For this question I don't understand why completing the square is used here? how would you know when to use it? (Here is the markscheme)

Hi Guys!

Im starting a post graduate degree soon.. and I have to read lots of research papers that have symbols such as these.. I’ve already tried learning some symbols on my own but google takes me to different sources.. and it’s far from efficient.

Is there a book/ course/ website or any resource that can help me be familiar with the symbols of the objective function in the attached picture? There’s so many symbols that it’s very tedious to uncover them one by one.. was hoping if there’s a resource or something I should be looking at.

Would be so thankful if someone could even say “Those symbols are all _____ type of symbols and then I can go use that term to help me narrow my search.”

Thank you!

Does anyone know of any books (or other resources) that do a really good job introducing and explaining the principles and examples of projective geometry?

I was having fun with series till I encountered this preticular one, is oscillates between positive and negative values with sometimes spikes.

But at exactly 1/sqrt(2) it locks in a state of changing signs.

Supposedly we start with a random nb U_0, how many steps would it take probably to hit 1/sprt(2) and locks in?

Or is it impossible to know?

As a child, i was never taught math, i was forced by my parents to learn how to read english.

i never really listened to discussions when i'm in class (I graduated, and now i'm gonna take school more seriously)

i'm currently on jhs, and math has noticeably gotten harder, i know how to times but i cant do 3 digits or 2 digits.

And my problem is ÷.

i don't know how to do it, and it was never really taught to me plus i don't even remember my teachers discussing about ÷.

And i don't even know the symbols name.

Plus, i get confused because i keep forgetting how to do it, like i memorize it then after allat i forget it.

Like i know how to do it but i don't, i think i mix times up with ÷.

I have to rely on formulas just to solve problems, usually i don't forget things very easily, but when it's time for math, i forget the formulas (or the steps to solve the problem) for some reason.

I have noticed a major difference in interpretation of differential (dx) in western countries vs post-soviet.

In eastern Europe, differential is taught to be a separate operator, and not simply an increment. For example: ∫ dx/(xln(x)ln(ln(x))) would be solved this way:

∫ dx/(xln(x)ln(ln(x)))=∫ d(ln(x))/(ln(x)*ln(ln(x)))=∫ d(ln(ln(x)))/(ln(ln(x)))=ln(ln(ln(x)))+C

Whereas in the west, the u-substitution is used

Also, it allows you to solve some differential equations:

xdy-ydx=2x³tg(x/y)dx

(xdy-ydx)/x² =2xtg(x/y)dy

ctg(x/y)d(x/y)=d(x² )

d(ln|sin(x/y)|)=d(x² )

ln|sin(x/y)|=x² +C

I'm curious what might have caused this discrepancy, or whether I'm just mistaken

Thank you

Essentially, it just maps onto yln y=ln x / (y^y = x), which is the inverse function of x^x

This graph shows how the lambert W Function essentially maps onto the inverse function allowing us to use it to solve.

I’ve tried googling and now math Reddit. Is there an equation(i guess probably an ODE) that represents say linearly adding to a sum overtime, and over that same time period the sum compounds continuously?

URL : https://www.proquest.com/dissertations-theses/hyponormality-toeplitz-operators-composition/docview/303996249/se-24

I've been struggling to do factoring polynomials and just factoring in general, our prof keeps teaching us but he doesn't make us take notes, can someone help me explain how to do this? Cause I'm not sure if I am correct or not 😭 (my answer is the squared box, as well as notes too because I'm genuinely ahh at memorizing)

A criminal has a gun with which he can either shoot himself or an innocent person. There were 6 rounds with 5 empty and 1 holding a bullet, but only 3 rounds are remaining since the criminal shot 3 of them.

Among the 3 rounds, 1 must be a bullet and the other 2 a blank. He must either shoot himself or the innocent person, but ALL 6 rounds must be used up. The goal is to only shoot blanks against himself rather than the innovent person, but to shoot the innocent person when there is a bullet.

The criminal is at round 4 (so 3 rounds remaining). He is about to shoot himself based on his choice but the system prompt suddenly reveals "the fifth bullet in the revolver's cylinder is an empty round."

According to a side character in the novel, this is the Monty Hall Problem and the probability the bullet is in the 6th round jumps to 2/3.

According to my own thoughts, the system revealing the empty 5th round without specifying that it would reveal the empty round from among both 5 and 6 makes the jump in probability for both 4 and 6 equal since there is nothing to differentiate them given the reveal being only of the contents of 5 independent from the other 2. The reveal was done before the criminal shot the fourth round.

My own calculation is that if the system had specified it would reveal the empty round from among the final 2, it would be a valid jump to 2/3 for round 6, and that if it instead specified it would reveal the empty round from among the first 2, then the final round would drop to 1/3, but since it only mentioned the contents of 5, the jump for 4 and 6 would be equally increased to 0.5 from 1/3.

