Mistake Master · AP Calculus · Unit 2 · Step-Through Animation
Two Rows Decide the Answer, So Take the Near One on Each Side
You'll learnto estimate a derivative from a table or a printed graph — bracketing the point with the nearest row on each side, running the centred difference quotient, and reading a slope off a tangent instead of off the curve.
Most functions do not arrive with a formula. They arrive as seven readings from a meter, or as a curve printed on a grid, and a rate still has to come out of them. What survives from the definition is its first half: a slope between two points. So everything now turns on which two. Reach to the far end of the table and you get the average of the whole run; take the nearest row on each side and the two one-sided errors cancel; halve the gap and what is left shrinks by a factor of four. Whatever you write down is an estimate, and saying so is part of the answer.
8 STEPS · 6 QUICK CHECKS · NEAREST ROW ON EACH SIDE · AN ESTIMATE IS NOT AN EQUALITY · v1
seven rows of data ⇄ V′(3) ≈ 5.8 litres per minute
Before you start
What you're looking at
A tank filling through a slowly closing valve. The meter was read once a minute for six minutes, and those seven numbers are the entire dataset — no formula, no curve between them. The columns of the table sit directly above the matching points on the plot.
The question
How fast is the tank filling at exactly t = 3 minutes — and which two rows should you divide to find out?
Watch for
The bracket under the chosen pair, the chords fanning apart as the window widens, and step 5, where the curve behind the data is finally shown so every estimate can be scored.
Mistake Master