Contextual Applications of Differentiation

Seven topics that point the derivative at something real. Interpreting a rate with its units and its moment, straight-line motion and the sign test for speeding up, applied rates where inflow meets outflow, related rates from the setup through to an answer, linearization and whether the estimate came out high or low, and L'Hospital's Rule, which finally resolves the indeterminate forms left open in Unit 1.

AB exam 10-15%BC exam 5-10%7 topics

Topics

4.1

Interpreting the Meaning of the Derivative in Context

Straight-Line Motion: Connecting Position, Velocity, and Acceleration

Rates of Change in Applied Contexts Other Than Motion

Introduction to Related Rates

Solving Related Rates Problems

Approximating Values of a Function Using Local Linearity and Linearization

Using L’Hospital’s Rule for Determining Limits of Indeterminate Forms

Lesson

Diagnostic

after diagnostic

Key forms For every problem in this unit

Units of a derivative

units of f PER unit of the input

Units of a second derivative

the input unit appears TWICE, e.g. gal/min/min

Full interpretation

when · what · increasing or decreasing · how fast + units

Motion chain

v(t) = s′(t); a(t) = v′(t) = s″(t)

At rest

v(t) = 0, NOT s(t) = 0

Speed

|v(t)|, never negative

Speeding up

v and a share a sign; opposite signs = slowing down

Average velocity vs speed

net displacement / time vs total distance / time

Net rate

dA/dt = (rate in) − (rate out), subtract never add

Amount at a maximum

the RATE curves crossing only makes a CRITICAL point. An interior max needs the net rate to go + → − there; for an absolute max, test the endpoints too

Relative rate

f′(t) / f(t), a percent per unit time, not a count

Related rates, the order

relate → differentiate d/dt → THEN substitute

Every varying letter

picks up its own dr/dt, dh/dt, dx/dt

Circle & sphere

dA/dt = 2πr(dr/dt); dV/dt = 4πr²(dr/dt)

Ladder

x² + y² = L² ⇒ x(dx/dt) + y(dy/dt) = 0

Cone constraint

similar triangles give r = kh, so V = πk²h³/3

Linearization

L(x) = f(a) + f′(a)(x − a), keep the (x − a)

Choosing the center

near the target AND exactly computable

Over or under

f″ < 0 concave down ⇒ OVER; f″ > 0 ⇒ UNDER

Not the test

the sign of f′ decides nothing here

L’Hospital, precondition

substitute FIRST: only 0/0 or ∞/∞ qualify

L’Hospital, the rule

lim f/g = lim f′/g′, top and bottom SEPARATELY

Other forms

0·∞, ∞−∞, 1^∞: rewrite as a quotient first

Unit 4 tools

Challenge bank

1 / 60

60 open-ended problems.

Read the question, work it out, then flip the card to compare your reasoning to the worked solution. Mark each card so you can return to the ones that still bite.

Question

Tap card to reveal explanation

Worked solution

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Cumulative assessment

Test the unit.

Twenty mixed items drawn from across all 7 topics, with guaranteed misconception-code coverage. Identifies which misconceptions still bite when you cannot see which topic the question came from.

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Check what stuck.

Units 1 through 4, drawn evenly so earlier units get the same share as this one. Twenty questions or a full 45-question section, your choice. Even coverage means this is a retention check rather than a score estimate. Follows the track you chose on the course page.

20 or 45questions

39topics

52codes covered

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