Mistake Master · AP Calculus · Unit 5 · Step-Through Animation
Solve It, Justify It, Then Answer the Question That Was Asked
You'll learnto carry a reduced objective all the way to a defensible answer — differentiate, solve, discard the roots the domain excludes, prove the survivor really is the extremum, back-substitute, and hand back the quantity the prompt actually named.
Topic 5.10 finished with a function of one variable and a domain. This topic runs it to a number, and then does the two things that decide whether the number counts. Differentiating and solving is the easy half — it produces a critical input, and almost nobody gets that arithmetic wrong. The hard half is showing that the critical input really is the extremum being asked for, which takes a different argument on a closed interval, on an open one, and on an unbounded ray. Then comes the last move: a solved fence problem holds four correct numbers, and three of them are wrong answers to any given prompt.
8 STEPS · 6 QUICK CHECKS · A CRITICAL INPUT IS NOT YET AN ANSWER · v1
A(x) = x(1200 − 2x) ⇒ A′ = 1200 − 4x ⇒ x = 300, y = 600, A = 180,000 ft²
Before you start
What you're looking at
The fenced-area function 5.10 built: 1200 feet of fence on three sides of a rectangular plot against a river, so A(x) = x(1200 − 2x) on 0 < x < 600. The horizontal axis is x in feet, and the gridlines stand every 100 feet.
The question
Solving A′ = 0 takes one line. What makes the answer worth full marks is the argument that the number it produced really is the maximum — and then naming the quantity the prompt asked for.
Watch for
Three domains, three different closing arguments: an open interval closed by concavity, a restricted domain where the critical number is not even available, and an open ray with no endpoints to evaluate at all.
Mistake Master