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
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
where the in and out RATE curves cross
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