Related rates
When two or more quantities change with time and are linked by an equation, differentiate the equation with respect to time to link their rates.
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Analogy
A balloon being inflated: radius and volume grow together, and one rate fixes the other.
Example
Height y = 8 m, since 6^2 + 8^2 = 10^2.x^2 + y^2 = 100, so 2x x' + 2y y' = 0.Base slides out at x' = 2 m/s.y' = -(6)(2)/8 = -1.5 m/s.
In practice
Used for sliding ladders, filling tanks and shadows that change length.
Practice
Flashcards
- Why differentiate before substituting in related rates?Changing quantities must keep their rates until the equation is differentiated.
Questions people ask
How do I start a related rates problem?
Draw a diagram and name every changing quantity. Write an equation that holds at every moment, such as Pythagoras for a ladder. Differentiate with respect to time using the chain rule. Only after that, substitute the known numbers for the moment in question.
Why not substitute the numbers before differentiating?
Quantities like the ladder's height change with time, so substituting a single value turns them into constants and removes their rates. Differentiate the general equation first, then put in the values at the instant you want. Only fixed lengths may be substituted early.