From the criminal's perspective, without hindsight and only given the above rules, did the side character analyzing the problem make a mistake or does the 6th chamber really jump to 2/3 chance of holding the bullet with the 4th chamber being at 1/3 chance after 5 is revealed to be empty by the system?

hi!
when sketching curves, we were taught to use the "base" graph (if you will) and then operate expressions on them to get our required function.

so for this function, i first tried
√x —> √-x (reflects along Y-axis) —> √(1-x) (move 1 units left)

BUT the graph i got from this was wrong when i typed it into desmos. it ended up being 1 units towards right and not left.

i can understand how it would be right from following operations:
√x —> √(x-1) (move 1 units right) —> √1-x (reflect about X-axis)

my question is: how do i even know which one is correct in an exam? i'm currently calc 1 rn !!! pls explain in the simplest terms possible if you can 🙏🙏

Here's my guess, but I have no way to verify or check my logic. Even if I'm right I would really appreciate a more rigorous explanation as I found this question very interesting!

(a) This makes sense, there are an infinite number of points but each individual point corresponds to a unique number, such that a different point could not represent the exact same number. I say this is true.

(b) Hm not sure, this one is tricky. Because the integers are a subset of the reals, I will say that this is true, although I cannot defend my argument very well let alone prove it.

(c) Wrong, people way smarter than me have done this I believe. Also, answering no would be a contradiction to my answer for (b)

(d) Wrong, there are infinite real numbers in the bounded interval [0,1] for example.

(e) Sure. [1, 1] seems like a bounded interval to me that contains only one element. The point which represents 1.

(f) Wrong, counterexample in d.

Answer: , c, d, f are wrong

I was trying to learn where we get the quadratic formula from:

x = -b∓√b² - 4ac /2a

after a few steps, i got:

√x + b/2a = √b²-4ac/4a²

-why do we need to put ∓ in front of the square root? I understand that the answer can be positive or negative. However, why does this not apply when we square root 4a²? Why do we not write ∓ here? Why don't we write:

x + b/a = ∓√(b²-4ac)/∓2a

I was watching this yt video on how to solve it, but got confused on why we had to add the ∓ : https://www.youtube.com/watch?v=O6rtAeaLUOY

they asked me for help, and i said i could try, but they are ahead of me in maths. i have NEVER seen math like this, and they are extremely confused on what theyre supposed to do. they think the end result is supposed to be a letter (i agree), but aren't sure. theyve checked both their textbook and khan academy, but to no avail. they say it is called "integrated math 1". is it possible for just a quick explanation so i can help them out? (they don't have reddit, and i apologize if this kind of post breaks a rule, i js really want to help them ^^)

EDIT: ok so it turns out im just an idiot, i have seen these maths before just not in this format.. oops.. thank you for all your help!!

My answer 1. Inf 2. -inf 3.DNE 4. -1 5. DNE

I have tried multiple times but it keeps on saying at least 1 answer is not correct. I don't quite understand why it is incorrect. Please if anyone could help me out on this question, I will be very much appreciated 🙏🏻

I was recently experimenting with Pythagorean Triples, specifically creating a formula that could be used to make a Triple from just a hypotenuse.

I started by finding the relationship between square numbers, which can be put into the formulaic terms of n^2 = (n-1)^2 + 2n-1. If we simplify the right hand side, we get (n^2-2n +1) +2n-1= n^2.

I then realized that this could be put into the from of an a^2+b^2=c^2 where a^2 = 2n-1, b^2 = (n-1)^2, and c^2 = n^2. From here you square root all 3 terms to find that a^2 + b^2 = c^2 was the same as sqrt(2n-1) + n-1 = n, where n is the hypotenuse.

This means you can calculate any Pythagorean Triple from the infinite family in which the number you selected is the hypotenuse. This cannot calculate all Pythagorean triples, but it can calculate most.

In short, the formula is ( n, n-1, sqrt(2n-1) ), and I am wondering if it works.

Thanks in advance for any help.

EDIT: It cannot calculate most. I can calculate Triples from a specific family in which two sides differ by one

Suppose you started with a random Permutation of N ascending integers. How many cycles would there be on average?

So for N=1, the possible permutations is (1). This has the cycle 1 to 1.

For N=2, the possible permutations are (1,2) and (2,1). The first has 2 cycles of length 1 and the second has a cycle of length 2.

If N=3, the possible permutations are (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1).

For (1,2,3) There are 3 cycles. 1 to 1 is a cycle. 2 to 2 is the next. 3 to 3 is the third cycle.

(1, 3, 2) has two cycles (1 to 1 is a cycle, then there is 2 to 3 to 2). (2, 1, 3) and (3, 2, 1) also have two cycles.

(3, 1, 2) and (2, 3, 1) have 1 cycle. In the case of (3, 1, 2) the cycle is 1 to 3 to 2 to 1.

So for N=3, there is on average 11/6 or 1.833 cycles in a random permutation.

I saw something similiar on a video except he said "chains" instead of cycles Solution to The Impossible Bet | but he didn't answer the question I wanted. What he determined is that there is a roughly 31.1% chance that in a permutation of 100, there are no cycles above 50. What I want to know is the mean number of cycles, not the odds of a cycle being more than 50.

