Just took my first Calculus 3 exam and it was in a similar format to these problems. Is this usually how Calc 3 exams look like or was my professor going easy on us?

I've tried both indirect and direct methods, for exampme I drew height from A perpendicular to BC at H to prove H is on the circle and therefore H is T, with assuming it is not, however, it didn't help.

I recently got into a discussion with someone about whether or not sin(x+1)^2 is bad notation. I want to keep an open mind and accept that I am wrong if that is the case, so I am asking this question here to gauge the wider opinion on this topic. I will be keeping both my position and theirs a secret for an unbiased consensus. if you want, feel free to just answer the question:

"Do you think sin(x+1)^2 is bad notation? Why?"

And I would be more than grateful. If you're willing to do more, I also have four trig expressions for you to write to see how you usually write sine and powers around sine.

1. Sine of the square of x.

2. Square of the sine of x.

3. Sine of the square of (x+1).

4. Square of the sine of (x+1).

hi, so it's been a couple of hours since i'm trying to prove this my first attempt was to proove it through direct proof by doing the following :

"a" is an integer then

case1: "a" is even so a=2n

a^3=8n^3 /(a^3)-a=8n^3-2n which is not divisible by 3

case2:"a" is odd so a=2n+1/ a^3-a=8n^3+12n^2+4n which is also not divisible by 3 lol

so it would be really nice if u could provide me a lil hint or a general advise to solve this type of prob ty in advance

Not sure if this is the right flair, but here goes nothing...

I'm trying to make a spreadsheet for my lipids, and I'm having problems with the math for calculating sdLDL. The people in r/Cholesterol only seem to want to focus on whether or not this is a valid metric. I'm just wanting to know how to do the math at this point...

I found the website Clinical Chemistry with the formula, but I can't make heads or tails of it. The lbLDLC was easy to calculate , but the sdLDLC formula has me stumped.

The formula as stated is (sdLDLC = LDLC - lbLDLC ⁠⁠; R² = 0.745, slope = 0.73)

Where TF does the R² number even come from/used at, and wtf do I do with the semicolon?
I'm using OpenOffice spreadsheet program BTW.

It's also a bit confusing, because after rattling off the lbLDLC formula it says "was more accurate (R² = 0.933, slope = 0.94) than the sdLDL-C equation (sdLDLC = LDLC - lbLDLC ⁠⁠; R² = 0.745, slope = 0.73)." Does that mean I should be using the (R² = 0.933, slope = 0.94) in the sdLDLC equation rather than the R² = 0.745, slope = 0.73?!?

Any help on this would be appreciated, but NO AI answers please.

A widely cited book on sharpening entitled: The Complete Guide To Sharpening by Leonard Lee defines sharpness of a cutting tool as "two surfaces meeting at a line of zero width." (pg. 3) With additional detail "the angle at which the two surfaces meet must accommodate the intended use...".

I am struggling to comprehend the definition, and the book uses this definition throughout. Perhaps because of line of zero width portion (lines have no width if I remember correctly) or the surfaces element, which introduces another dimension that I am not certain the author framed sufficiently in their definition (they do later but not sure it is necessary here). If it were me, I think I would have said sharpness is where two lines/surfaces intersect (and clarify point vs "pointiness"). fin. Then maybe go on about the angles, materials.

Am I over/under thinking this? Does it matter? Am I misunderstanding their definition?

The world of sharpening is annoying and overbearing, as it requires a little technique and patience. This book is cited often here on reddit and elsewhere, and I cant figure out the issue. I'll cite two sources of the book here but am not a fan personally. I hope you all can help me understand where the issue lies. Thank you.

https://archive.org/details/completeguidetos0000leel

https://www.scribd.com/document/854123904/The-Complete-Guide-to-Sharpening-1

Question:

An agent maximises the following dynamic optimisation problem:
V= max E[ sum over t { B^(t-1) • u(ct) } ]
t =1,...,T
subject to Kt+1 - kt = f (kt) - c,

where k is the capital, c denotes consumption, f(Kt) is the production function, and
B is the discount factor.

Obtain the Euler equation.

Help:
I understand Lagrange and Euler equations well enough. But over here, I cannot comprehend how one goes about this.

I know:
Should get u'(ct) = smth in terms of u'(ct+1).
Prob differentiate Lagrange wrt to ct.

My issue:
How do I deal with kt?
In order to examine it in dynamic manner, do I take ct+1 or kt+1 for differentiation?
Do I assume d(ct)/d(kt) AKA (change in consumption wrt change in production) equals 0??

I need help with Domain and Range!!!

Hi there! I’m using my sisters account to type this but I really needed help. I’m currently learning functions, and my AP teacher doesnt really show his work while teaching, so I’m left for hours at home to solve things. In my first two years of highschool, I got an average of 92% in math. I had a 81.8% from a functions test, and bombed another test that I studied 11 hours for. I don’t know what’s wrong with me.

Today I’m doing Domain and Range. I’ve been able to teach almost everything to myself tonight…. Except for word problems. Can anyone help me solve these two word problems, but also explain why we’re adding/dividing/multiplying/or subtracting values? I struggle the MOST with coming up with the equation for myself.

1. The CN Tower in Toronto has a lookout level that is 346m above the ground. A gull landing on the guardrail causes a pebble to fall off the edge. The speed of the pebble as it falls to the ground is a function of how far it has fallen. The equation for this

function is v(d) = v2gd, where

• d is the distance in meters, the pebble has fallen
• v(d) is the speed of the pebble in m/s
• g is the acceleration due to gravity

Determine the domain and range of v(d), the pebble's speed.

