[math-fun] Calculus conundrum
I just invented this paradox, though I doubt I'm the first: "On the one hand, the indefinite integral of 2x+2 w.r.t. x is x^2 + 2x + C; on the other hand, putting u = x+1, we can write the integral as the integral of 2u w.r.t. u, which is u^2 + C = x^2 + 2x + 1 + C. Equating x^2 + 2x + C with x^2 + 2x + 1 + C we get 0 = 1." Here's my question for you guys: For those students who are stymied by this paradox, what would be a good hint? I've tried "Think about what indefinite integration IS", but that just seems to confuse them more. Is there a good hint that doesn't give away the game? Jim Propp
I imagine "what is the sea" would give it away and "don't confuse different variables for each other" would confuse half and give it away for the balance. Have you tried waving your arms around and shouting, "MATH IS BROKEN!"? Charles Greathouse Analyst/Programmer Case Western Reserve University On Thu, Mar 13, 2014 at 4:24 PM, James Propp <jamespropp@gmail.com> wrote:
I just invented this paradox, though I doubt I'm the first:
"On the one hand, the indefinite integral of 2x+2 w.r.t. x is x^2 + 2x + C; on the other hand, putting u = x+1, we can write the integral as the integral of 2u w.r.t. u, which is u^2 + C = x^2 + 2x + 1 + C. Equating x^2 + 2x + C with x^2 + 2x + 1 + C we get 0 = 1."
Here's my question for you guys: For those students who are stymied by this paradox, what would be a good hint? I've tried "Think about what indefinite integration IS", but that just seems to confuse them more. Is there a good hint that doesn't give away the game?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On 3/13/2014 1:24 PM, James Propp wrote:
I just invented this paradox, though I doubt I'm the first:
"On the one hand, the indefinite integral of 2x+2 w.r.t. x is x^2 + 2x + C; on the other hand, putting u = x+1, we can write the integral as the integral of 2u w.r.t. u, which is u^2 + C = x^2 + 2x + 1 + C. Equating x^2 + 2x + C with x^2 + 2x + 1 + C we get 0 = 1."
Here's my question for you guys: For those students who are stymied by this paradox, what would be a good hint?
What does C represent? Brent
I've tried "Think about what indefinite integration IS", but that just seems to confuse them more. Is there a good hint that doesn't give away the game?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I agree with Brent's suggestion. I’ll add that the usual way of writing an indefinite integral as "F(x) + C" leaves it quite unclear to the average student that this stands for the *set* of functions {F(x) + C | C in R}. And as I think I’ve mentioned before, this is particularly confusing in cases like the claim (*) Integral (1/x) dx = ln|x| + C , which misleadingly imply the set of indefinite integrals is 1-dimensional. I maintain that (*) is just plain wrong. —Dan On Mar 13, 2014, at 1:48 PM, meekerdb <meekerdb@verizon.net> wrote:
"On the one hand, the indefinite integral of 2x+2 w.r.t. x is x^2 + 2x + C; on the other hand, putting u = x+1, we can write the integral as the integral of 2u w.r.t. u, which is u^2 + C = x^2 + 2x + 1 + C. Equating x^2 + 2x + C with x^2 + 2x + 1 + C we get 0 = 1."
Here's my question for you guys: For those students who are stymied by this paradox, what would be a good hint?
What does C represent?
I would simply remind them that it is the derivatives of the indefinite integrals that have to be equal, not the indefinite integrals themselves. After all, they are "indefinite"? Veit
On Mar 13, 2014, at 4:27 PM, "James Propp" <jamespropp@gmail.com> wrote:
I just invented this paradox, though I doubt I'm the first:
"On the one hand, the indefinite integral of 2x+2 w.r.t. x is x^2 + 2x + C; on the other hand, putting u = x+1, we can write the integral as the integral of 2u w.r.t. u, which is u^2 + C = x^2 + 2x + 1 + C. Equating x^2 + 2x + C with x^2 + 2x + 1 + C we get 0 = 1."
Here's my question for you guys: For those students who are stymied by this paradox, what would be a good hint? I've tried "Think about what indefinite integration IS", but that just seems to confuse them more. Is there a good hint that doesn't give away the game?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I think I'll adopt a modified version of Veit's suggestion and say "Remember why it's called an INDEFINITE integral." Jim On Thursday, March 13, 2014, Veit Elser <ve10@cornell.edu> wrote:
I would simply remind them that it is the derivatives of the indefinite integrals that have to be equal, not the indefinite integrals themselves. After all, they are "indefinite"?
Veit
On Mar 13, 2014, at 4:27 PM, "James Propp" <jamespropp@gmail.com<javascript:;>> wrote:
I just invented this paradox, though I doubt I'm the first:
"On the one hand, the indefinite integral of 2x+2 w.r.t. x is x^2 + 2x + C; on the other hand, putting u = x+1, we can write the integral as the integral of 2u w.r.t. u, which is u^2 + C = x^2 + 2x + 1 + C. Equating x^2 + 2x + C with x^2 + 2x + 1 + C we get 0 = 1."
Here's my question for you guys: For those students who are stymied by this paradox, what would be a good hint? I've tried "Think about what indefinite integration IS", but that just seems to confuse them more. Is there a good hint that doesn't give away the game?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
"JP" == James Propp <jamespropp@gmail.com> writes:
JP> For those students who are stymied by this paradox, what would be a JP> good hint? It probably wounldn't work in a classroom setting, but a little humour might help some students: http://www.netfunny.com/rhf/jokes/old90/constant.html (The first thing I though of when reading your question.:) -JimC -- James Cloos <cloos@jhcloos.com> OpenPGP: 1024D/ED7DAEA6
participants (6)
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Charles Greathouse -
Dan Asimov -
James Cloos -
James Propp -
meekerdb -
Veit Elser