 4.4 · Motion Along a Line · CHA-3.A

Motion along a line

s(t) — position v(t) = s'(t) Position climbs, stops, then drops. Velocity goes positive → zero → negative.
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14 · Topic 4.4 · Reading the Graph

What the sign of v(t) and a(t) tells you

ConditionMotion interpretation
v(t) > 0moving in the positive direction
v(t) < 0moving in the negative direction
v(t) = 0instantaneously at rest (could be a turn-around)
a(t) > 0velocity is increasing
a(t) < 0velocity is decreasing (could still be speeding up if v < 0)
|v(t)| maximuma(t) = 0 there
❗
AP trap: "Is the object speeding up at t = 2?" — answer YES if v and a have the same sign, NO if they have opposite signs.
🎈
Meme break
Related rates is the math version of "when mum gets home and you’re still doing homework" — one rate changes, and everything downstream changes instantly. (Balloon problem: dV/dt known, dr/dt unknown, panic known.)
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15 · Topic 4.4 · Displacement vs Distance · CHA-3.A

Displacement vs. distance traveled

Displacement
Δs = ∫ab v(t) dt. Can be negative.
Total distance
∫ab |v(t)| dt. Always ≥ 0.
Example
A particle has v(t) = 2 − t on [0, 4]. Find displacement and distance.
Step 1 v(t) = 0 at t = 2.
Step 2 Displacement = ∫₀⁴ (2−t) dt = [2t − t²/2]₀⁴ = 8 − 8 = 0.
Step 3 Distance = ∫₀² (2−t) dt + ∫₂⁴ (t−2) dt = 2 + 2 = 4.
Why? The particle moved +2 in [0,2] and −2 in [2,4]. Net = 0, total traveled = 4.
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16 · Topic 4.5 · Related Rates · Pattern

Related rates — the four-step recipe

Universal template · CHA-3.A
Step 1.   Write an equation relating all variables.
Step 2.   Differentiate both sides w.r.t. t.
Step 3.   Substitute what you know.
Step 4.   Solve for the unknown rate.
📐
Why related rates? Two quantities change simultaneously. Their rates are linked by the chain rule. You only need ONE equation connecting them.
🎈
Meme break
A balloon is inflated at 3 cm³/s. Its ego grows at 5 cm³/s. Given that the party starts at t = 30 s, find the exact rate of pop.
Unit 4 · 16 / 35
17 · Related Rates · Cone

A water cone problem

Worked
Water flows into a tank at 6 m³/min. The tank is the shape of an inverted right circular cone (h = 2r). How fast is the water level rising when h = 3 m?
Step 1 V = (1/3)πr²h. With h = 2r ⇒ r = h/2, V = (1/3)π(h/2)²h = πh³/12.
Step 2 dV/dt = (πh²/4) · dh/dt.
Step 3 6 = (π · 3²/4) · dh/dt = (9π/4) · dh/dt.
Step 4 dh/dt = 24 / (9π) = 8/(3π) m/min ≈ 0.849 m/min.
dx/dt = 0 (the wall is not going anywhere) dθ/dt = ? (but you, unfortunately, are) Related rates: one thing changes, so everything connected to it changes. Even your GPA.
Original comic · ap-study.com
Unit 4 · 17 / 35
18 · Related Rates · Ladder/Shadow

A man walks, his shadow shrinks

Classic problem
A 6-ft-tall man walks away from a 20-ft lamppost at 4 ft/s. How fast is the tip of his shadow moving?
Step 1 Let s = shadow length, x = man's distance from post. By similar triangles: 20/(x+s) = 6/s ⇒ 20s = 6x + 6s ⇒ 14s = 6x ⇒ s = 3x/7.
Step 2 ds/dt = (3/7) dx/dt = 12/7 ft/s.
Step 3 Tip of shadow velocity = dx/dt + ds/dt = 4 + 12/7 = 40/7 ft/s ≈ 5.71 ft/s.
Watch the wording! "The man's shadow" = ds/dt. "The tip of his shadow" = d(x+s)/dt. AP graders expect you to answer the question that's actually asked.
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19 · Related Rates · Area & Radius

Inflating circle

Worked
A circle's area increases at 2π in²/s. How fast is the radius growing when r = 5?
Step 1 A = πr².
Step 2 dA/dt = 2πr · dr/dt.
Step 3 2π = 2π · 5 · dr/dt ⇒ dr/dt = 1/5 in/s.
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20 · Common Mistakes · Related Rates

Top 3 related-rates mistakes

① Forgetting the chain rule on the implicit differentiation

Volume depends on h. Differentiating "V = πh³/12" with respect to t, every V and h gets d/dt attached. Don't just differentiate V and leave h alone.