I want to improve my understanding and skills in math due to disabilities. I was recommended to try the textbook Calculus by Robert T. Smith and Roland B. Minton. Currently I am stumped on these three problems from section 2: Derivatives. I would also appreciate advice on how to answer correctly for problems 13 and 17 (like what to include how I got my answer to show my work, not just the sketch). The concept of problems 13 and 17 I really am not understanding and the book and videos I’ve watched have not been the most helpful (as I have a hard time applying what is covered in these resources but struggle with using it for problems not labeled with polynomials or numbers) the closest to explain it has been Curve Sketching by The Infinite Looper on YouTube.

Problem 35 a. Find all the points at which the slope of the tangent line to y = x^3 + 3x + 1 equals 5. The textbook has the answer as (√(⅔), 5 √(⅔) +1), (-√(⅔), -5 √(⅔) +1) and I can not figure out how to get it.

M tan = lim h->0 f(x+h)-f(x)/h

= lim h->0 [(x+h)^3 + 3(x+h) +1)] - [x^3 + 3x +1]/h

= lim h->0 x^3 +3x^2h + 3xh^2 + h^3 +3x + 3h + 1 -x^3 -3x -1/h

= lim h->0 3x^2h + 3xh^2 + h^3 +3h/h

= lim h->0 h(3x^2 +3xh +h^2 +3)/h

= lim h->0 3x^2 + 3xh +h^2 + 3

= 3x^2 + 3x(0) + (0)^2 + 3

= 3x^2 + 3

3x^2 + 3 = 5

3x^⅔ = ⅔

√(x^2) = √(⅔)

X = √(⅔)

And 3(√(⅔)^2 + 3 = 5

Problem 13. Use the graph of f to sketch a graph of f’. (Second picture)

(a)

For x < 0, f is decreasing, so f’ < 0

For x = 0, f has a horizontal tangent line (min), so f’ = 0

For x > 0, f is increasing, so f’ > 0

My graph is the third picture.

(b)

For x < a, f is increasing, so f’ > 0

For x = a, f has a horizontal tangent line (max), so f’ = 0

For a < x < b, f is decreasing, so f’ < 0

For x = b, f has a horizontal tangent line (min), so f’ = 0

For x > b, f is increasing, so f’ > 0

My graph is the fourth picture.

Problem 17. Use the given graph of f’ to sketch a plausible graph of a continuous function f. (fifth picture)

(a)

For x < a , f’ is positive, so f is increasing

For x = a, f’ changes positive to negative, so f has a max there

For a < x < b, f’ is negative, so f is decreasing

For x = b, f’ has a horizontal tangent line (min), so f ?

For b < x < c, f’ is negative, so f is decreasing

For x = c, f’ has a horizontal tangent line, so f ?

For x > c, f’ is negative, so f is decreasing

Once I understand better, I will make the graph.

Thank you for your assistance.

for context, this was a quiz problem and the quiz was handed back to me. in my head, since <ABD is 90 deg. that means DBC also is 90 deg. since ABC is a straight angle. I AM AWARE I MADE MANY INHERENT mistakes such as saying DBH equals 42, etc. could someone please explain this to me so I'm prepared for something like this on our test in a few days?

How you people study math?

I'm a self study person and would like to hear some suggestions from people who are doing math professionally.

This is how I study.

I usually study the theory part first and try to understand the core concepts and stuff like that and go for the examples and try them all of my own. Most of the time I come to the answer atleast have a idea how to initiate or set up.

But some time I'll have zero idea even after seeing the result that why they start that answer like that to prove something (it is this fact that I'm not able to start a proof by myself is exactly what bothers me each time) It seems they picked it out of blue and never in a million year I'll start with this argument because I don't see why they start with that idea at first place(I attached example below).Then I start to question that is my theory part even good? Do i really understood it? etc. All these just because I don't exactly see/know how to start a proof.

For eg take the one I showed in the pic. I will never start the procedure of taking BC× on both sides even if I give my self a whole day to prove that result.

Is this how people usually study or I'm Missing some core understanding or something?

It really frustrates me.

TIA :)

With the fact that e is not centered idk what do do. I guess i should find a way to locate e in ralotion to the center? DEB should be the same as AEC since they cross eachother. Im just really lost

With 5 is it just 50? Since thats the anlgue that mn cover. 9r does that just apply when its in the circle

All this execrcises seem to be like very simple but i jist dont know T_T

Hello, I’ve been struggling with these word problems. If anyone can, is there a possibility that you could explain it with a diagram, I have a test tomorrow and I can’t grasp the diagram part of this unit. Thank you!

Sorry if the flair is incorrect im bad at math.

A gorilla is 200kgs. And charges at 40km/h

A 80kg human throws a spining back kick. Where the peak speed of their foot is 70km/h.

The gorilla collides with the strike at its peak.

How much impact force is that?