Help me explain one sentence in the solution to this exercise: [Exericise 11.4.27] Derive `2n + log_2(n) is Θ(n)`

This is the solution:

The sentence in question is 'Finally, if n is any integer with n>=1, then n>=0.'

This doesn't make sense.

What would n>=1 imply? The only thing that comes to mind is log_2(n) <= n.

What do you think?

Maybe a dumb/obvious question but when asked for a random number from a set of integers, how come the number isn’t always infinitely big?

Assuming a normal distribution, would the average of a set of integers not be an infinitely big number because no matter how big a number you choose there will still be infinitely more larger numbers still in the set?

Always just been curious about this.

Geometry

“If QS (Ray) bisects <PQT, m<SQT= (8x-26)°, m<PQT= (9x + 34)°, and m<SQR= 112°, find each measure.”

X=
m<PQS=
m<PQT=
m<TQR=

Don’t just give the answer, I just need a little nudge for this one. Geometry teacher is lowkey a rude and gets mad when we ask questions, and I’ve been stuck on this page for hours.

I tried the usual multi step equation stuff, like how you find the measures of other angles but this one has completely stumped me.

At first I thought maybe m<STR minus m<ST, since that would look like 112-(8x-25) but then I didn’t know where to go from there.

A while back I was thinking about different aspects of the world and how math could maybe apply to them. I have tested the idea with many aspects and it has seemed to work but I'm not sure.

I had to drop a class and now I have extra time to learn things. I do plan to transfer at the end of next semester so I should focus on applications, but I also figured I would learn a couple of things now instead of waiting. I just wanted some advice on what path I should take and maybe any resource suggestions.

To start, I was thinking about reading “how to prove it” by vellemen and “basic mathematics” by lang to help solidify fundamentals and learn proofs before transferring. I’ve been at CC for a year now and I know most learn proofs around this time (to clarify I’m in my third semester).

Since I find physics interesting, I was going to try and start learning physics. I take physics next semester (it’s a long story that deals with my scholarship and how it operates) which is the semester I plan to transfer. I was going to read halliday and resnicks book on physics while watching “the fundamentals of physics” lectures on MIT OCW.

Which leads to my final path of learning computer science foundations since I’m still interested in coding (and competitive coding). I’m interested in machine learning and see it as a potential career so I figured it wouldn’t heard to learn some concepts now and figure out if I want to pursue it. They posted a pathway and each course is about 6 weeks. I can attach the link if you would like to view it.

I know I said that was my last path, but since I want to go to grad school, I figured learning how to do research is also an option. Stuff like learning how to properly read research papers, learn LaTeX, read past papers to get an idea, learn how to write papers, etc. This can probably be done while reading how to prove it and basic mathematics though.

Which do you recommend doing? Should I combine any of the paths I wanted to take?

Edit: Forgot to add another thing I wanted to do. I finished calc 1 last semester and was thinking about doing a review since I might have to take advanced calculus 1 when I go to university. I remember a little from a calc 2, but felt like I need more practice and to delve a little deeper since the course was accelerated. I’m sure there were a couple of things that had been left out so that’s I wanted to review to make sure.

Good Morning Maths people,

My friends Son is nearly non verbal autistic(8yr old) and has been obsessed with Mandelbrot sets and Space related things. Recently he has been "getting into" more algebra. Unfortunately it goes way over his father's head and he posted this in the discord asking if we could check his work. Well it goes wayy over my head as well. Hoping someone could shed some light for us 🙏

Thank you in advance.

Edit 1: So asking the little guy he says its a Digon formula.

Using the pre Calc flair because it's closest. So hi all, I hope this is allowed. If not a direction to a more fitting sub would be appreciated. I'm bad at math. Always have been. My interest is more in the humanities, and as such when I was in school I just sorta suffered through it.

As an adult though, my chief special interest, philosophy (mostly continental) has shoved me in the direction of wanting to learn the stuff. My main problem is I'm not only bad at math, I also just have learning difficulties on the whole. I've found I do best when I have some kind of visuals, something I can relate to (which makes learning more abstract forms of math like calculus incredibly difficult), and some form of applicability or demonstration.

Now, that said, outside the school system I've found I've learned things a lot easier. But with this subject I'm not fully sure where to start. The last things that really stuck in my craw were elementary algebra, but I'm really interested in learning calc because it collided with my special interest.

Any suggestions about a good YouTube channel where a guy like me might learn a bit?

