Mistake Master · AP Calculus · Unit 4 · Step-Through Animation
Differentiate First. Substitute Last. The Order Is the Whole Topic
You'll learnto run a related rates problem all the way to a signed number in the one order that works, and to recognise on sight the answer of exactly zero that a frozen variable always produces.
Topic 4.4 built the setup. This one runs it to a number, and the difference between a correct solution and a confidently wrong one is the order of two steps. Differentiate the relationship with respect to t first; put the instant's values in afterwards. Reverse them and a variable becomes a number, a number cannot move, and the derivative reports — correctly — that nothing is changing. That is why this particular wrong answer is almost always exactly zero. A 13-foot ladder whose foot slides out at 2 feet per second gets a top that is not moving at all, and a balloon being inflated is reported as holding its volume.
8 STEPS · 6 QUICK CHECKS · DIFFERENTIATE FIRST, SUBSTITUTE LAST · A FROZEN VARIABLE ANSWERS ZERO · EVERY MOVER BRINGS A RATE FACTOR · v1
A 13-foot ladder against a vertical wall. Its foot is x feet from the wall and slides out at a steady 2 feet per second; its top is y feet up the wall. The 13 never changes. Everything else on the screen does.
The question
How fast is the top sliding down at the instant the foot is 12 feet out — and what happens to the answer if you put the 12 in before you differentiate rather than after?
Watch for
The step 7 ladder drawn twice. One of them is sliding; the other has had its numbers substituted first and cannot move at all, which is the wrong answer of zero made visible.
Mistake Master