##c = sup \{x \in [a, b]: f(x) \le 0 \}##
f(c) < 0
c ∈ [a, b] a≤c≤b
a≤x≤b
c ≥ x
x≤c≤b
f(b)>0
b∉S
Intuitively it makes so much sense and I feel really close but it hasn't clicked yet
I see the logic of the first statements, here's my attempt
There is also a possibility that there are several c's for which f(c) = 0 correct? But we are trying to show if there is at least one.
If ##f(c) < 0##, then it's in an interval (a,c]
I am thinking this is where p comes in and there is...
I can't figure out the rest of it either
The hint says show f(c) = 0 by contradiction and consider two cases. Apply the fact that if f is continuous at 'a', ∃ an open interval I centered at 'a' such that f(x) >0for all x ∈ I. Explain why 'c' not equal to 'a' which will mean a<c. Then let 'p' be...
Thank you so much, Can't believe it was right there and I couldn't see it. So for showing a ≤ c from the set def we know that
a≤x≤b and as c in an upper bound it's
c ≥ x and by transitivity
a ≤ a
Now on to the rest of this problem :)
Homework Statement
We've been given a set of hints to solve the problem below and I'm stuck on one of them
Let f:[a,b]->R , prove, using the hints below, that if f is continuous and if f(a) < 0 < f(b), then there exists a c ∈ (a,b) such that f(c) = 0
Hint
let set S = {x∈[a,b]:f(x)≤0}
let c =...
Homework Statement
2. Relevant equation
Below is the definition of the limit superior
The Attempt at a Solution
I tried to start by considering two cases, case 1 in which the sequence does not converge and case 2 in which the sequence converges and got stuck with the second case.
I know...
I had recently upgraded my version of python from 2 to 3. I had a program that encrypted a text file by converting a character to its Unicode value, altering it and then changing it back to a character using the ord() and chr() methods. This does not seem to work with python 3 and I was...
These look really helpful I will have a look into them. I had been reading that the Littrow configuration you mentioned had high efficiency. However wouldn't the order with the most energy be reflected back on the incidence beam and due to the conservation of energy the other orders be less intense?
On another note, I'm likely to make this another thread but my aim is to uses the grating to combine two laser beams of different wavelengths. I want to do this in an energy efficient manner. Do you know any good resources I could use that explains the math of a blazed grating well?
I am trying to understand how a blazed diffraction grating works and came across a deduction I don't understand. I believe that you don't need to know much about optics to answer this question as it is more geometry related.
I have the diagram below of a diffraction grating with all the relevant...
Yes, I can get the error from the spectrophotometric measurement which is ±0.0005 from the readings which is around 0.09% for my first reading and 0.48% for my last. So I don't need to bother with calculating the error in the concentration. Got it thanks :)
Homework Statement
Some Background - We are calculating the amount of acetylsalicylic acid in a sample using spectrophotometry. We were told to make sure to include the error in our answer. So first to calculate the moles of acetylsalicylic acid in a measured mass.
0.1620 ± 0.0005g measured...
Oh I see. The period is, ##\dfrac{2\pi}{\omega}##, which means that the equation becomes zero at this period meaning that the peaks occur at this period which is independent of ##\varphi##.
Thank you all.