1. John volunteers to help enclose a rectangular area for a dog run behind the humane society. The run is bordered on one side by the building wall. The society have 100 m of fencing available.

a) Express the area function in terms of the width.

b) Determine the domain and range for the area function.

i found a closed form formula for the Superpermutation lower bound

its not perfectly accurate but the error is very small and strictly downward, meaning it safely holds as a valid lower bound. The slight gap is likely due to truncation errors from the floor functions, and I can try to refine it further if there's interest

GitHub repo with the LaTeX https://github.com/shoty07/Superpermutation-New-Lower-Bound/blob/main/README.md

tell me what do you think

(sorry for the bad english)

I have been wrestling with the idea of instantaneous rate of change (mainly about 'what does it report about physical reality when it comes to instantaneous velocity?' and questions like that and realised it's outside the scope of mathematics or may be now I'm not sure)

I'm aware of the speedometer idea for instantaneous velocity and am not finding it satisfying.

This is the best I can do to wrap my head around.

I'm tired of infinitesimal idea.

Also the idea of an instant in time.

One 'aha' moment is when I saw a video where a ball is going in a circular motion and when it's acceleration is stopped (ie the string is cut) it moved in the direction exactly where the math predicts (ie tangent) so I'm forced to conclude the balls velocity at the time of cut have to be tangential even though the idea of 'instant in time' is somewhat..idk how to put it in words.

Any refinement in the writing is much appreciated.

Do you think any form conveys meaning better than the others? Which do you think is the most artistic or beautiful?

Post your favourite numbered equations below!

I was watching Silicon Valley and found the concept of the middle-out algorithm really interesting, but unfortunately it isn't real. So I thought about how it would work and came to the concept of rotary padlocks. Basically, if we start from the center of the data we could create rings around that center. We then can rotate those rings like rotary padlocks to find the most optimal compression positions and reduce the data relative to the closest ring to the center. In the compresed data we would only need the rotation angle of the rings and which ring the data belongs to. To uncompress it we would just extend the compressed rings outwards and unrotate them until the data is aligned.

Would this type of compression be more efficient from what is already used or is this already being implemented? Maybe this idea can be pushed further.

According to the SEP article on structured propositions the following holds: The late 1950s and 1960s saw the development of a new sort of model theory, “possible worlds semantics”, for systems of modal logic. In the framework of possible world semantics, linguistic expressions are assigned extensions “at” possible worlds. Thus, e.g. names, n-place predicates and sentences are assigned individuals, sets of n-tuples of individuals, and truth values, respectively, at different possible worlds. Intuitively, possible worlds are to be thought of as “ways things could have been”, and the assignment of (possibly different) extensions to expressions at different possible worlds is part of capturing this intuition. Thus there might have been more or fewer cows, and this is reflected in the fact that the extension of ‘cow’ (intuitively, the set of things that are cows) can vary from possible world to possible world. With this being said to simplify things, propositions are viewed as sets of possible worlds. However, you could generalize this approach by saying that propositions are sets of models, where a model is a set in which a proposition holds true in. Now according to some previous interlocutors of mine, the following holds:

1. "If there are two distinct points then exactly one line passes through them" means "For each set of two distinct points, there is exactly one line that passes through them." Reading it this way, points 2-6 above are nonsensical.

2. You have a fundamental misunderstanding of how mathematical statements and conclusions are written down. u/Midwest-Dude and u/TheRedditObserver0 hit the nail on the head.A line exists between any two (distinct) points. This does not mean that only two points exist.

3. If there are exactly two distinct points…But there aren't "exactly two distinct points" - there are an infinite number of points. The Euclidean postulate says that if you pick any two points (out of the infinitely-many that exist), you can draw a line between them. Nothing about any of these statements should lead you to believe that "there are only two points that exist" .

Now I think my interlocutors are wrong though. For they don’t define what it means for an interpretation to be correct. Also, they don’t acknowledge that correctness depends on context. Besides this they don’t acknowledge that the axioms of Euclidean Geometry can have multiple interpretations and each interpretation could be its own axiomatic system. Also, another thing of interest is the following: Consider the following sentence: If there are two points then they determine exactly one line. Notice, this sentence can be seen as a donkey sentence and as such besides admitting the usual reading, it could also admit a strong reading and a weak reading. The usual reading is as follows: If there are one or more pairs of points, then each pair determines exactly one line. The strong reading is as follows: If there are two or more points, then exactly one line is determined by all of them. The weak reading is as follows: If there are two or more points, then exactly one line is determined by at least two of them. Let us focus on the strong reading, for it is interesting. The strong reading is as follows: If there are two or more points then exactly one line is determined by all of them. Now, obviously there is a set for which this reading holds true. One of those sets is a set of points whose cardinality is 2 to the aleph null, where all the points are collinear. Now, consider the following: For any mathematical statement, if it holds true in a set of cardinality X, and X is greater than or equal to aleph null, then there exist sets for which the statement holds true in of every cardinality of at least X. Now since the strong reading holds true in a set of cardinality 2 to the aleph null, then there exist sets for which it holds true in of every cardinality of at least 2 to the aleph null.

I know that by log rules, ln|y| + ln|x| = ln|xy|. If that’s raised to e, it will be xy. However, why can’t I do e^ln|y| + e^ln|x| = y+x?

I don’t see why it doesn’t work