Unit 4 Related Rates: Connecting Changing Quantities in Time

Related rates

Problems where two or more quantities change over time and are connected by a known relationship, so their rates of change are related.

Rate of change (with respect to time)

A derivative like d(variable)/dt that measures how fast a quantity changes per unit time, including units (e.g., cm/s).

Implicit differentiation (in related rates)

Differentiating an equation relating variables with respect to t, even if t is not shown explicitly, because the variables depend on time.

Chain Rule in related rates

When differentiating a quantity like r^2 with respect to t, you must multiply by dr/dt (e.g., d/dt(r^2)=2r·dr/dt).

Constraint equation

A single equation (often from geometry) that relates the changing variables in a related rates problem (e.g., x^2+y^2=L^2).

“Differentiate with respect to time”

The core move in related rates: apply d/dt to both sides of the constraint equation to connect the variables’ rates.

Instantaneous rate

A rate evaluated at a specific moment (“at the moment when…”), using the variable values at that instant.

Variable vs. rate

Distinguishing a quantity (like r or y) from its time derivative (dr/dt or dy/dt); they are not interchangeable.

Units check

Using units to verify correctness (e.g., (cm)(cm/s)=cm^2/s for dA/dt).

Sign of a rate

Positive means increasing in the chosen positive direction; negative means decreasing or moving in the opposite direction.

“Equation first, derivative second, numbers last”

Workflow memory aid: write the relationship, differentiate, then substitute numerical values to avoid breaking the variable relationship.

Standard related rates workflow

Define variables, write one constraint equation, differentiate w.r.t. t, substitute values at the instant, solve for the unknown rate (with sign/units).

Diagram (related rates)

A labeled picture of the situation that helps choose correct variables, identify constants, and write the correct geometric relationship.

Constant (in related rates)

A quantity that does not change over time (e.g., ladder length), so its derivative with respect to t is 0.

Pythagorean theorem constraint

For a right-triangle situation: x^2 + y^2 = L^2 (often used for ladders, distances, and perpendicular motion).

Differentiated Pythagorean equation

From x^2+y^2=L^2 (L constant): 2x·dx/dt + 2y·dy/dt = 0.

Ladder sliding interpretation

In the ladder problem, as the bottom moves away (x increases), the top moves down (y decreases), so dy/dt is negative.

Circle area formula

A = πr^2, relating a circle’s area A to its radius r.

Area related-rates equation (circle)

Differentiating A=πr^2 gives dA/dt = 2πr·dr/dt.

“Plugging in too early” mistake

Substituting numerical values before differentiating (e.g., r=12) can turn a changing variable into a constant and destroy the rate relationship.

Cone volume formula

V = (1/3)πr^2h, relating the volume of a cone to radius r and height h.

Similar triangles (cone problems)

A proportionality relationship (like r/h = constant) used to eliminate one variable so the volume can be written in terms of a single changing variable.

Cone setup example ratio

If the full cone has radius 4 and height 12, similar triangles give r/h = 4/12 = 1/3, so r = h/3 for the water cone.

Perpendicular motion distance relationship

If one object is x east and another is y north, the distance between them satisfies s^2 = x^2 + y^2.

Distance-between-objects rate equation

From s^2=x^2+y^2: 2s·ds/dt = 2x·dx/dt + 2y·dy/dt, so ds/dt = (x·dx/dt + y·dy/dt)/s.