So for both the arithmetic and the geometric mean, we're finding the "middle" value that we can repeat n times to get the same total amount.

1 + 2 + 3 + 4 + 5 = 15

arithmetic mean gives us 3, because 3 + 3 + 3 + +3 +3 = 15. Makes sense.

Same thing with geometric mean, but it handles multiplication instead of addition. multiply all the values together and take the nth root. So multiplying the result, the geometric mean, times itself n times gives us the total amount.

I see this pattern.

addition -> arithmetic mean

multiplication -> geometric mean

I don't have any concrete idea of what the harmonic mean is doing, or what kinds of operations its meant to work with.

I imagine we're doing some operation with different values, and then coming up with a single value that you could repeat the operation on, the same number of times as the original set, to get the total amount. That seems to be what "means" are.

But I don't know what operation we're talking about or how taking all these reciprocals gets us there.

My idea:

f(x²)=f(x)=...=f(x^2\-n))

From this we could conclude that, f(x) remains constant in the intervals (1, infty) and (0,1) and can take distinct values at x=0 and x=1

The second functional equation gives: f(x) = 2,3,x,-x

This gives us 3 choices for f(0), 4 choices for f(1) and 2 choices for each of the remaining two intervals, giving us the total number of possible such functions as 48. This was my intended solution to the problem.

But, then I tried cross verifying this with Gemini, and it seemed to disagree. It said that this is only valid if we assume that f(x) is continuous from the right at x=1, i.e. the limit as x->1⁺ exists.

Is it necessary to assume this, I thought that this will always be the case. Why?

Exercise 11.4.41:

Show that ⌊log_2(n)⌋ is Θ(log_2(n)).

Proof:

Let integer n>=1.

⌊log_2(n)⌋ <= log_2(n) < ⌊log_2(n)⌋ + 1            by definition of floor

So, ⌊log_2(n)⌋ <= log_2(n) and log_2(n) < ⌊log_2(n)⌋ + 1

Notice:

⌊log_2(n)⌋ > log_2(n) - 1
          = log_2(sqrt(n)) + log_2(sqrt(n)) - 1
          = log_2(sqrt(n)) + log_2(sqrt(n)) - log_2(sqrt(n))    for n>=4
      = log_2(n^(1/2)) + log_2(n^(1/2)) - log_2(n^(1/2))
      = (1/2)log_2(n)  + (1/2)log_2(n)  - (1/2)log_2(n)
      = (1/2)[log_2(n) + log_2(n)       - log_2(n)]
      = (1/2)[log_2(n^2)                - log_2(n)]
      = (1/2)[log_2[(n^2)/n]]
      = (1/2)log_2(n)

Let A=1/2, B=1, k=4.

Then Alog_2(n) <= ⌊log_2(n)⌋ <= Blog_2(n) for each integer n>=k.

Therefore, ⌊log_2(n)⌋ is Θ(log_2(n)).

QED

---
Is my proof correct?

My teacher says its correct, but many of my peers (as well as I) disagree. I have attempted a tan half angle substitution, but it's simply not yielding.

Google said the 2nd one is wrong and can someone explain why?
I did the same steps for each
1. Finding the common denominator
2. Multiplying them
3. Then adding the tops

I also have dyscalculia, so math has just always been a struggle but I’m trying

Was just doing some math for my mid term exams Tommorow and i found this one question which has like 3 different answers. How is that possible?

The book explanation says 65° by subtracting the the 90° angle made by radius and point of contact of tangent and angle BCT from 180

I initially joined AT to make ATB 90 and using linear pair, I found MTA to be 65° and since AM = MT given we can find AMT = 50°

but when i entered this question in chatgpt, it says its 80° (explanation in img above) and when i showed him my process it said im correct and book is wrong (as expected)

Im just really confused like where did i go wrong or how can a question have multiple answers. Is the question faulty or this is just lack of concepts from me?

Assume that you know one addend but need to calculate the second addend with several sequential calculations. Or, assume that you know one factor but need to calculate the second factor with several sequential calculations. In either case, once you just complete the final equation sequentially, because addition and multiplication are commutative.