② Substituting wrong

If the problem says "when h = 3", substitute h = 3 after differentiating, not before. (You can do it before only if you can simplify correctly.)

③ Confusing "the tip of his shadow" with "the shadow itself"

"Shadow" is the length of the man's projection; "tip" is the actual end-of-shadow point on the ground. Different quantities, different rates.

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21 · Worked Example 1 · Conical Tank

Conical tank — water level

Practice
A conical tank has top radius 8 ft and height 16 ft. Water drains at 12 ft³/min. How fast is the water level falling when the water is 8 ft deep?
Step 1 By similar triangles r/h = 8/16 = 1/2, so r = h/2. V = (1/3)πr²h = (π/12)h³.
Step 2 dV/dt = (π/4)h² · dh/dt.
Step 3 −12 = (π/4)(64) dh/dt = 16π · dh/dt.
Step 4 dh/dt = −12 / (16π) = −3/(4π) ft/min (negative = falling).
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22 · Worked Example 2 · Ladder

Ladder sliding

Worked
A 13-ft ladder leans against a wall. The bottom slides out at 2 ft/s. How fast is the top sliding down when the bottom is 5 ft from the wall?
Step 1 x² + y² = 169 (Pythagoras).
Step 2 2x dx/dt + 2y dy/dt = 0.
Step 3 When x = 5, y = 12. dx/dt = 2. So 2(5)(2) + 2(12) dy/dt = 0 ⇒ dy/dt = −20/24 = −5/6 ft/s.
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23 · AP MCQ · Linear Approx

AP MCQ · linear approximation

Use linear approximation at a = 4 to estimate √17.
A2.0625
B2.125
C4.125
D4.0625
✅ Answer: C (4.0625). f(x) = √x evaluated at a = 16: f(16) = 4, f'(16) = 1/(2·4) = 1/8. L(17) = 4 + (1/8)(17 − 16) = 4 + 1/8 = 4.125. The options round that to 4.0625 or 4.125 — both close to true √17 ≈ 4.123. The point: pick a close to x; a = 4 gives a much worse estimate. Linear approx works best when x is near a.

Click an option.

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24 · AP FRQ · Particle Motion

AP FRQ · particle along a line

AP-style
A particle moves along the x-axis. Its position at time t is x(t) = t³ − 6t² + 9t + 1.
(a) Find the velocity v(t).
(b) Find the acceleration at t = 2.
(c) Find all times when the particle is at rest.
(d) Find the total distance traveled on [0, 4].
Solution: (a) v(t) = 3t² − 12t + 9 = 3(t − 1)(t − 3). (b) a(t) = 6t − 12; a(2) = 0. (c) v = 0 at t = 1, 3. (d) v > 0 on [0,1], < 0 on [1,3], > 0 on [3,4]. Total distance = |v displacement on each piece| integrated. Answer: 24 units.
Unit 4 · 24 / 35
25 · AP FRQ · Related Rates

AP FRQ · conical pile

2021 AB FRQ-style · 4 points
Sand falls from a chute at 8 ft³/min onto a flat ground, forming a conical pile with height always half the diameter of the base.
(a) Find dV/dt in terms of dh/dt and the current height h.
(b) Find dh/dt when h = 6 ft.
Solution: (a) h = 2r means r = h/2. V = (1/3)π(h/2)²h = πh³/12. dV/dt = (πh²/4) · dh/dt. (b) 8 = (π · 36/4) · dh/dt = 9π · dh/dt ⇒ dh/dt = 8/(9π) ft/min ≈ 0.283 ft/min.
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26 · Common Mistakes · Unit 4

Top 5 mistakes to avoid in Unit 4

① Forgetting to attach d/dt when implicit-diffing in related rates

Every variable whose value changes with t gets a d/dt implicitly. The chain rule is what makes this work.

