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## Forum Posts

Ian Fowler
Jan 07, 2022
In Video Ideas
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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Ian Fowler
Nov 15, 2021
In Video Ideas
???
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Ian Fowler
Nov 15, 2021
In Math Problems
???
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Ian Fowler
Nov 01, 2021
In Video Ideas
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.
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Ian Fowler
Jun 16, 2021
In Video Ideas
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Ian Fowler
Jan 18, 2021
In Video Ideas
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
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Ian Fowler
Jan 17, 2021
In Video Ideas
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
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Ian Fowler
Oct 05, 2020
In Math Problems
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?
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Ian Fowler
Sep 18, 2020
In Video Ideas
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.
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Ian Fowler
Jun 13, 2020
In Math Problems
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.
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Ian Fowler
Jun 13, 2020
In Video Ideas
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.
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Ian Fowler
Dec 24, 2019
In Math Problems
Pretty cool. Evaluate each integral. int(0--> inf) [1/(1+x^2)^2]dx int(0--> inf) [x^2/(1+x^2)^2]dx
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Ian Fowler
Oct 20, 2019
In Math Problems
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Ian Fowler
Oct 20, 2019
In Math Problems
Why was "Odd intgration" removed? I put a lot of time and effort into that.
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