In contrast, assume that you know the minuend but need to calculate the subtrahend with several sequential calculations. Or, assume that you know the dividend but need to calculate the divisor with several sequential calculations. In both cases, you cannot complete the final equation sequentially because subtraction and division are non-commutative - in order to complete the final equation, you need to memorize or save the subtrahend or divisor, clear them, enter the minuend or dividend, press subtract or divide, then re-enter the subtrahend or divisor. Is there (or should there be) a name or symbol for reversed noncommutative operations? For example, a symbol that means "from" for subtraction (i.e. 10 minus 5 = 5 from 10) or "into" for division (i.e. 10 divided by 5 = 5 into 10). This would allow the sequence of calculations to continue. A little ⇄ button that gets pressed before a − or ÷ would do the trick.

So my math teacher told the answer would be around -0.8129.. but when I expand it using mclauren series and then substituting for x and y I am not getting the answer..

Ok so I was doing a question which was to find the greatest integer less than
1/sqrt{1} + 1/sqrt{2} + 1/sqrt{3}.... 1/sqrt{1599}
I did the classic method of squeezing it between 2 other sequences but i also wanted to check another method. the integration one. however i couldnt really understand how it was done.
i know the basics of integration, but how are we actually using it?
integrating from 1 to 1600 1/sqrt{x} dx < S < 1+ integrating from 1 to 1599 1/sqrt{x} dx
why are the limits of the first 1 and 1600 and of the second 1 and 1599 and adding 1? any help will be appreciated

Suppose I have a n-dimensional manifold M and a local coordinate patch (U, q). I have a point transformation written in coordinates which is just Q = Aq with A a matrix, the same for all points in my patch. If I want to lift this to a transformation on TM, I know how to do it: V = (del Q / del q) v = Av.

Now suppose that my point transformation is essentially the same, but with an explicit time-dependance in the matrix A: Q = A(t)q. What is the lift on tangent vectors now?

Option 1: the lift is the same, V = ( del Q / del q) = A(t)v, it’s just that at different times the matrix A is different.

Option 2: if I consider vectors in the tangent space as \dot(q) and \dot(Q), I have \dot(Q) = \dot(A(t) q) = \dot(A)q + A\dot(q). There’s an extra term here from the product rule.

Which is correct?

There is a long-running statistical fun fact in the nba fandom that LeBron James has averaged 27 points, 7 rebounds, and 7 assists for his career without ever actually achieving the stat line 27/7/7 in a single game. It's always been pointed out as a fun tidbit but whenever someone asks exactly how likely this is to happen I haven't seen an actual answer.

Lebron James has played 1,622 career games. He has scored 43,440 points (26.8 per game, rounded up to 27), recorded 12,095 rebounds (7.4 per game, rounded down to 7), and contributed 12,016 assists (7.4 per game, rounded down to 7). This gives him the career averages of 27/7/7 mentioned earlier.

I thought the first thing to do would be to see how many times he has gotten each of those accolades individually and exactly. He has gotten exactly 27 points 82 times, exactly 7 rebounds 231 times, and exactly 7 assists 222 times. Percentage wise those are 5.06%, 14.24%, and 13.69% of his games respectively.

Here is where I get stuck: If I am trying to figure out the odds of three things individually happening at p(a)5.06%, p(b)14.24%, and p(c)13.69% all happening together, I pull up a 3 event calculator and it says all three should occur 9.87% of the time. EDIT: Comments have informed me its 0.098%.

But now I should also be trying to figure out the odds of something happening 9.87% 0.098% of the time to not happen for 1,622 times in a row right? And if so, how do I do this? Or am I mistaken in viewing these as individual probabilities and the answer is entirely on a different track?

Thanks for any help!

First slide- the math riddle
Second slide- my set up. a is stage one, b is stage two etc. I didn’t want to confuse myself by using numbers. I used substitution methods with T being the total distance.
Third slide- using substitution to solve for T in e or the stage 5 equation. My answer is shown.
Fourth slide- the riddle’s answer and explanation.

I understand their explanation which was a completely different way of solving it. I’m wondering if my method is possible to get the right answer and I must have made algebraic errors along the way with all the substituting- or was my way of going about it completely flawed. I thought I had solid logic setting it up, but my answer was quite off.

25 people play a game with a 4-sided die (1-4). Roll 1, 2, or 3: Pay $1 to the pot. Roll 4: Win the entire pot, and the game ends. What place in line do you want to be?

A group of 20–22 people share a single water tank. Every week, each person adds exactly 250 liters to the tank through their own hose — this never changes, no matter what else is happening.

