The Flying Colours Maths Blog: Latest posts

Ask Uncle Colin: Parametric Second Derivatives

Dear Uncle Colin, I have a pair of parametric equations giving $x$ and $y$ each as a function of $t$. I'm happy with the first derivative being $\diff{y}{t} \div \diff{x}{t}$, but I struggle to find the second derivative. How would I do that? - Can't Handle An Infinitesimal Nuance Hi,

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Sweet and Sour Limits

Eagle-eyed friend of the blog @robjlow spotted an error in Uncle Colin's last answer. As I'm forever telling my students, making errors is how you learn; Rob has graciously delivered a lesson for us all. Thanks for keeping me honest! Recently, Uncle Colin gave a couple of ways to see

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Ask Uncle Colin: A Short, Sweet Limit

Dear Uncle Colin, What is $\lim_{x \to \infty} \left\{ \sqrt{x^2 + 3x} - x\right\}$? - Raging Over Obnoxious Terseness Hi, ROOT, and thanks for your very brief question. My approach would be to split up the square root and use either a binomial expansion or completing the square, as follows:

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Wrong, But Useful: Episode 43

In this month's installment of Wrong, But Useful, our special guest co-host is @mathsjem (Jo Morgan in real life) from the indispensable We start by talking about and how Jo manages to fit such a punishing blog schedule around being a nearly-full-time maths teacher. Colin wonders how writing

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The Mathematical Pirate and The Quotient Rule

"Arr, that be a scurvy-lookin' expression!" said the Mathematical Pirate. "A quartic on the top and a quadratic on the bottom. That Ninja would probably try to factorise and do it all elegant-like." "Is that not the point?" "When you got something as 'orrible as that, it's like puttin' lipstick

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Ask Uncle Colin: Approximating an embedded exponential

Dear Uncle Colin, Help! My calculator is broken and I need to solve - or at least approximate - $0.1 = \frac{x}{e^x - 1}$! How would you do it? -- Every $x$ Produces Outrageous Numbers, Exploring New Techniques Hi, ExPONENT, and thanks for your message! That's a bit of a

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The Mathematical Ninja lets the student investigate… cube roots

"Sensei! I have a problem!" The Mathematical Ninja nodded. "Bring it on." "There's a challenge! Someone has picked a five-digit integer and cubed it to get 6,996,364,932,376. I know it ends with a six, and I could probably get the penultimate digit with a bit of work... I just wondered

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Announcing: Ninja|Alpha

After several months of high-intensity development, we're very happy to announce the launch of Ninja|Alpha. Go ahead! Ask it anything you like.

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Ask Uncle Colin: Why does the constant term vanish?

Dear Uncle Colin, When I differentiate $y=2x^2 + 7x + 2$ and apply the $nx^{n-1}$ rule, why do I only apply it to the $2x^2$ and the $7x$ but not the 2? -- Nervous Over Rules, Mathematically A Liability Hi, NORMAL, and thanks for your message! There are several ways

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How the Mathematical Ninja approximates $\ln(5)$

"Isn't it somewhere around $\phi$?" asked the student, brightly. "That number sure crops up in a lot of places!" The Mathematical Ninja's eyes narrowed. "Like shells! And body proportions! And arrawk!" Hands dusted. The Mathematical Ninja stood back. "The Vitruvian student!" The student arrawked again as the circular machine he

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