The Official Math/Physics/Whatever Homework questions thread

[Wyman]

Originally Posted by Sven2812

lol english is tough
ok we got x^y=y^x

so now we wanna write the equation in terms of:
x=f(y) or y=f(x)

but not everywhere just in a small area around (e,e) and (2,4)

A ha.

So first of all, realize that we are not writing an exact formula for y = f(x) or x = g(y). We're writing an approximation.

Remember that if we take a function that is differentiable, it looks like a line if we zoom in close enough. In fact, at (x0,y0), f(x) is approximated locally by the tangent line to f(x) at x0.

So, my questions for you are:

(1) At each of the points you listed, what is the slope of the tangent line to the curve x^y = y^x?
(2) Can you use that slope to write down the linear approximation (tangent line) to the curve at each point?

[Cueballmania]

Originally Posted by Wyman

Yeah, some regs in this thread are more tolerant of this than others (I think I vary from day to day).

Yeah, I have a pretty low tolerance for "do it for me". Especially when it's "Do it for me and show your work."

[Sven2812]

ok I got:

y'=(logy*y^x-y*x^(y-1))/(x^y*logx-x*y^(x-1))

now if i put in (e,e) my y' is not defined(what kinda makes sense if u look at the plot).
for (2,4) i get (log(16)-4)/(log(4)-1) but I am not really sure what to do from there.
u mean that y(x)=(log(16)-4)/(log(4)-1)*x+ some constant

[Wyman]

Originally Posted by Sven2812

ok I got:

y'=(logy*y^x-y*x^(y-1))/(x^y*logx-x*y^(x-1))

now if i put in (e,e) my y' is not defined(what kinda makes sense if u look at the plot).
for (2,4) i get (log(16)-4)/(log(4)-1) but I am not really sure what to do from there.
u mean that y(x)=(log(16)-4)/(log(4)-1)*x+ some constant

Yeah, I'm not checking your derivative at the moment or your ability to plug in points, but you've got the idea.

Also, to save yourself a bit of trouble, you might recall that there are many forms in which you can write down the x-y relationship of a line.

y = mx + b seems to be the one you're using, but

y-y0 = m(x-x0) is called point-slope form (since it requires you to have a point (x0,y0) and a slope (m)).

[non-self-weighter]

"Find a cubic function f(x) = ax^3 + bx^2 + cx + d that has a local maximum value of 3 at -2 and a local minimum value of 0 at 1."

Holy moly. What do I have to work with here? I know is can be found at points where f'(x) = 0 or does not exist (not applicable to this polynomial). So I need a derivative equal to something like this?

f'(1) = 3ax^2 + 2bx + c = 0 and f(1) = 0
f'(-2) = 3ax^2 + 2bx + c = 0 and f(-2) = 3

I'm a little stuck. Where do I go next? Hints?

Should I just make c something arbitrary (like 1)? Then solve f'(1) = f'(2) for a and b? Then I'm a bit lost how to make sure I get the correct values for f(1) = 0 and f(-2) = 3.

This is a question with infinitely many solutions, right?

Thanks!

[Cueballmania]

How do you know if something is a local minimum or local maximum? The critical value of the first derivative tells you that it's one or the other. Hence, you need to consider how you know which is which.

[non-self-weighter]

The way I was taught:

Find critical values by solving f'(x) = 0 or where f'(x) is undefined. Draw a number line with the critical values. Solve each interval on the number line for the sign (+/-) of f'(x). In this question, it is given that -2 and 1 are critical numbers. The local max is at x = -2 and local min is at x = 1, so f'(x) is positive on (-inf,-2) and on (1, inf). Also, f'(x) is negative on (-2, 1). But, now how do I use this information to create a function that satisfies all the given conditions? Hmm... I'm thinking about it...

[non-self-weighter]

Let me add, I also know the function goes through points (-2,3) and (1,0). What do I need to do here though? Take the graph y = x^3 and modify the vertical/horizontal/phase shift, and amplitude? I'm kind of at a loss. I'm only familiar with doing this for sinusoidal functions, so I must be on the wrong track.

[Cueballmania]

Oh, this is actually easier than I thought. You have four equations with four unknowns. EZ game.

[non-self-weighter]

Okay, okay. I can do that. So, let me just state the formulas I'm starting with:

1) 3a(1)^2 + 2b(1) + c = 0
2) 3a(-2)^2 + 2b(-2) + c = 0
3) a(1)^3 + b(1)^2 + c(1) + d = 0
4) a(-2)^3 + b(-2)^2 + c(-2) + d = 3

I'm going to work with a ton of scratch paper and check back if I get stuck.

Thanks!

[Wyman]

Originally Posted by Cueballmania

Oh, this is actually easier than I thought. You have four equations with four unknowns. EZ game.

Yes, though he should still learn the "second derivative test" for checking whether a critical point is a local min or local max. Even if the 4 equations in 4 unknowns produce exactly 1 solution, it would be nice to verify that the solution is consistent with the parameters of the problem.

Edit: not that my feelings on the above have changed, but I guess it's pretty clearly impossible for the critical points in this problem to be min/max.

[Mariogs37]

Hey guys,

Final on Monday and I have a couple questions from ch. 6 of Baby Rudin:

1) On p. 121, Rudin says that the integral from a to b_ = inf U(P,f)) and that the integral from _a to b = sup L(P,f).

But he defines U(P,f) = sum from i = 1 to n of Mi * delta xi ("i" is subscript). So say we had f(x) = x^2 and we had unit long partitions so [0,1], [1,2], etc. Then we'd get:

U(P,f) = sup f(x) on [0,1] + sup f(x) on [1,2] +...+ sup f(x) on [x(i-1), xi]. So for f(x) = x^2 this is just 1 + 4 + 9 +...+(xi)^2.

So what does he mean then, when he asks for the inf of U(P,f). Isn't U(P,f) just one number (the sum from above)?

He says to take the inf over all partitions of our interval but I don't know what he means given that we're taken inf (the sum from above)...

2) Later on he refers to alpha as a function and also f as a function. Ex: Thm 6.9: If f is monotonic on [a,b], and if alpha is continuous on [a,b], then f is Riemann integrable (we're assuming, of course, that alpha is monotonic).

What is alpha here and what is it being used for? Strangely lost on this one also.

Other than that, I feel pretty comfortable with chapters 1-5.

Get this: Prof says he'll curve the class to between B- and B+. Our midterm was out of 150. Last year no one above 100. This year:

Average = 67.7

I get 70.

Find out online that 3 kids dropped from our 24-person class before taking it. Also find out that the median is 53...

I get a B...lasdkjfl;asjf;lkdsjf;kljflk;asfj;klajwslkdfgj

[Swwiinn]

So since I don't go to school this isn't actually a homework question, but I didn't feel it warranted an entire thread so just thought I'd post it in here.
'If an unfilled circle equals 53, a filled triangle equals 56, an unfilled square equals 62, a filled circle equals 65 and an unfilled triangle equals 68, what does a filled square equal?'
Also if you could explain why it equals what it does that would be great.

[DeucesAx]

Is this a interview question from an investment bank?

[Swwiinn]

Nah, one of my friends just asked me because I'm usually reasonably good at maths/problem solving, but this one has me stumped. I have a feeling I'm over-thinking it though.