You are not logged in.
No, that is not possible. We have f(x)≤g(x) in the interval (a,b), except at x=c. This means that it is possible to have f(x)≥g(x) ONLY at the point x=c. If f(x) and g(x) are both continuous, then you can have f(c)>g(c) only if at some point, d, with d<c, f(d)>g(d). But we said that f(x) can only be greater than or equal to g(x) at x=c, not at x=d. So if f(x) and g(x) are both continuous, then f(x) cannot be greater than g(x), and it should make sense that the limit would hold. But, it is still possible to have f(c)>g(c). Although this can only happen if either f(x) or g(x) are discontinuous at x=c. So imagine f(x) is smooth, but jumps up at the point x=c. In this case we have only f(c)>g(c). Now we must consider the limit of f(x) as x tends to c.
So, let
. Because f(x) is not continuous at x=c (it jumps), then f(c) ≠ L. In fact, in the case of this example, f(c)>L. So, the inequality still holds since the limit of f(x) as x tends to c remains less than or equal to g(x).Hi Al-Allo,
If f(x) ≤ g(x) for all x ∈ (a,b), except maybe at x=c where a≤c≤b, that means that f(x) will never be greater than or equal to g(x), unless at x=c. For this to be true, at x=c, f(x)=g(x), or one of the functions is discontinuous at that single point (this will not change the limit). This should make sense, because if f(c)>g(c) and f(x) is continuous, it must have surpassed g(x) at some point other than x=c, so it must be true that either f(x)≤g(x), or f(x) has a disconinous point at x=c, making f(c)>g(c). The single disconinuous point will not change the limit of f(x) as x tends to c.
In other words, either f(c)<g(c), or f(c)=g(c), or f(c)>g(c) only occurs at a single point which will not alter the limit of f(x) as x tends to c.
Thus,
Does this help?
This is just a question I thought up, but couldn't figure out how to answer:
A parabola whose vertex lies along the the graph of
What value of c will maximize the area between f(x) and g(x), where f(x)>g(x)? In other words, the area enclosed beneath f(x) and above g(x).
The resistors are in series, so their equivalent resistance is simply the sum of the three resistances,
Then we simply plug in our numbers:
Remember, this is power - the rate at which the energy is being converted. So if we want to look at heat, we can say that 18.52 Joules of heat are being produced every second that current is running through the resistors. For example, if you left the circuit connected for 10 seconds, 185.2 Joules of enerygy would be dissipated.
Does this make sense?
I do not know the reason why, but 1/7 = 0.142857 repeating. Then any multiple of that gives the same recurrence, but may start with a different place. For example, 3/7 = 0.42857142857, and the 142857 repeats. Notice it's the same recurrence, but it begins at a different place this time. Try this: for all integers n, with
, find n/7. You should find that you get the same 6 digits repeating in the same order, but starting at a different value. Once you get to n=7, then 7/7=1. Beyond that, for n=8, 8/7= 1+1/7, so you just repeat the same pattern, but with a 1 in front of the decimal place. Now, I do not know why the decimal repeats for every integer divided by 7, but maybe you can figure it out by showing what happens when you add the digits of two multiples of 1/7?Also, notice that there are 6 integers between 0 and 7, and for each, the decimals begin at at a different number, but repeat in the same order. If for two integers less than 7, n/7 and m/7 began at the same number and repeated in the same pattern, then n/7 would be equal to m/7 where n,m<7. That is impossible - it's like saying 1/7=4/7.
Of course this still doesn't answer why those 6 repeat in the first place, or why any integer divided by 7 repeats those same 6 in the same order.
I know nothing about number theory, so I may not have been able to help at all. I just thought it would be fun to examine this and think about it with you! Good luck, I hope I helped somewhat..
No problem, demha!
This thought has always amazed me. After the big bang, there was energy. As the universe expanded, the energy became less dense as there became more space to occupy. Eventually, forces became more significant as energy became less dense. This gave rise to the possibility of matter particles, which creates protons and neutrons, which thus created atoms. With the universe still expanding, more forces became able to overcome the decreasing energy density. The weakest, gravity, caused atoms to "clump," and over time, stars, star clusters, galaxies, and galaxy clusters formed. In the stars, the most basic atoms (hydrogen) fused, releasing energy and creating heavier elements. Supernovas spread this new matter across the universe. These new elements, all with vastly different properties combined, giving us molecules. Some special molecules, over billions of years, formed - amino acids, nucleotides, and carbohydrates. These molecule are capable of forming larger molecules - molecules able to multiply. And somehow - through the mystery of abiogenesis - life formed. Life evolved over billions of years to produce sentient beings.
From pure energy, randomness, and (for us) luck, came collections of matter capable of observing and unerstanding the universe from which it sprung. We truly are the universe experiencing itself. And as our knowledge and understanding of the universe grows, so does our ability to manipulate it. Some day (if we don't wipe ourselves out, of course) we will be the universe manipulating itself, and destroying the randomness created over billions of years.
We are the same as any other "thing" in that we are part of the universe and a product of the laws governing the universe. I think the only thing making us any 'different' is our complexity - the highly specialized arrangement of specific types of matter. But this complexity has given us a way of processing and storing information - it results in our sentience. But other than that, we are no different.
Thank you for bringing this up. The universe and life as a result are truly marvelous and mysterious. It mesmerizes me to think of how we became us.
The fact that 2n is even is based on the definition of an even number. An even number is any number divisible by 2, and 2n is always divisible by 2. An odd number is any number which is not divisible by 2, and 2n±1 is never divisible by 2.