unit-4 – HighFiveAP

4.1

Interpreting the Meaning of the Derivative FRQ Language

The derivative f'(x) is an instantaneous rate of change. On FRQs, you’re usually graded on your sentence, not your computation.

FRQ “Mad Libs” Template (copy this)

At x = c, the [context quantity] is [increasing / decreasing] at a rate of [value] [units of y] per [units of x].

Score saver: if f'(c) is negative, say “decreasing at magnitude” (don’t say “decreasing at −3 gallons/min”).

Graph: Tangent Line = Instantaneous Rate

Tangent vs. Secant (why derivative is “instantaneous”)

As Q moves toward P, the secant slope approaches the tangent slope → that limit is the derivative.

AP Exam Moves

State the input (x=c), the quantity, the direction, and units.
“Rate of change” must be written as y-units per x-unit.

4.2

Particle Motion Classic FRQ

AP loves connecting position, velocity, and acceleration — plus interpreting “moving right/left,” “speeding up,” and “stopped.”

The Chain (memorize)

v(t)=s'(t) and a(t)=v'(t)=s''(t)

Position → derivative is velocity → derivative is acceleration.

Graphs: Moving Right/Left + Speeding Up/Slowing Down

Case A: Speeding Up (v and a same sign)

If position is increasing and concave up, slope is positive and increasing → speed increases.

Case B: Slowing Down (v and a opposite signs)

Slope stays positive, but decreases → moving right while slowing down.

Question	Use
Moving right/left?	Sign of v(t)
Stopped?	v(t)=0
Speeding up/slowing down?	Compare signs of v(t) and a(t)

Common Mistake

Negative velocity does NOT mean slowing down. It means moving left. Speeding/slowing needs both v and a.

4.3

Rates of Change in Context Units

Unit 4 rates are often “a rate of a rate.” Your job is to match the derivative to a sentence and unit.

Common “Derivative Translations”

Given	Meaning	Units
f'(x)	Instantaneous rate of change of f w.r.t. x	(y-units)/(x-units)
f''(x)	Rate of change of the rate of change	(y-units)/(x-units)2

Graph: f'' tells “rate is increasing or decreasing”

Concave Up (f'' > 0)

The tangent slopes get steeper as x increases → the rate of change is increasing.

Concave Down (f'' < 0)

The tangent slopes flatten as x increases → the rate of change is decreasing.

4.4 - 4.5

Related Rates Chain Rule in Disguise

Related rates problems link multiple changing quantities. The trick is: write a geometry equation first, then differentiate with respect to time.

Graph/Diagram: Ladder Sliding (the most common picture)

Right triangle model (x, y change over time)

Write the equation first (x² + y² = 100), then differentiate w.r.t. time: 2x·dx/dt + 2y·dy/dt = 0.

The 4-Step Protocol (do this every time)

Sketch & Label: constants as numbers, variables as letters.
Equation: connect variables (Pythagorean, area, volume).
Differentiate: take d/dt of both sides. Add dx/dt, dy/dt, etc.
Substitute & Solve: plug “snapshot” values AFTER differentiating.

Big FRQ Trap: Do NOT substitute snapshot values before differentiating.

4.6

Linearization Tangent Approx

Linearization uses the tangent line at x=a to approximate values near a.

Tangent Line Approximation

L(x)= f(a) + f'(a)(x−a)

This is point-slope form with slope f'(a).

Graph: Overestimate vs Underestimate

Concave Up → Tangent Underestimates

If f''(a)>0, the tangent line sits below the curve → underestimate.

Concave Down → Tangent Overestimates

If f''(a)<0, tangent line sits above curve → overestimate.

4.7

L’Hospital’s Rule Limit Tool

L’Hospital’s Rule is used only for indeterminate forms 0/0 or ∞/∞.

The Rule

If lim f/g is 0/0 or ∞/∞, then lim f/g = lim f'/g'

Differentiate numerator and denominator separately, then re-evaluate the limit.

Graph Intuition: 0/0 near x=a

Both numerator and denominator approach 0

When both hit 0 at the same x-value, the quotient’s behavior depends on how fast each approaches 0 — derivatives capture that local behavior.

FRQ Scoring Note: Show numerator → 0 and denominator → 0 separately, then state L’Hospital.

Unit 4

Everything You Need for a 5 (Unit 4 Checklist)

If you can do ALL of these, Unit 4 FRQs become “free points.”

Interpret f'(c) with a full sentence + units.
Particle motion: connect s, v, a; decide right/left, stopped, speeding up.
Know what f'' means in context (rate increasing vs decreasing).
Related rates: sketch → equation → differentiate w.r.t t → substitute & solve.
Linearization: build L(x)=f(a)+f'(a)(x−a); concavity → over/underestimate.
L’Hospital: only for 0/0 or ∞/∞, justify on FRQ.