##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.
I tried to take what both of you said into account and it seemed to work. I was able to solve for t but it is not what I expected as is seems to still depend on the dampening constant ##\Upsilon##. Did I make a mistake somewhere?
Homework Statement
[/B]
Question 3.9
Homework Equations
equation for dampened ocillation[/B]
The Attempt at a Solution
In case this might appear confusing, I derived(with respect to t) the equation for dampened oscillation given above and tried to solve for when it equaled zero...
Cities need to become larger as more people want to live there. Building down will provide that space while being less obstructive. There could be added benefits that would lower the cost of living by saving energy. A heating and cooling system may not be needed because the ground will provide...
True will definitely consider that, I was just worried our scope would become to big. Can you tell me which idea you would prefer to see in a project @sankonpure6.
Hello, Physics forum community.I have a project for an intro to engineering class in which I must design a system to solve an environmental problem. This class is only an introduction so I'm not sure how detailed and specific our work has to be. I must have been feeling very sci-fi at the time...
Thank you so much for your replies I found them extremely helpful. I didn't know about axle friction @rcgldr and that link you sent was very helpful @haruspex.
interesting, so when the load increases the surface of the rubber changes making it more slippery? that makes sense. Our track was a kind of metal and the wheels were a hard plastic. Is it safe to say these materials do not have an interesting property like rubber?
Firstly, thank you so much for your reply.
here's a little sketch of our spring launcher system before the spring had been compressed at all. We can alter how much it's bee compressed(x) by moving the second purple block back and forth. so in our first test with just the cart we compressed the...
Homework Statement
I know this is long, but that is only because this question requires me to be thorough, conceptually I think it isn't difficult. To make it easier to read I bolded the important parts.
In Physics lab, we were told to conduct an experiment to investigate the effect off mass...
Homework Statement
Homework Equations
V=IR and kirchoff's laws
The Attempt at a Solution
Number three is my attempt at finding a solution but I got stuck when I had to find an expression for loop one on the diagram. Thanks in advance for your help...
Homework Statement
In an experiment modeling point charges I had to find equipotential lines. I did so using a voltmeter. The results looked like this:
Not a good quality picture, but the equipotential line on the right says 1.3 mV and the one on the left says 2.3mV. From this I had to...
1. Homework Statement
I'm trying to integrate this, the only variable is y the others(x,w) are all constants.
Homework Equations
The ways of integrating that I am familiar with are substitution, trigonometric substitution, by parts & partial fraction decomposition.
The Attempt at a Solution...