It is a beautiful Spring morning. Everywhere you look, people are happily going to their classes, or coming from their classes. “College is fun!” calls out one student, and about twenty more yell “Sure is!” in unison. Someone else calls out, “I love History!” A bunch of other students call “Great subject!” in response. Swept up in the spirit of things, you call out, “Calculus is wonderful!” “Lies! Lies!” calls out one familiar voice. You wheel around and directly behind you is a wild-eyed hungry-looking stranger.
“Oh, don’t be silly,” you say. “I just learned about trigonometric integration. It wasn’t that hard a section, and there isn’t a single lie in it.”
He looks up at you and says, “Oh, really? Perhaps you can take a quick true-or-false test, and see how easy the section is.” The stranger then whips out a sheet of paper with this on it:
“Both are clearly true!” he shouts. And you realize that he is correct. His first integral is a simple u substitution (Let u = 2x) as is the second (Let u = cos(x)). He continues: “AND we know that -2sin(2x) = -4sin x cos x!” You realize that this is also correct because sin(2x) = 2sin(x)cos(x). “Thus cos 2x=2cos2x! Ho ho!”
“Ho ho?” you ask.
“Ho ho, I say! Ho, ho I mean! Because at x = 0, cos 2x = 1, and 2cos2x=2. Once again, your “Calculus” gets you into trouble! 1=2! 1=2!” At that, the stranger skips off into the distance.
We clearly have a problem. But the conclusion… If 1=2, then how can you tell odd numbers from even ones? Would 1 still be the loneliest number? Or is there a possibility that there is an error somewhere in the stranger’s reasoning? Find the error.
- Find the error: Differentiation
- Find the error: L’Hopital’s Rule
- Find the error: Related Rates
- Find the error: Fundamental Theorem of Calculus
- Find the error: U-substitution
- Find the error: Integration by Parts (part 1)
- Find the error: Integration by Parts (part 2)
- Find the error: Trigonometric Integration
- Find the error: Taylor Series
- Find the error: Improper Integrals and Taylor Series
- Find the error: Separation of Variables
- Find the error: Separation of Variables – Exponential Growth
- Find the error: Credits