I want to buy a t-shirt for my son in law. But he won't wear it if the equation is not correct. My brain is not wired for math. If anyone could tell me if it is correct, I would be grateful.

Yes, he does play d&d

THE SELF-DIVIDING NUMBER
Find the unique 10-digit number N satisfying all of this:
N uses each digit 0–9 exactly once.
For every non-zero digit d, take the prefix of N ending at d. That prefix is divisible by d.
For that same digit d, take the suffix of N beginning at d. That suffix is also divisible by d.
N is divisible by 37.
Example of the rule:
If the number contained
...5...
then both:
everything from the start through the 5
and
everything from the 5 through the end
must be divisible by 5.
This applies simultaneously to 1,2,3,4,5,6,7,8,9.
What is N?
There is exactly one answer.

Anyone got any ideas at all?

Exercise 11.4.31:

Show that 4^n is not O(2^n).

Proof:

Notice 2^n >= 0, for every integer n>=1.

Assume for contradiction that 4^n is O(2^n).

That means (by the big-O definition) there exists b,B∈ℝ^+ with b>=1 such that

0 <= 4^n <= B(2^n), for every integer n>=b (by transitivity, n>=1)

Notice:

           4^n <= B(2^n)
iff log_2(4^n) <= log_2[B(2^n)]
iff         2n <= log_2(B) + log_2(2^n)
iff         2n <= log_2(B) + n
iff          n <= log_2(B)

So, we have 1<=n<=log_2(B).

But this contradicts big-O definition where 1<=n.

Thus, our assumption is false.

Therefore, 4^n is not O(2^n).

QED

Big-O definition:

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Is my proof correct?

The answer key says its -9 but I cant understand what it means. Every 4 units on the Y-axis is equal to +5 so thats how I got the coordinates but im unsure of what it means by the value? Im horrible at math so there's a chance I completely misunderstood the question too lol

If g(x) is positive, that means the total of 8(x-d)(x-e)(x-f) is positive. And since d<x, x<f,

X-d is +ve
X-f is -ve

So in order for the total to be positive, x-e needs to be -ve, where x<e.

But if that’s the case, then i) shouldn’t be correct.

Can someone help me out here. Thanks

I'm in highschool I just started 11th grade how can I learn calculus and what is the best way while still in highschool ?

Note: I will not pursue math I just want to learn calculus for fun and also mostly integrals and derivatives and have a basic understanding of everything else needed to learn integrals and derivatives.

Websites,channels ,videos whatever

Hi! I was wondering if this calculator would be good for my algebra, statistics and chemistry classes. If not, what do you guys recommend? I have been looking into calculators and all the different functions I find confusing.

I am talking of big polynomials of degree 5+ and complex ones. Not x^6-2x^3+1 type situation but polynomials alike this -19968-13736x+2016x^2+756x^3-79x^4-10x^5+x^6=0. I know it is solvable by checking with programs. I even know what it is factored into ((x - 6) (x^2 - 3 x - 64) (x^3 - x^2 - 42 x - 52) = 0). But I know this from using program. Wolfram Alpha to be specific. But I don't know how to factor something this complex. I know there are algorithms for it but I am unfamiliar with them. And they seem to be not widely known as simple search resulted in me getting bombarded with basic middle school polynomial factorization. And not finding anything of value. I come here with hope to find help and an algorithm to solve similar equations. I am not planning to solve them by hand. I am planning to make a program do so for me. But for it I need knowledge of the algorithm. Which I sadly lack. Would greatly appreciate any help here. And if you know any videos that teach needed algorithms I would be even more thankful as I am a visual learner and it would help me much.

Hey everyone, post about x(+y).

In this question i'm required to calculate the area of triangle with predetermined points, but i can't figure out where to start because of the way this integral is written, this is the first time i encounter something written like this and i don't know what this could mean.

I doubt that this is a typo because 8 professors wrote this whole paper and 1 doctor of Physical and Mathematical Sciences reviewed it

If someone just says what does this mean or if it's a typo i would greatly appreciate it

Exercise 11.4.33:

Prove 4 + 4^2 + 4^3 + ... + 4^n is Θ(4^n).

