Related Rates Problems

AP Calculus · Axiom Academy

Connect rates of change between related variables In a related rates problem, two or more quantities change with respect to time, and they're connected by some relationship. Your job is to find how fast one quantity changes given how fast another changes. Draw & label: Sketch the situation, label all variables and fixed quantities Find the relationship: Write an equation connecting all variables (often using geometry or physics) Differentiate with respect to time: Use implicit differentiation on both sides of the equation Substitute & solve: Plug in known values (including rates) and solve for the unknown rate A 10-meter ladder leans against a wall. The base slides away at 0.5 m/s. How fast is the top sliding down when the base is 6 m from the wall? Step 1: Set up variables Let = distance of base from wall (in meters) Let = height of top on wall (in meters) Step 2: Write the relationship By the Pythagorean theorem: Step 3: Differentiate both sides with respect to time Step 4: When x = 6, first find y Step 5: Substitute known values We know Principle 1: Time is Always Implicit Every variable is a function of time, even if we don't explicitly write it. This is why we can use the chain rule. Principle 2: The Relationship Stays the Same If is always true (the ladder doesn't change length), then its derivative is always true too! If x is in meters and changes in m/s, then dx/dt is in m/s. Always track your units!

This is the written version of the interactive lesson above. See the full AP Calculus course.