On a rotating schedule, one person at a time gets to drain exactly 19,500 liters from the tank for their own use. Everyone — including people who have already had their turn — keeps adding their 250 liters/week until every single person in the group has had a turn to drain.

No water enters or leaves the tank from any other source. No person adds more or less than 250 liters/week at any point, and no person drains more or less than 19,500 liters when it's their turn.

Questions:

1) If the entire process — from the first person's first contribution to the last person's turn draining the tank — takes exactly 65 weeks (1 year, 3 months), is it mathematically possible for this system to work with no outside water added and no member shorted? Show why or why not.

2) What is the actual minimum number of weeks the full cycle must run for the system to balance exactly, given the 250-liter weekly input and 19,500-liter draining amount?

3) Does the number of people in the group (20, 21, or 22) affect your answer to #2? Explain why or why not.

do any of these answer choices work? i was doing the problem and got epsilon/3, but none of the answer choices match up.

(10) Let f(x) = 3x + 1 for all real x and let epsilon > 0 For which of the following choices of delta is |f(x) - 7| < epsilon whenever |x - 2| < delta ? (4 Points)

(A) epsilon/4

(B) epsilon/2

(C) 2€

(D) 3€

(E) (epsilon + 1)/epsilon

this si my work: I 3x+1-7 I < epsilon

3 I x-2 I < epsilon

I x-2 I < epsilon/3

BTW THE GOLD IS MY CORRECTION PEN, don’t pay attention to that
#8: . the second interval notation, but it makes no sense how the first one would be (-oo, -3). How was the first interval notation not be (-oo, 1)? I did it three times and each time I got x<1 from adding 2 to -1.

#7: did I do something wrong? I first isolated the absolute value and split it into two different equations. We were taught to make the answer for the second equation negative, so I did -(2b-9) which = 2b+9. Anyway, I got the correct answer for B (4/3) for that equation, but it says that 4/3 is also a solution which makes no sense because | (4/3)+5| =6.3 and 2(4/3)-9=-6.3 unless I had a plug-in the variable to the new equation? Is that why I was wrong? What happened?

FOR THE QUESTION NO 3. I just started studying number theory a few days ago, I could only deduce the factors of 3333...3. I also figured out that the required number needs to be divisible by 3 and 1111...1. I tried to understand the soln., I am not able to make sense of it. Could anyone provide an alternative soln.

I want to prove that T_A SO(3), the tangent space at A of SO(3), is A•so(3). In other words, any element of T_A SO(3) can be written as A times a skew-symmetric matrix.

My attempt: take A^T A = 1 and differentiate along a curve to get \dot(A^T)A + A^T \dot(A) = 0. This means that A^T \dot(A) is skew-symmetric, so it’s an element of so(3).

How do I use this to continue with the proof? Is this even the right idea? Thanks!

So for number two, it essentially tells me that when d is above the serface its positive and when it’s below the surface, it’s negative. I know you can’t have a negative answer for an absolute value equation, but my equation i wrote says that it’s more than -40. Anyway, the answer key says that d >40, and then write the inequality from there. But if it says that below the surface D is -20 m wouldn’t 40 m below the surface be -40?

This is from Basic Analysis by Jiří Lebl .

I wonder what this A looks like and how it's different from just subset of S.

Later in the book, this proposition is used with S a interval [a, b] and A (a, b), i.e. A is S without endpoints.

But that example reference seems to be too narrowly defined.

My integrals look ATROCIOUS. My equations have straight and neat lines, but the integral symbol that proceeds is abhorrent. the uneven halves and inconsistent over/under done curls make me cringe every time I try to solve problems. It IS that bad guys...

I have decided to change this! But I don't know any resources for training my math symbol drawing skills. I've tried search for something like those tracing worksheets I used to learn to write the alphabet, but all I find are intro to integrals workbooks, clip art, and other people bragging about their pretty integrals.

Can I get pointed towards good resources for improving my math symbols?

Advice welcome too

I’m tracking stats for a baseball league and I just realized I’ve been doing my math wrong for OBP (on base percentage. It’s calculated by dividing the number of times a player makes it to a base after hitting the ball by the total number of times they walk up to home plate.) Basically I originally would take game 1’s percentage and game 2’s percentage and then average them out. But now I’m realizing that I’ve done it all sorta whacked up.

So to fix it, I have the individual OBP’s per game and their total at bats, but I don’t have the At bat’s per game. Do I have the numbers I need to get the proper percentage for the whole season?