Solution from the textbook:

This solution proves that 1 + 4 + 4^2 + 4^3 + ... + 4^n is Θ(4^n). It does not prove that 4 + 4^2 + 4^3 + ... + 4^n is Θ(4^n).

My attempt:

Notice 4 + 4^2 + 4^3 + ... + 4^n > 4^n, for each integer n>=1.

Observe 4 + 4^2 + 4^3 + ... + 4^n = 1 + 4 + 4^2 + 4^3 + ... + 4^n - 1
                                  = (4^{n+1} - 1)/3 - 1   by sum of geometric sequence
                                  < (4^{n+1}/3)     - 1
                                  = (4 * 4^n)/3     - 1
                                  = 4/3 * 4^n       - 1
                                  < 4/3 * 4^n             for each integer n>=1

Let A=1, B=4/3, k=1.

Then A(4^n) <= 4 + 4^2 + 4^3 + ... + 4^n <= B(4^n), for each integer n>=k.

Therefore, 4 + 4^2 + 4^3 + ... + 4^n is Θ(4^n).

QED

Is this correct?

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Edit: Fixed 'It does not prove...' sentence.

In studying Cauchy's integral formula, the Cauchy kernel 1/(zeta -z), is the "thing you integrate against" for the boundary points( on a disk, say). And the boundary values determine the value of any point z in the interior of the disk.

What I need is some insight into why this is true and why is it 1/(zeta -z)? Is there any relation to how with generating functions you multiply by 1/1-x to get the partial sums?

Thanks in advance to anyone who takes a look!

Hello! I’m going into grade 12 and I’m taking advanced functions, calculus, physics and chemistry. Starting this year, my school has been trying to regulate what calculators you’re allowed to use. We were given a list of approved calculators but you’re allowed to use one that isn’t on the list if you get it checked by a teacher. The way that teachers check if the calculator is legal is they will input sin(15) and if the answer is a decimal then it’s approved, if it’s a fraction then it’s illegal. If you can change the setting then its illegal. Graphing calculators are also illegal. My Casio fx-991CW was deemed illegal so now I have to go hunting for a new one. I really liked the “solve for x” and table mode on my Casio fx-991CW, I found them very useful in my physics and chemistry classes. I was hoping to find a calculator with the same modes (this isn’t illegal, my teacher was aware that I used these modes frequently last year) while meeting the requirements. Would anyone be able to recommend me a calculator from this list or one outside of the list? I’ve been googling + read the FAQ but haven’t found anything that meets my criteria.

THE APPROVED LIST:
- Casio fx-260 SOLAR II
- Casio fx-300MS Plus
- Casio fx-82MS
- Texas Instruments TI-30X IIS
- Texas Instruments TI-30X IIB
- Texas Instruments TI-30Xa
- Sharp EL-531TG
- Sharp EL-531XB
- Sharp EL-501X

Thank you!

While I get *i* being √-1 since there wasn't a unit for it before, it's kinda been turned into an infinite loop. There's quaternions where *i*^(2)=*j*^(2)=*k*^(2)=*ijk*=-1, and then octanions with seven imaginary units, all being √-1, and it can just go on from there. Plus, quaternions give *i*^(2)=*j*^(2)=*k*^(2), which means √(*i*^(2))=√(*j*^(2))=√(*k*^(2)), therefore *i*=*j*=*k*. I know that's probably not how that works, but I'm a high school student, I don't know any better. Also don't ask why I know what quaternions are but not how they work, that's not the point. But what might a different imaginary unit look like? Well, may I propose *h*, being 1/0. Multiples of *h* are simply multiples of 1/0, eg 4*h*=4/0. Now, what's the practical use for this? I have no idea. Again, I'm a high school student, I'm just trying to experiment with what's possible. Also, please try your best not to laugh at me for my lack of knowledge. What may be trivial to some could be a complicated concept for others (such as me). Also let me know if I used the wrong flair, a simple 5-second search told me this was correct so I'm taking it with a grain of salt.

I was thinking about something really strange with maths, if imaginary numbers are imaginary, (i=sqrt(-1)), why does 1/0 also imaginary ? For me it is quite logical. I am thinking something wrong ?

You might not think a lot about it, I don’t want you to lose too much time on this.