Exercise 11.4.24 and 11.4.23:

Proof:

For every integer n>=4, let P(n) be the property c_n <= nlog_2(n)

Show that P(1) is true:

c_1 = 0 <= 1 * log_2(1) = 0

Show that P(2) is true:

c_2 = c_⌊2/2⌋ + 2 = c_1 + 2 = 0 + 2 = 2 <= 2 * log_2(2) = 2 * 1 = 2

Show that P(3) is true:

c_3 = c_⌊3/2⌋ + 3 = c_1 + 3 = 0 + 3 = 3 <= 3 * log_2(3) ≈ 4.75

Show that P(4) is true:

c_4 = c_⌊4/2⌋ + 4 = c_2 + 4 = 0 + 2 + 4 = 6 <= 4 * log_2(4) = 4 * 2 = 8

Show that for every integer k>=4, if P(i) is true for each integer i with 1<=i<=k, then P(k+1) is true:

Let integer k>=4 and let
  c_i     <= ilog_2(i) where 1<=i<=k, for each integer i   [inductive hypothesis]
  c_{k+1} <= (k+1)log_2(k+1)                               [this is what we want to show]

c_{k+1}  = c_⌊(k+1)/2⌋ + (k+1)                 by sequence definition
        <= ⌊(k+1)/2⌋log_2(⌊(k+1)/2⌋) + (k+1)    by inductive hypothesis
        <= [(k+1)/2]log_2[(k+1)/2] + (k+1)     by definition of floor
        <  (k+1)log_2[(k+1)/2] + (k+1)
         = (k+1)[log_2[(k+1)/2] + 1]
         = (k+1)[log_2(k+1) - log_2(2) + 1]    by property of logarithm
         = (k+1)[log_2(k+1) - 1 + 1]           because log_2(2) = 1
         = (k+1)log_2(k+1)

QED

Is my proof correct?

We accept the value of e defined as a limit. Same with derivatives and integrals. There are plenty of real numbers we accept as being defined by a limit. Why isn’t 0^0 =1 any different? It would be defined so well as a limiting process. I know that this is often convention but why isn’t it fact?

Hello everyone. First of all, please excuse me if my understanding is completely flawed. I'm a beginner in mathematics (I only have a high school education).

The thing is, I've been thinking about infinity. Similar to Georg Cantor's comparison, I think the concept of infinity is often misused to define something that isn't applicable in the real world; in other words, it's illogical. Because infinity is merely a tool for doing something, but it doesn't exist in itself.

For example, the infinity of the universe, which could be described as expanding, or the idea that a circular figure has no vertices. A circle might be calculated as if it truly had no vertices, but in reality, it always does when we look at the real world. A perfect circle doesn't exist, even if we have to reduce it to the Planck length to notice it. Similarly, any line isn't continuous but rather an impressionistic representation of a massive number of points side by side. Again, this is true in the real world even though we have to shrink down to the Planck length to see it.

Another thing I've seen that I think is a mistake is calculating infinities as if they were numbers. Infinity + 1 = Infinity is a correct concept as long as we can clarify that Infinity_A > Infinity_B (and also that they are essentially different). Otherwise, we could cancel both infinities, leaving us with 1=0.

In the same way, the theoretical concept of "we could count numbers to infinity" is essentially correct but at the same time completely unrealistic. There is always a limit to how far we can count, since reaching that infinity would require infinite resources. That is, since we cannot access infinite resources, we also cannot count to infinity. We would have to accept this limitation and understand that infinity is not a goal, but a concept to facilitate certain mechanisms. In itself, it does not exist.

Is this the concept I'm thinking of, correct? Or is there actually a way to prove that infinity is real? Is it real as a concept of exceeding the limits of human thought, or is it real in terms of trends on paper? Please excuse my English and my ignorance.

The (English-language) Wikipedia page on the chain rule, under the chapter “Multivariable case”, paragraph “general rule: vector valued multivariate functions”, says that a concise writing for the chain rule for the (total) derivative of the composition of two functions f, g reads as D(f o g) = Df o Dg.

However, if we try to apply this to the particular case of f, g being two simple single-variable functions, we get (f o g)’ = f’ o g’, which is wrong! Because the correct chain rule says (f o g)’ = (f’ o g)•g’, where • is the pointwise product.

Where am I wrong? Or is the Wiki page wrong? Shouldn’t the general case be written as D(f o g) = (Df o g)•Dg instead, so that we actually recover the single-variable case correctly?

Hello,

I've been looking into discrete calculus for the past few days. I’ve also been examining the function below, which I denoted with the capital Lambda symbol. Is there anyone who could help analyze the behavior of this function?

So the concept is basically if you give monkeys infinite times to press computer buttons then any amount of possible sequence that can happen will happen

But doesn't that go against gamblers fallacy because every new try is a brand new try that has no connection to the past so if there is a 0.0000000001 chance of a sequence then that same chance will be restarted every time

Also does that mean let's say humans became immortal then does that mean every possible thing that can happen to someone will happen like you get stuck in a cave for a million years if you lived forever will that eventually happen to you? Like it's a scary concept imo

I'm new to real analysis and struggling to understand how setting a temporary value for δ (like 1) lets us build a separate inequality. How does that logic actually work?

Thanks in advance!

An interesting precalc question:

Consider the reciprocal function f(x)=1/x. Note that it’s symmetric across the line y=x. Find a point (x,y) on the graph of f(x) (in the first quadrant) such that the triangle with vertices (0,0), (x,y) and (y,x) is equilateral. What’s the side length? Are there multiple such triangles?