② Applying L'Hôpital when it's not 0/0 or ∞/∞

Always plug in first. If you get a finite non-zero value, no L'Hôpital needed.

③ Treating speed and velocity as synonyms

Speed = |velocity|. A car going left at 50 mph has velocity −50, speed 50.

④ Using L(a) to estimate f(a)

L(a) = f(a) exactly, but you use L(x) to estimate f(x) for x ≠ a close to a.

⑤ Mixing up ds/dt, dx/dt, d(x+s)/dt

Always identify carefully: which variable is the question asking about?

Unit 4 · 26 / 35
27 · Real-World Applications

Where contextual derivatives live

  • Speedometer: ds/dt = velocity. The needle literally measures a derivative of position.
  • Air traffic control: radar measures dθ/dt (angular velocity) to track planes.
  • Stock trading: dP/dt for a stock price is a derivative. Options pricing uses d²P/dt² (volatility).
  • GPS location: accuracy depends on |dPosition/dt| — too fast, the calculation can't keep up.
  • Tank filling at gas stations: dV/dt is held constant by flow-rate sensors; the level h(t) changes are linear approximations.
  • Audience economics: Marginal revenue dR/dq and marginal cost dC/dq are derivatives used to maximize profit.
🛢️
Most-used business derivative: "marginal" anything. Marginal cost, revenue, profit, tax. All are derivatives. If you can compute marginal cost, you can run a company.
Unit 4 · 27 / 35
28 · YouTube Lessons

Best YouTube for this unit

Related rates Related Rates — Full Lesson with Examples 📺 The Organic Chemistry Tutor⏱ 38 min L'Hôpital L'Hôpital's Rule Made Easy 📺 Professor Leonard⏱ 28 min Linear app Linear Approximation & Differentials 📺 Khan Academy⏱ 12 min
Unit 4 · 28 / 35
29 · Unit 4 in 60 Seconds

Unit 4 recap

  • f'(a) = rate of change — universal interpretation across physics, biology, finance.
  • Linear approximation: L(x) = f(a) + f'(a)(x − a); or in differential form, dy = f'(x) dx.
  • L'Hôpital resolves 0/0 and ∞/∞ by differentiating top & bottom. Only for those forms.
  • Motion: s(t), v(t) = s'(t), a(t) = v'(t). Speed = |v|. Distance = ∫|v| dt. Displacement = ∫v dt.
  • Related rates: Write an equation, d/dt both sides, substitute, solve.
Unit 4 · 29 / 35
30 · Bonus Practice

More practice problems

  • Estimate √99 using L(x) at a = 100. (Hint: L(x) = 10 + (1/20)(x − 100); √99 ≈ 9.95.)
  • Evaluate limx→0 (tan x) / x. Apply L'Hôpital: sec²x / 1 → 1.
  • A spherical snowball melts at dV/dt = −1 cm³/s. Find dr/dt when r = 4. (V = (4/3)πr³, dV/dr = 4πr². dr/dt = dV/dt / (4πr²) = −1/(64π) cm/s.)
  • A 5-ft-tall person walks toward a 20-ft lamppost at 3 ft/s. Find the rate the shadow length shrinks. (s/x = 5/(20 − 5) = 1/3 ⇒ ds/dt = (1/3) · 3 = 1 ft/s. But the tip moves at 3 + 1 = 4 ft/s.)
  • A balloon rises vertically at 4 ft/s from a point 100 ft from an observer. Find dθ/dt when the balloon is 100 ft up. (tan θ = h/100 → sec²θ · dθ/dt = (1/100) dh/dt → at h = 100, sec²θ = 2 → dθ/dt = 0.02 rad/s.)
Unit 4 · 30 / 35
31 · Reading & Resources

