Recent content by QuietMind

My algebraic solution: Let D be the number of steps total and let s be the speed of the escalator. It took 20 seconds in the first case and 16 in the second case. ## D = (s + 1) 20 ## ## D = (s+2) 16## ##(s+1)20 = (s+2) 16 ## ##4s = 12 ## ## s = 3 ## Then ## D = 4*20 = 80 ## To do this in...

Homework Statement Sonia walks up an escalator which is going up. When she walks at one step per second, it takes her 20 steps to get to the top. If she walks at two steps per second, it takes her 32 steps to get to the top. She never skips over any steps. How many steps does the escalator...

Yes, if we raise it to the natural log of some rational number. ## e^ {\ln{ \frac{c}{d} }} = \frac{c}{d} ##. Applying this to the original statement, we would have ## b = \log_{a} \frac{c}{d} ##. Do we have to prove that ## \log_{a} \frac{c}{d} ## is irrational to finish this? My attempt to...

Homework Statement True or false and why: If a and b are irrational, then ##a^b## is irrational. Homework Equations None, but the relevant example provided in the text is the proof of irrationality of ##\sqrt{2}## The Attempt at a Solution Attempt proof by contradiction. Say ##a^b## is...

Thank you! I revised the formatting for anyone who happens to come across this later.

Homework Statement Let ##\prod_{n=0}^{1996} (1 + nx^{3^n}) = 1 + a_1 x^{k_1} + a_2 x^{k_2} + ... + a_m x^{k^m} ## , where ##a_1, a_2, ..., a_m ## are nonzero and ##k_1 < k_2 < ... < k_m ##. Find ##a_{1996}##. From Art and Craft of Problem Solving, originally from Turkey, 1996 Homework Equations...

I think I see what you are saying with this holding for any function f, but I'm wondering if you'd need any additional constraint of it being "nice" and/or smooth. The argument hinges on if p(x) goes negative, there must necessarily be a local minimum at a negative value of p(x) to guarantee...

So in order to have the correct behavior at large positive or negative x, if the function crosses below the x axis, it must come back up and cross the x-axis again. This guarantees that a local minimum must exist somewhere on this interval I where this function goes below the x axis, but if the...

I did out the argument for ##p'(x_1) > 0## for some ##x_1 \in I## by considering decreasing x from ##x_1##. The derivative must be even more positive for these points, and because the second derivative is negative, this means that p(x) must approach negative infinity as x decreases from ##x_1##...

The second derivative determines the concavity of a function. At a zero of f(x), ##x_0## , the function has ##f''(x_0) \leq 0 ## . The problem is I don't know anything about the first derivative, so I can't use the second derivative test to say if a point is a local max/min. If the second...

I'm really not sure how to go about proving the first one, but the second derivative being zero should mean that the first derivative is not changing at ##x_0##. This tells me that the first derivative is not changing at this point, but does this also tell me that the function never crosses the...

Homework Statement Prove that if p(x) has even degree with positive leading coefficient, and ## p(x) - p''(x) \geq 0 ## for all real x, then $$ p(x) \geq 0$$ for all real x Homework Equations N/A Problem is from Art and Craft of Problem Solving, as an exercise left to the reader following a...

I'm a sophomore in college studying physics and computer science, and I'm in the process of considering a transfer. I have offers to Brown and Northwestern, and I'd like to get some more opinions on the quality of each school for these fields. I currently attend Vanderbilt. In physics, beyond...

This is a very similar problem that I'd like to check my work on to see if I understand the Markov Chain approach. If it's inappropriate to follow up with a question like this, I apologize, but I think the question is very similar and starting a new topic would be redundant. Homework Statement...

Could you elaborate a bit more on this? When would you decide to multiply by the number of flips? N+2 I understand because you are adding 2 to account for the flips that already occurred, but I don't see where multiplication comes into this.