In my freshman year of high school, I took geometry and covid knocked out the trig unit. As a sophomore, the trig unit of precalc was online and a total disaster. I am taking Calc III now and it is assumed that we understand the trig concepts applicable to calculus. I would like to know the general rules that are the building blocks of that understanding, and a good resource for delving into the subject more thoroughly so that I can stop winging it whenever I see theta in a question. If you have any information it would be very appreciated.

In my freshman year of high school, I took geometry and covid knocked out the trig unit. As a sophomore, the trig unit of precalc was online and a total disaster. I am taking Calc III now and it is assumed that we understand the trig concepts applicable to calculus. I would like to know the general rules that are the building blocks of that understanding, and a good resource for delving into the subject more thoroughly so that I can stop winging it whenever I see theta in a question. If you have any information it would be very appreciated.

I’m aware of the arithmetic mean (which is invariant under x -> cx) and the geometric mean (which is invariant under x -> x^c). Is there a mean that is invariant under x -> e^x?

Hi all

Background - I'm in bioinformatics/structural biochemistry, and I'm currently invested in a type of analysis known as Protein Energetic Network analysis. It basically treats proteins like 3D graphs with weighted edges corresponding to their interactions.

I'm hoping to get some insight regarding resources for better understanding the grapb part. Do you know of any youtube or text resources that describe graph theory fully? Additionally, other than shortest paths and centrality measures (betweenness, closeness, eigenvector, etc.), what other analysis techniques can we analyze graphs with, and what information would we be able to get?

Many thanks in advance c:

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I think this is missing info like the directional relationship between the vectors. The solution says vector A is 7.05 m at 28.4 degrees. How do I solve this??

Suppose X is an arbitrary element of G. Suppose a is an arbitrary element of UF. Then there must exist a set Y such that Y ∈ F and a ∈ Y. Since every element of F is a subset of every element of G, a ∈ X. Since X was arbitrary, a ∈ ⋂G. We can thus conclude that ⋃F ⊆ ⋂G.

Any tips/stuff to fix? In particular did I condense the "Since every element of F is a subset of every element of G, a ∈ X" too much? I wrote the logic stuff out on paper and thought it was too messy for the write up. Any input is much appreciated, thanks!

The question is:

Find k such that y=x^2 +kx-x+k+2 has exactly one real root

I can't figure out how to get only one real root

All I can think to do is this:

y=x^2 +x(k-1) +k+2

a=1 b=k-1 c=k+2

b^2 -4ac=0

(k-1)^2 -4(1)(k+2)=0

k^2 -2k+1-4k-8=0

k^2 -6k-7=0

(k-7)(k+1)=0

k=7

k=-1

This is two different roots right? How do I get exactly one out of this

Good day sirs,

I want to prepare for a Master's Degree in Data Science. I know that the first courses are Intro to Algorithms/Trees/etc.

However, it's been a while since I've been in any academic situation, and I want to brush up on my math before I go into this. What math courses (coursera/udemy/whathaveyou) do you experts suggest? Please, I beg you, don't just name concepts because there are thousands of materials online and I wouldn't know good from bad. I understand that I need to know certain concepts, but it would be great if you could tell me the courses that offer said concepts.

Any and all help is appreciated!

I’m wondering if there’s a way to do a Taylor series approximation to prove the approximation above. I’ve tried to do the derivative using tangent inverse and making a Taylor series that way but I just can’t seem to make it equal to any Taylor series function using e^6x.

Hi everyone , I hope you are all doing well. I know this problem relates to parabolas but im not sure how to solve it. Thanks for your help!

Damien went skeet shooting. The trajectory of the clay pigeon followed a second degree function. If the machine were placed at the origin of a cartesian plane, then the pigeon reached a maximum height of 25 m at a distance of 50 m from the machine. Damien's shot followed a linear path whose equation was y=-0.2x+35. At what distance from the machine did he hit the pigeon if he hit it on its way up?

I'm trying to work on this problem on my homework: https://imgur.com/a/aykI58x

(c) should follow trivially from (b), but currently I'm stuck with thinking about (a) and (b).

To show that T is injective, I'm trying to show that null(T) = {0}. So I set 0 = rp + sq, and now I'm trying to show that this equation implies that r, s = 0 (then null(T) ⊆ {0}, and since {0} ⊆ null(T), null(T) = {0}). This is more or less where I get stuck. I suppose I can consider the division algorithm knowing that p, q have no zeros in common? I'm not sure what the best approach is. Thanks for any suggestions.

Hi all, this might be a stupid question, I searched but couldn’t find an answer to this.

Suppose we have a directed graph G=(N,A) with the source node denoted as ‘s’, arc weights are c_ij for (i,j) in A. There’s a directed path from s to any other node.

I’ve searched online, and almost all the sources say the shortest path question become meaningless if there’s a negative directed cycle in the graph because the negative cycle can be traversed again and again to reduce the cycle length to practically negative infinity.