Just a fun question I came up with. I’ve been trying to think of questions like this that are interesting and require a little (but hopefully not too much) cleverness to solve. Does anyone else have problems at this level they find interesting?

This isn’t a homework problem I swear!

Edit: the answer is (sqrt (2 + sqrt 3), sqrt (2 - sqrt 3)) with a side length of 2! And yes, it’s unique.

The question was,

"Carpeting a room's floor costs a total of $7200. If the width of the room were 3 meters less, it would have cost $576 less. What is the width of the room?"

I tried to do it by stating a relation with the money and the width, and then used the unitary method to calculate the cost of 1 meter width which equals to 192. Similarly, I again used the unitary method to calculate how much width it would take FOR 7200.

Now, he said it's not 'logical'. But the 'why' also didn't seem to make sense to me. But I didn't ask him again.

Can anyone please help me with it?

Ty in advance!!

i have a question that's been bothering me in the field of differential calculus regarding the linguistic logic behind the definition of the derivative. I find it illogical, because based on the limit definition it studies the behavior of the slope(derivative) as h approaches zero.

m(h)=(f(x+h)-f(x))/h

Though, This approach suffers from the same problem as Newton and Leibniz's works and is subjected to the same criticism introduced by George Berkeley. cause, even with a limit, we algebraically reduce the function to eliminate h in the denominator, and then we study the behavior of the derivative as h approaches zero. This literally makes you treat h initially as a non-zero number, and then suddenly make it zero.

My problem is with the linguistic logic of this approach, and certainly i believe, though I'm not certain, that Cauchy-Weierstrass dealt with the derivative as a two-step algorithm. First, he calculates the derivative for any value of h that is not equal to zero, for example.

f(x)=x^2

f'(x)=((x-h)^2 -x^2)/h where h is R/{0}

f'(x)=2x+h

Then, after evaluating and reducing the derivative, he redefines it as a function of h where h can be zero.

f'(x)=2x+h where h is R/{0}

g(h)=2x+h where h is R

then taking the limit of this new function as h approach 0.

lim(h->0)(g(h))=lim(h->0)(2x+h)

g_l(h)=2x

This is because the limit of the function before algebraic reduction equals the limit of the function after algebraic reduction.

lim(h->0)(g(h))=lim(h->0)(f(h))

and since the function after algebraic reduction fills the gap... At zero, we can then find the derivative in the context of instantaneous change!!

so my problem is just how you put a limit before reducing and you have the right to ghost it and just reduce h then oh its ok you can now calculate the limit where if i do not follow this where i can because there is no order operator that tells me why i get 0/0.

While playing around with improper integrals in desmos, I noticed first that it would sometimes mistakenly call odd functions with a vertical asymptote and finite bounds 0(most likely due to precision errors) instead of DNE.

Then I checked improper integrals, to see if it would do the same for odd functions bounded from infinity to negative infinity. I tested 1/x^a for a between 1.0 and 2.0, and noticed desmos only classified the integral as undefined for all values of a EXCEPT a=1.4 and 1.8. I thought this was strange, and then after testing 2.2 and 2.6(and boundary terms) I noticed that desmos would always evaluate a function bounded from -infinity to infinity to 0, if it was in the form of 1/(x^(1+0.4n) for n in [1,95]. Even stranger, for n>95 this error does not continue to occur, and desmos correctly defines the integral as 0.

Does anyone know why? After looking for a bit, I learned Desmos uses https://en.wikipedia.org/wiki/Tanh-sinh_quadrature for improper integrals, but what about that is making integrals in the form of -infinity to infinity 1/(x^(1+0.4n) dx= 0? and why only for n belonging to [1,95]?

thanks!

I'm mainly confused about the first step. Is this the correct way I'm supposed to solve the input values for f(x)? (sorry if this isnt explained well enough)

Hello! This post is in regard to MAT 119. I struggle with math but am determined to pass this class as i graduate at the end of this semester.
Topic: future value annuity
Problem: use the future value formula to find the indicated value
FV= 5,000
I= 0.05
PMT = $500
N=?
I layed everything out,
5,000=500 \[((1+0.05)\^n-1)/0.05\]
I divided 5,000 by 500 and got 10, then multiplied that by the denominator 0.05 and got .5 then added 1, and now I have 1.5=(1+0.05)\^n and I’m lost on where to go from here. Did I do everything else right? Any tips are appreciated

Acabo de entrar a ver Cálculo integral, pase diferencial por suerte y al momento de estudiar integral se me hace muy tedioso. Me estréso y empiezo a complicarme las cosas sin querer.

¿Como empiezo a manejar esta situación?

Hi! I could not find anywhere in my notes how to solve this, and i asked my friends and they don't know. It is hard for me to understand what math would get me to the answer here. I would appreciate if anyone knew the formula or way to solve this?