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Math101learn.math101.caRelated Rates
A rigorous, example-driven guide to related rates, including hypotheses, method choice, verification, and practice.
The central idea
Related-rates problems involve quantities linked by an equation and changing with a common time variable. If $F(x(t),y(t))=0$, differentiation with respect to $t$ produces terms such as $F_x dx/dt+F_y dy/dt$. Rates are evaluated at an instant; the geometric relation must remain true throughout the motion.
Definitions, hypotheses, and notation
The sign convention should be chosen before algebra. A distance increasing can be assigned a positive rate and a height falling a negative rate; the differentiated constraint then relates them automatically. If the question asks 'how fast' as a speed, report the magnitude after first finding and interpreting the signed derivative.
Geometry often supplies an intermediate quantity not mentioned in the final question. In the ladder problem, the missing height is found from the Pythagorean relation at the snapshot. In cone or shadow problems, similar triangles may be needed before differentiating. Eliminating extra variables early usually produces one constraint with exactly one unknown rate at the end.
Conceptual meaning
The chain rule transmits change through a constraint. A quantity can be fixed at the instant while still having a nonzero rate, so numerical values should be substituted after differentiating, not before.
A dependable method and decision rule
- Draw and label a diagram; declare which quantities depend on time.
- Write one equation relating the changing quantities.
- Differentiate the entire equation with respect to time.
- Substitute the instantaneous values and all known signed rates.
- Solve for the requested rate and interpret its sign and units.
Fully worked example
Graphical or geometric meaning
The ladder is a radius-constrained point $(x,y)$ on a quarter-circle. Its velocity vector is tangent to the circle and hence perpendicular to the radius, which is the vector form of $x x'+y y'=0$.
Common mistakes and why they fail
Verification and reasonableness checks
- Verify the snapshot satisfies the original constraint.
- Check units after implicit differentiation.
- Use the diagram to confirm whether the sign matches the physical motion.
Differentiate before inserting the snapshot
All connected quantities vary with time, so implicit differentiation produces terms such as $2r\,dr/dt$. Substitute numerical values only afterward; earlier substitution can turn a changing quantity into an apparent constant and erase its rate. Draw and label the configuration, choose one relation containing the supplied and requested rates, and establish positive directions. A shrinking length has a negative derivative even if its speed is quoted positively. Attach units and interpret the final sign. If a geometric relation has several branches, ensure the chosen lengths and angles match the physical picture. A magnitude alone is incomplete when the problem asks whether something is increasing or decreasing.
Practice
- A circle's radius grows at 3 cm/s. Find $dA/dt$ when $r=2$.
- For $x^2+y^2=25$, if $x=3,y=4,x'=2$, find $y'$.
- Why differentiate before substituting?
Answers and brief solutions
- $12\pi$ cm$^2$/s.
- $-3/2$.
- Substitution can erase the time dependence and its rates.
Connections and next steps
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
A circle's radius increases at 2 cm/s. How fast is its area increasing when r=3 cm?
- dA/dt=2πr·dr/dt.
- Substitute r=3 and dr/dt=2.
- dA/dt=12π cm²/s.
End of lesson
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