Browsing category core 4

Why I don’t buy that $1 + 2 + 3 + … = -\frac{1}{12}$

Thanks to Robert Anderson for the question. I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + · · · = −1/12 under my theory. If I tell you this you will at once point out to

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Integration by parts: how you do it, and why it works

Integration by parts is one of the two important integration methods to learn in C4 (the other is substitution1 ). In this article, I want to run through when you do it, how you do it, and why it works, just in case you're interested. When you use integration by

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There’s More Than One Way To Do It: Equations of a circle

Depending on your AS-level maths exam board, you might encounter the equation of a circle in C1 (OCR) or C2 (everyone else). It's really just a restatement of Pythagoras' Theorem: saying $(x-a)^2 + (y-b)^2 = r^2$ is the same as saying "the square of the horizontal distance between $(a,b)$ and

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Why the binomial expansion is stupid and you shouldn’t use it

The Mathematical Ninja looked offended. For once, it wasn't a student that was the guilty party, it was me. "You're considering a series on the WHAT?!" "The binomial expansion," I said, brightly. "Drivel!" said the Mathematical Ninja "Drivel, piffle and poppycock! Newtonian claptrap, not worth the space made for it

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The Mathematical Ninja’s C4 Paper

"I stayed in ALL DAY waiting for that delivery of exam papers in Amsterdam," said the Mathematical Ninja. "I could have been out pillaging, but no, sodding Yodel told me the package was out for delivery and it was only when I called them up for the eighth time they

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C4 Integration Quiz (tough stuff)

Here's a quick multiple-choice quiz about the tough stuff in C4 integration. Ready?1 Question 1: squared trig functions What method do you use to calculate $\int \sin^2(x) dx$? (Give me all four answers!) a) Parts ($u = \sin(x),~v'=\sin(x)$) b) Trig substitution ($u=\cos(2x)$) c) Split-angle formula ($\sin(A)\sin(B) = d) Parts ($u

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Maths in the real world: flowerpot volumes

A reader asks: how do I figure out the volume of soil I need to fill a flowerpot? A flowerpot is a slightly peculiar shape: it's not a cone, it's not a cylinder, it's somewhere between the two. Luckily, we have a word for such shapes: it's a frustum of

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Why is the dot product the way it is?

(Thanks to Barney Maunder-Taylor for teasing me with this one.) This question interests me for two reasons: Firstly, it's a neat proof in its own right, and I'll start by giving a little sketch of it. Secondly, though, even after Barney gave me the crux of the proof, it still

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"How do I know which method to use?" – How to Integrate

My student frowns as I write down $u = \cos(x)$. "Wait wait wait", he says, "how do you know to do a substitution?" The honest answer is, I just do. I've done integration sum after integration sum for the last 20 years and it comes naturally. However, I know that's

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