The problem is:Take a limit of these sums as n approaches infinity to calculate the area under the curve over \[0,1\]
I get stuck on the Σ calculation…

Can I proof them without using integration? I ask ChatGPT and it said I have to make t=hx and take out x…

My instructor wants us to display 15 Divided by four on a number line using subtraction. I’ve tried countless ways to display it but all the ways I’ve tried my teacher I said I was wrong. He said he wants it in a subtraction form, but I’m still struggling how to display it. Maybe this is a no brainer but all the ways I’ve tried he’s told me it’s wrong. He wants it specifically to be subtraction. The first one is my latest attempt but it doesn’t feel right so I haven’t submitted it yet. Help and explanations is appreciated please.

I am trying to solve the following problem:

Let f be a differentiable function on [0,1] such that f(0)=0 and f(1)=1. Prove that there exist two distinct numbers a,b in (0,1) such that

f'(a)f'(b)=1

By the Mean Value Theorem, I know that there exists some c in (0,1) such that

f'(c)=1

since

(f(1)-f(0))/(1-0)=1

But I do not know how to prove that there exists another point d different from c in (0,1) such that

f'(d)=1

Any hints or ideas?

This is a question my linear algebra teacher left for us, this is my first linear algebra course and im a 3rd semester math student, we are currently studying bases, dimensions and linear independence. I know that each base needs to have 4 vectors, I can't use (0,0,0,0) in any of the bases and there are 15x14 options for the first two vectors of any of the base, but im stumped beyond that. How could I check the linear independence of all possible subsets of S? Is my only option manually checking? Any help is appreciated as this is the first time I post here.

In lecture they didn't teach us that type of questions
the only thing i know is when given a set of these vectors we have to compressed them using a representation vector + data point but i don't know how to calculate no. of data points and number of features

we can choose from that line any point right or am i interpreting this in wrong way

I have tried to substitute the points (-7,15) and (0,299) in f(x) but I only get two-4th degree polynomials, which is not enough to solve. Perhaps manipulating the properties of functions could yield some insights, but I'm lost.

I'm a bit confused on how to start this question, our teacher has been very dismissive and haven't delivered us the pre requiste knowledge required to solve this question, I'll be grateful if an expert can assist or guide me on this. Thank you

How did they go from the top equation to the bottom? Why did they seem to swap around the roots? I thought the a - a on the bottom would just cancel out so the fraction is all just over 1, so that you’d get /a - /a+1 (cant find root button lol) but it seems they’ve done something else that I can’t see?

Here is the mark-up from r(6) to r(12) assuming the equation for 'r(k)' represents the exponential rate of change per month.

Is there another answer, this could be?

thank you for the help!!

Is this right? I put it into google after the test and it gave me something completely different

Got to the point where I tried AI and was still getting it wrong.

i only managed to do it until this point but i don’t know how to continue

Shouldn’t angle E correspond to D chatGPT says other wise

i thought i would be able to find the matrix A and compare elements accordingly to solve the simultaneous equations but based on this it feels like i’m on the wrong track