But, isn’t the definition of a path such that no node is visited more than once? Wouldn’t traversing the negative cycle not result in a path anymore? My thought was if there are |N| nodes, and if path doesn’t visit a node more than once than there should be finitely many paths and one of them has to be the shortest one.

The sources I’ve found were also talking about the algorithms to find the shortest path, so maybe the negative cycles make it difficult for some of those mentioned algorithms to work? But in any case, shouldn’t the shortest path exist?

Edit: I guess I forgot to write it but by shortest path I meant from s to any other node in the graph.

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Would this be a valid answer if x represents students, y represents some food, and C(x,y) represents x being allergic to y?

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So far, I have found that triangle FAB is isosceles, and angle FAB is 120 degrees, so angle AFB = angle ABF = 30 degrees. It is possible to divide triangle FAB into two congruent right 30-60-90 triangles, and solve such that FB is 10sqrt(3).

Because FB is the hypotenuse of right triangle FQB, you can also find that QB = 10sqrt(2).

I am stuck solving for QP. I don't know what else to do.

There is an adage that “anything is possible”. This saying has always annoyed me because it is obviously, intuitively, not true. Surely! More things are impossible than possible.

Is there a way to calculate what is possible vs impossible?

This is possibly an excel question…

I know that I can get the sum of a progressive series where each number is progressively larger by the initial value by calculating first and last, adding them together, dividing the number of data points by 2 and multiplying… but I have no idea what the syntax of that function is.

Help?

Math is known to be universal but if another intelligent species saw an equation like 2x^3-5/19(4x^2+6) would they equally solve it left-to-right or right-to-left or using a completely different order of operations that that species agreed on?

(Sorry if I didn't flair right, I didn't know what I should post under)

Edit: thanks for all the answers everyone, convention seems to be the agreed upon answer. That also answers my question about another intelligent species not using PEMDAS.

Hello! I am not sure if this question is appropriate for the subreddit, but I figured it was worth a shot given its mathematical nature.

I have been trying to figure out how to calculate the internal volume of a cube in Minecraft. That is to say, how many blocks can a cube hold, not counting the borders? I believe the solution might be some kind of logarithmic or exponential function.

As you can see below, I have included a visual model of what I am trying to solve. From left to right I built a 5x5x5, 4x4x4, and 3x3x3 cube. They can each store 27, 8, and 1 block respectively. It seems like they store their volume -2 to each dimension. However, I am not sure how I would model this mathematically. I am sure there is a way though, right? I tried base 2 or 4, but it doesn't seem to match.

https://preview.redd.it/so9jrozi6lmc1.png?width=1920&format=png&auto=webp&s=621a2146ff736b59b6f3a586994705e47e403ad1

How to prove for A and B square matrices, if AB invertible then A and B invertible without det

now i know that AB=C

A(BC^-1)=I and (C^-1A)B=I but this only show that B is left inverse and A is right inverse and not inverse left and right., i have been stuck here trying to find a way but couldn't reach to anything

I have a deck of 60 cards. 42 are blue, 18 are red. If I draw 8 cards to start a game, how does that formula look?

It’s surely not quite as simple as 5.6 blue and 2.4 red, right?

Hey, , since my hobby is ultra distance cycling I was wondering if there is an index on how wel you've ridden your bike in comparison to other contestants in a race, kind of like body mass index.

This is what i found:

"Bike Performance Index" (BPI). We'll consider the factors you mentioned: distance traveled (D), time ridden (T), total time elapsed (Te), and average speed (S). Here's a formula that combines these factors:

BPI = D/Te x T/Te x S

Explanation:

The first fraction, D/Te , represents the proportion of the total distance covered by the contestant.

The second fraction, T/Te , represents the proportion of the total time spent riding by the contestant. Multiplying these two fractions gives us a measure of the efficiency of distance covered and time spent riding.

Finally, we multiply this efficiency measure by the average speed (S) to incorporate the speed factor.

Here's an example: Let's say a contestant covers a distance of 400 kilometers in a race that spans 5 days (120 hours). They spend a total of 30 hours riding their bike during this time and maintain an average speed of 20 km/h.

BPI = 400/120 x 30/120 x 20 BPI = 1/3 x 1/4 x 20 BPI = 1/6 x 20 BPI = 20/6 BPI ≈ 3.33

So, the Bike Performance Index for this contestant would be approximately 3.33. The higher the BPI, the better the performance relative to other contestants.

End

Since this is way over my head math wise I don't know if this is right

My question is , Is this equation valid (usable) in the sense of real comparison between riders? And what do you think of this all?

I have 7 cards in my draw pile The game "randomly" draws 5 cards What's the probability of drawing both "pummel strike" and "bash" in that draw so I can kill the boss?

This thing is bothering me so much And also why following logic doesn't work:

Let's say the game draws 1 card... The probability is 2/7, right? Let's repeat that 5 more times 2/7 + 2/6 + 2/5 + 2/4 + 2/3=... Why does it add up to more than 1? What did I do wrong?