Given a polynomial function, f(x), with degree n>=3. For any real number, a, divide f(x) by (x-a)^(n-1) This will ensure that the quotient, q(x), is linear and that the remainder, r(x), is of degree n-2. 1) f(a) = r(a) and 2) f '(a) = r '(a) giving the tangent as y - r(a) = r '( a)[x - a]. See if you can prove this - it's pretty easy. This really struck me as f(x) and r(x) are clearly not the same polynomial and yet! f(a) and r(a) yield the same value as well as f '(a) and r '(a). This all started with Dr. Barker's post: The Method No One Taught You - Finding the Tangent to a Curve - YouTube

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Here's a challenge. Find the probability that, in a group of 4 people, at least 2 have the same birthday. The catch is - try doing it the DIRECT way with cases and do NOT use the indirect method that has been flogged endlessly and everyone has seen. BTW this is NOT a paradox. So when you see this listed as "The Birthday Paradox" you can politely correct them.

Some problems that are popular on the internet are in correctly classified as paradoxes but are not paradoxes at all - they are simply problems that have answers that might seem to defy common intuition and therefore have answers that come under suspicion. These are not paradoxes First. A true paradox is a logical contradiction that CANNOT be resolved by the current axioms. Famous paradoxes that are well known include: 1) Russel's Paradox involving set theory and the paradox arising from the set of all sets that do not contain themselves. 2) A more popular paradox that's been around for many years arises in the Barber Problem and asking "Who shaves the Barber?" 3) Another popular one arises from 2 statements written on opposite sides of one sheet of paper. On one side is the statement "The statement on the other side of this sheet paper is true" and on the flip side is the statement "The statement on the other side of this sheet of paper is false" All 3 of these are not resolvable and thus lead to a logical contradiction and are , in fact, true paradoxes. On the other hand there are problems often described as paradoxes which are not. 1) The Birthday Problem- what is P(that at least 2 people have the same birthday in a group of n people) ? When n = 35 the probability > 80% . A surprising result to a lot of people but the answer is totally correct and therefore not a paradox. 2) The False Positive Problem which arises when a rare disease is present in the population. Percentages are given concerning the reliability of blood tests. When you calculate P(you actually have the disease given that you have tested positive) the result always turns out to be so low that it goes against popular intuition. Again, not a paradox. Maybe this can be an informative video.

Given the following definitions: The differential of x, denoted by dx, is defined by: dx = delta x The differential of y, denoted by dy, us defined by: dy = f'(x)dx For the curve f(x) = sin(x) between x = pi/6 and x = pi/4, find the exact values of: 1) The point P1 on the curve at x = pi/6 2) The point P2 on the curve at x = pi/4 3) f'(pi/6) 4) The equation of the tangent at P1: x = pi/6 5) The point P3 on the tangent when x = pi/4 6) dx 7) dy 8) delta y between P1 and P2 (note that dy > delta y) 9) the ratio, dy (from 7) divided by dx (from 6) and compare to (3) A picture is worth a thousand words. Cheers - Ian

We are all very familiar with the statement: dy = f'(x)dx - yes, that's "equal to". However, there seems to be a very deep rooted misconception about what this actually means and how that meaning is based on the actual definitions of these "differentials". And there is a good reason why "dy by dx" behaves like a fraction (i.e. diff(y) "divided by" diff(x)) when, in reality, "dy by dx" is not a fraction. In the "differential" definitions the variables dy and dx are NOT - yes you heard it right- NOT restricted to be infinitely small. dx can be as large as you like, and when you find the corresponding value of dy (also quite likely to be large) and then divide them ( dy "divided by" dx) - by the magic of similar triangles - you get the slope of the tangent. In fact, depending on f(x) and the value of dx, dy can actually be larger than "delta"y. Now having said that, the definitions of differentials, do not prevent dx and dy from being infinitely small, but the key is that they are not required to be so. Newton knew this and it was key to his wonderful discovery. As strange as this may sound to some, it is not fake news, but really is true. BTW Newton used the term "moment of x". "differential of x" is a Leibniz term. But they are very similar. "Moment" is a "differential" that is infinitely small. Cheers - Ian

I get a sneaking suspicion that a minority of the posts here are from people just wanting someone to do their homework for them so they can copy it, hand it in and take the credit for it. Does anyone else here share the same suspicion or am I out in left field?

As you might think, maximum curvature does occur at the stationary points, but it does not. You can find the exact values of x which are quite surprising and give you an insight as to how the curve behaves.

Given: cos(t) = 2/[5*sqrt(5)]. Find the exact value of cos(t/3). I would have given you the answer, but it's too easy to work backwards using the triple angle formula for cos(3t). I stumbled across the answer when dealing with a different problem.

Given: cos(t) = 2/[5*sqrt(5)]. Find the exact value of cos(t/3). I would have given you the answer, but it's too easy to work backwards using the triple angle formula for cos(3t). I stumbled across the answer when dealing with a different problem.

It was the name of a recent post listed in these discussions to which I responded several times and finally solved. Then, for no apparent reason, the post and all of the replies just disappeared. Below is a copy of the last email notification I received from discussions. When I tried "Check it out" I noticed it was removed. No explanation. Just gone. Maybe someone here can give me an explanation. I have tried the "Let's Chat" at least 5 times but no one ever responds. So much for that. I seem to be hitting a dead end. I must admit it's rather annoying that there seems no way to contact anyone in charge and get an honest answer. I put in a lot of time and effort in this. I tried to contact "blackpenredpen" himself but that also seems impossible, or at least beyond my means. So backpenredpen, if you are out there - please respond. I will repeat this post on the main thread. Thanks for taking the time to reply - Ian Mohd E.21 Likes Your Comment Hi Ian Fowler,Mohd E.21 likes your comment in “Odd integration” in the forum, Discussions. Dear Mohd E.21, Well, well well. You know, sometimes, the workings of the brain is a real mystery. I took your lectu... Check It Out

Why was "Odd intgration" removed? I put a lot of time and effort into that.