Calculus Cases: the Vektor calculator
Three weeks ago Argen Labs was broken into, and the prototype of the AE-3 was taken: an experimental calculator that estimates \(e^x\) by adding up terms. Yesterday a device that does the same turned up at Vektor Mechanical, and its technical director says it is Vektor’s own design. The Chief Inspector, a role played by the teacher, will not hear of theft before knowing how alike the two methods are and where they part. Each team brings together three specialists: the archive holds Argen’s design note and rebuilds its polynomial; inspection holds the readings of the seized device, but no drawing; validation holds the reference values and the tolerance. No sheet is enough on its own.
The twist of the case is that the supposed copy passes the tests and the original does not: the Vektor readings carry one more Taylor term. And still, three readings do not identify a mechanism, let alone say who built it.
The case follows the format of every Calculus Case:
- Launch, at the end of a class (18 min). Two entry questions and the projected presentation: the case, the Chief Inspector’s request, the three roles and the class’s teams. Each student receives the sheet of their role (A, B or C).
- At home (45–60 min per team). Each specialist works on their sheet alone, and then the team meets, in person or online, and exchanges the data aloud: the polynomial \(P_3\) and its outputs, the differences from the readings and a correction that fits them, the errors and the verdict of the tolerance.
- Closing class (55 min). Pooling; the term strips (sheet D, one per team), which turn each term into something laid on the table and added with its sign; a hearing in which each team says what it certifies, what it conjectures and what is missing; the formalization, the team’s opinion and an individual check.
Materials generator
The configurator guides you through four steps. The only essential setting is the number of students: from it, the website forms the teams of three specialists (with one or two of four if students are left over) and works out how many copies of each sheet to print. Everything else already has a recommended value, which is the book’s.
Activity documents
Start with the printing and distribution guide: it says what to print, how, how many copies and when each document is handed out. The teacher packet contains the solution and must not circulate. In each tab, the red row downloads a ZIP with everything in it; the icons let you view or download each file.
What is studied and why the result is right
| Sheet | Role | Mathematics | What it brings to the team |
|---|---|---|---|
| A · The Argen archive | Archive | Matching value and derivatives at 0: \(a_j = f^{(j)}(0)/j!\) | The polynomial \(P_3\) and its outputs at the three inputs |
| B · The seized device | Inspection | Differences \(d(x)\) and the quotient \(d(x)/x^k\) | The correction \(x^4/24\), which fits the readings |
| C · Validation | Validation | Absolute error and tolerance | Which machine passes the tests, and how far that goes |
| D · The strips and the opinion | Team | Signed partial sums; a different rule with the same readings | The opinion: what is established, what is suggested and what cannot be proved |
Since every derivative of \(e^x\) equals 1 at zero, Argen’s polynomial is \(P_3\). With the tolerance of 0.012, the AE-3 fails at \(x=-1\) and at \(x=1\), while the readings of the seized device, which agree with \(Q_4=P_3+x^4/24\), pass all three tests. But three passed tests are a statement about three inputs: the rule \(Q_4(x)+c\,(x+1)(x-0.5)(x-1)\) gives the same readings for any \(c\), and only a remainder bound, such as Lagrange’s, says something about the whole interval. No number, however sharp, says who designed the machine.
Argen Labs, Vektor Mechanical, the AE-3 calculator, the people and the records are fictional and purely educational. The header images are drawn by a program.