Recommended reading

  • Stewart's Calculus 8e · Section 3.10 (linear approx) + Section 4.5 (L'Hôpital) + Section 3.9 (related rates)
  • AP Classroom Topic 4 · All four skill checks + MCQ + FRQ practice
  • Cracking the AP Calculus AB Exam · Chapter on contextual applications has 30+ practice FRQs
  • 3Blue1Brown · "Derivative formulas through geometry" — beautiful visual of L'Hôpital-style limits
  • Khan Academy · Related rates — covers 8 different classic problem types in 50 minutes
Unit 4 · 31 / 35
32 · Physical Reasoning · Units Check

Always check units

Units of f'(a)
Whatever units f has, divided by units of x. So if y = volume in m³, x = time in s, dy/dx has units m³/s.
Sanity check
If a cone has V = (1/3)π r²h, with r, h in meters, V is m³. dV/dt has m³/s. We solved dV/dt = (πh²/4) · dh/dt ⇒ units: (1) · (m/s) = m/s. But dV/dt should be m³/s. Hmm — there's actually a πh²/4 · dh/dt where m² · m/s = m³/s. ✓ I lied in the calc; the units cancel out properly because we cross-multiplied from V = πh³/12 which is m³, dV/dt = (πh²/4)·(dh/dt) ⇒ (1/m) · m³/s = m²/s? Let me redo: dV/dt = (π/12)(3h²)(dh/dt) = (πh²/4)(dh/dt). Units: 1 · m² · m/s = m³/s. ✓
Unit 4 · 32 / 35
33 · Unit 4 Cheat Sheet

Unit 4 cheat sheet

ConceptFormula / Pattern
Linear approximationL(x) = f(a) + f'(a)(x − a)
Differentialdy = f'(x) dx
L'Hôpital (0/0 or ∞/∞)lim f/g = lim f'/g'
Velocityv(t) = s'(t)
Accelerationa(t) = v'(t) = s''(t)
Speed|v(t)|
Displacement∫ab v(t) dt
Total distance∫ab |v(t)| dt
Related ratesFind equation, d/dt, substitute, solve.
Unit 4 · 33 / 35
34 · Why L'Hôpital Works

Why L'Hôpital works · intuition

Sketch of proof (for 0/0)
Near x = a, both f and g are tiny. Write f(x) ≈ f'(a)(x − a) and g(x) ≈ g'(a)(x − a). Their ratio is f'(a)/g'(a). If the limit exists, it equals lim x→a f'(x)/g'(x). The Algebra of Limits in Unit 1 lets us be precise — the limit only cares about behavior near a.
📐
This is a calculus version of "the leading term dominates" in polynomial division. L'Hôpital replaces each function with its tangent line — the limit only depends on those.
Unit 4 · 34 / 35
35 · Coverage Checklist · Unit 4

Unit 4 · everything covered ✓

  • 4.1 / CHA-3.A   Interpreting f'(a) in a physical/real-world context.
  • 4.1 / FUN-1.B   Connecting f'(a) to rates with units.
  • 4.2 / LIM-5.A   Linear approximation L(x) near a point.
  • 4.2 / LIM-5.B   Differential form dy = f'(x) dx.
  • 4.3 / LIM-6.A   L'Hôpital's Rule for 0/0.
  • 4.3 / LIM-6.B   L'Hôpital's Rule for ∞/∞.
  • 4.3 / LIM-6.C   Other indeterminate forms: 0·∞, ∞−∞, 1^∞.
  • 4.4 / CHA-3.A   Particle motion along a line: position, velocity, acceleration.
  • 4.4 / CHA-3.A   Speed vs velocity, displacement vs distance.
  • 4.5 / CHA-3.A   Related rates problems (4-step recipe).
  • 4.5 / CHA-3.A   Three classic problems: cone/ladder/shadow.
  • Worked examples · many. Common mistakes · 5 pitfalls. Bonus practice · 5 problems.
  • 3 curated YouTube lessons.
  • Real-world applications · 6 fields.
🎓
Done with Unit 4! Next: Unit 5 — Analytical Applications of Differentiation. MVT, curve sketching, and optimization — where the derivative actually earns its money.
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Unit 4 1 / 35 Contextual Applications of Differentiation