Yet Another Calculus Question (actually two)!

Colleges & Universities Massachusetts Institute of Technology
1

<p>Now that someone has already posted a Calc question, I feel more comfortable posting some problems that have/had stumped me. </p>

<p>I have captured the two problems in jpeg format.</p>

<p>Please feel free to post any interesting solution methods.</p>

<p>

<a href=“http://img43.imagevenue.com/img.php?loc=loc14&image=086ec_calcproblems.jpg[/IMG]”>ImageVenue.com - 086ec_calcproblems.jpg

</a></p>

2

<p>For (b), write a + ib = r’ e^(i t’) for appropriate r’ and t’ (which you can easily solve for), and then
a - i b = r’ e^(-i t’), whereupon the rest is trivial. (After you simplify appropriately using this trick, and have some new complex numbers, you can go back to “rectangular coordinates” by using r e^(i t) = r cos t + r i sin t, for any r and t. You can hopefully see how this would kill off the imaginary part.)</p>

<p>(a) </p>

<p>dy/dx = (dy/dt) / (dx/dt)</p>

<p>then use</p>

<p>(d^2 y) / (d x^2) = d/dx (dy/dx) = d/dt (dy/dx) dt/dx</p>

<p>(chain rule)</p>

<p>and that, if you just want it in terms of t, then dt/dx = 1/(dx/dt)</p>

<p>provided the latter expression is well-defined.</p>

3

<p>Thanks for the help! </p>

<p>I can’t believe I overlooked the simple solution for part a!</p>

<p>For b., I would never have stumbled upon your method. I’ve heard of another way of solving it (will also dig that up), but it doesn’t compare to your suggestion.</p>

4

<p>No problem. BUT, I have a feeling the CC folks will soon tell us that the MIT forum is not really a place for math problems… it would just get cluttered up too fast with people’s calculus homework. The Art of Problem Solving site is great and though I’m not a regular, I’m pretty sure they have a forum for tricky problems from high school classes which plenty of people are happy to help you with. (tokenadult, if you’re around, I am pretty sure this is true but can you confirm?)</p>

<p>: )</p>

5

<p>Yes, [url=<a href=“http://www.artofproblemsolving.com/Forum/index.php]AoPS[/url”>http://www.artofproblemsolving.com/Forum/index.php]AoPS[/url</a>] definitely has forums on which math problems from school homework are on-topic, and of course it has LOTS of forums on which math problems from math competitions are on-topic. </p>

<p>What I find most annoying here on CC are the math problems that are multiposted on the Caltech, MIT, Stanford, etc. forums, having no connection to any of those schools.</p>

6

<p>I didn’t post it in multiple forums. But, rest assured, I won’t post anymore problems in the forums–just couldn’t resist the chance :).</p>

<p>Anyway, point taken.</p>