A reliable method
Reduce an optimization objective to one independent variable. In related rates, differentiate the relationship before substituting the instant values.
Worked foundation example
A rectangle has perimeter 8 units. What is its greatest possible area in square units? If one side is x, the other is 4−x. A(x)=x(4−x); its derivative is 4−2x. The maximum occurs at x=2 and has area 4.
The common trap
Do not substitute a changing quantity as a fixed constant before differentiating. Check endpoints and physical restrictions in optimization.
Connect the representations
A derivative equation uses rates with respect to the same time variable. Attach units so an area rate is not confused with a length rate.
Explain before checking
Write the defining equation, state its assumptions, and attach units before calculating. After checking the numerical answer, compare every step with the worked reasoning. These short tasks build selected foundations; a complete AP free-response solution also needs written justification, correct notation and attention to the rubric.
Compare a changing variable with an instant value
For a circle A=πr², related rates differentiates the relationship first: dA/dt=2πr·dr/dt. If r=3 and dr/dt=2, the area rate is 12π. Substituting r=3 into the original formula before differentiating would make area appear constant and would destroy the changing-radius model. The numbers describe one instant, not an identity for all time.
Transfer and explain
Transfer: a rectangle with fixed perimeter 20 has area x(10−x). Its maximum is 25 at x=5. Compare this optimization task, which searches an input, with the circle rate task, which evaluates an instantaneous derivative.
A rectangle has perimeter 8 units. What is its greatest possible area in square units?
- If one side is x, the other is 4−x.
- A(x)=x(4−x); its derivative is 4−2x.
- The maximum occurs at x=2 and has area 4.
Compare the other answer choices
Report the optimal side instead of area.
Forget that both side lengths share the semiperimeter constraint.
Treat half the perimeter as both side lengths.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
AP Calculus AB courseCalculus Volume 1: Fundamental Theorem of Calculus