Conditions before computation
A ladder has fixed length, so x²+y² is constant while both x and y vary. A sphere’s volume rate is different from its radius rate. Water in a similar cone changes radius and depth together; substitute the radius-depth relationship before differentiating. For a closed-cylinder optimization, both circular ends belong to surface area, and positivity of radius and height restricts the domain. A stationary point needs a minimum or maximum argument.
Worked example with a method check
A 15-m ladder leans against a vertical wall. Its foot is 9 m from the wall and moves away at 2 m/s. Find the signed vertical speed of the top. The height is 12 m by the Pythagorean relation x²+y²=225. Differentiate while the ladder length stays fixed: x x′+y y′=0. Thus y′=−(9/12)(2)=-3/2 m/s; the negative sign means downward.
Compare two methods
Holding a cone’s radius fixed while its water level changes describes a different container. If r=h/2, V=πh³/12 and V′=πh²h′/4. Treating r as fixed would yield V′=πr²h′/3 and an incorrect depth rate. Similarly, minimizing curved surface alone for a closed cylinder omits two physical surfaces and changes the optimization problem.
Transfer: solve before reading
A 10-meter ladder has foot distance 6 meters and moves away at 2 meters per second. Find the signed vertical rate of the top. Then explain why substituting x=6 and y=8 before differentiating the fixed-length equation destroys the rate calculation.
Transfer reasoning and answer
The height is 8, so 6(2)+8y′=0 gives y′=−3/2 meters per second. If the instantaneous lengths are first treated as constants, differentiating produces only 0=0 and erases the dependence on time. Measurements locate an instant; they do not make the changing quantities constant.
Practice the next representation
Work all four task families linked to this lesson. For the illustrated items, state which numerical labels and relationships you used. For a written extension, compare each line of your solution with its original self-assessment criteria; these criteria are practice feedback, not an official AP score.
A 15-m ladder leans against a vertical wall. Its foot is 9 m from the wall and moves away at 2 m/s. Find the signed vertical speed of the top.
- The height is 12 m by the Pythagorean relation x²+y²=225.
- Differentiate while the ladder length stays fixed: x x′+y y′=0.
- Thus y′=−(9/12)(2)=-3/2 m/s; the negative sign means downward.
Compare the other answer choices
Omit the specified horizontal speed factor.
Give the downward speed as positive despite the signed vertical-rate request.
Reverse the ratio x/y.
Explain your thinking.
Optional written extension. Self-assessed only; it does not change your automatic practice result. These are teaching criteria, not official awarded marks.
(a) State the relationship or theorem that makes this calculation valid, including its relevant conditions. (b) Show the intermediate computation and give the requested result. (c) Explain why one plausible competing method would fail for this problem.
This draft stays on this page only. Use paper if you want to keep your working.
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 and BC Course and Exam DescriptionCalculus Volume 1: Fundamental Theorem of Calculus