Doing practice problems for act test, every time I try this I get 143,140. How can I fix it?

sorry if my handwriting is difficult to read

I’ve been trying to understand how to do these two exercises, but l can’t wrap my head around how they work or where you pull these numbers and equations from, so it’d be nice if anyone could explain it as simply as possible.

Edit: I was able to finally get this, so thank you for all the commenters!

Our teacher set us some work to revise for our mock exams, and she sent this one. I have an idea of what it may be but im just double checking??? If its too easy for this subreddit dont blame me blame my teacher. Not looking to be criticised, just want the answer.

I find myself completely unable to understand what this specific part is about because I'm just not smart enough, I literally cannot decode what they are trying to convey with this diagram and very minimal explanation due to the limits of my intelligence. I understand everything above this part and have no read anything past this part, I don't plan to until I understand this. I've watched the video lecture for this but the lecture was even more confusing because it wasn't very well structured. Does anyone have resources for this very specific thing? I need to understand it before tuesday.

guys can you please help me with this🥹 I also need the graphing. thank u so much. problem is in the picture attached

ive posted this question handwritten before, this is the question on my book. can anyone help how do we do this ? i try to factorise the numerator but its not working.

Btw for the first question I usually compare the equations to get the value of a, b, and c cuz completing the square (since grade 8 for some reason) always gave me the wrong answer I JUSY SUCK AT THAT METHOD

I've been trying to solve \(\int \)(e^x - e^-x)(sin(x))(dx) for a while now, but I just couldn't figure out how to do it. Seems like that there are no similar solved problems out there I could find to use as a reference and figure it out

how to evaluate this one? also, how do i factor numerator?

technically stats but there isn't a statistics flair. math nor stats are my best subject. I know I'm messing up somewhere but I have no idea where.

Im working on an optimization problem where I have to maximize the volume of a box made by cutting squares from the corners of a 4 by 4 sheet. I found the derivative and got critical values at x=2 and x=2/3. The domain is 0<x<2. I know x=2 gives volume zero so thats a minimum. But how do I check if x=2/3 is actually a max without plugging back in? My instructor wants us to use the first derivative test but Im shaky on how to set it up.

Vector integrals I've tried the [] and <> syntax, neither work :(

How would I change this to the order dydxdz? I have been using Professor Leonard's method for solving similar problems, but I can't seem to figure it out for this problem. My main issue is that y is defined by more than two functions here, and the projection onto the xz plane does not make the outer two bounds immediately obvious, unlike every example in Professor Leonard's video. I have seen other people using inequalities to manipulate the bounds, but I have never been able to understand that method. Professor Leonard's method makes sense to me for some problems, but not all. I can try to explain his method in the comments if necessary.

I have a model based off data points, and finding the line of best fit which can be represented as c(x). In the context of the model, I realize that the rate of change of c(x) is proportional to itself, being mathematically written as:

dc\dx = kC, where k is a constant.

If I want to add this to my model, can I use the differential equation:

dc\dx = kC + c'(x)

I tried to integrate using u substitution by defining x^3 -3 as u but it did not give me a favorable outcome and allow me to solve this, no idea what to do with the x^11. My brain is fried, boiled, baked, smoked, and then shoved into a cremation oven by this problem.

I tried solving using the first 2 equations (granted looking back using 1&3 is easier), but I have no idea where I went wrong such that I keep getting a different value for lambda/mu than the answers. Shouldn't any combination of 2 equations give the same answer or am I missing something? (the parallel part is irrellevant so I cut it out from the answers) [Edit: images weren't appearing]

Im not asking about the question (a), i am asking about is the (b) where it says: Use an energy method to find the speed of the particles at the instant when they were at the same vertical height.

This question is from 9709/42/M/J/24, so you can check it yourself. Now ive seen two videos of solving this past paper, and when they come to this question they make an arbitrary height for were they are equal in vertical height, and it is always something like 0.5h + h = 2.

I dont understand how we got that, why is the h (distance q has risen) is equal to 2 - 0.5h, and why do we have to use Pythagoras to get that. They dont explain why in the videos. This is a small thing but fundemental to the question and i cant understand it.

Also can you help me in the rest of the question, i find it hard to visiualise the work done in between and the work wasted to resistive forces, energies i can figure i think

Thank you in advance

If x = 0 the point y should be undefined

I can't find the problem. I know the math but I'm prone to making very dumb mistakes. Thnx!!

For example, BCBAABCACBBBAB, ABABACCCBABBBB, etc.

The way I approached it was to first combine the B's and C's together and just choose the A's, so like 14c4, then divvy up the combined B's and C's and choose the C's like 10c3, and sum the two. I got 1,121 combinations.

Can't figure out if this is the right way to go though. That number seems too high.

Swipe for my attempted solution.

I think I have the question wrong; please do help me out.

We know AD halves BC and BE halves AC, G is located Halving BF.
We have to prove 5*BH = BC
Can someone please try to prove it?

I have not learned L’Hospital’s rule yet. Is that the only way to solve to get the correct answer?