Posted in ask uncle colin.

Dear Uncle Colin, I have to show that \(-\frac{x}{2} = \ln (\sqrt{1+e^x} – \sqrt{e^x }) + \ln (\sqrt{1+e^{-x}} + 1)\). I can’t get it anywhere near the right form! – Proof Of It Not Coming – Any Reasonable Explanation? Hi, POINCARE, and thanks for your message! That’s a bit of

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Posted in ask uncle colin.

Dear Uncle Colin, I’m told that \(x\sqrt{x} – 5\sqrt{x} = 2\) and I have to find \(x – 2\sqrt{x}\). Everything I try seems to make it worse! Any ideas? Mastering A Cubic – Help Is Needed Hi, MACHIN, and thanks for your message! At first glance, that’s a strange one.

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Dear Uncle Colin, As I progress through my maths education, I notice that the people around me are getting smarter and smarter. How do I keep my head up when everyone is brighter than me? I’m Mightily Put Off Seeing Their Outstanding Results Hi, IMPOSTOR, and thanks for your message!

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Posted in ask uncle colin.

Dear Uncle Colin, Why is it called “completing the square”? To me, it looks like you’re taking something away from a square. – Some Quadratics, Understandably, Are Requiring Explanation Hi, SQUARE, and thanks for your message! Completing the square involves taking a quadratic such as $x^2 + 6x + 5$

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Posted in ask uncle colin.

Dear Uncle Colin, I struggled with a problem where you had 5 blue balls and x green balls, and the probability of picking two blue balls out of the bag was $\frac{5}{14}$. I can’t really see where to start! -Baffled About Likelihood, Lacking Startpoint Hi, BALLS, thanks for your message!

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Posted in ask uncle colin.

Dear Uncle Colin I just bought a new set square and noticed it had a couple of extra marks – one at seven degrees and one at 42 degrees. Have you any idea what those are for? – Don’t Recognise Extra Information Engraved on Calculus Kit Hi, DREIECK, and thank

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Posted in ask uncle colin.

Dear Uncle Colin, I need to show that $\sqrt{7}$ is in $\mathbb{Q}[\sqrt{2}+\sqrt{3}+\sqrt{7}]$ and I don’t really know where to start. We Haven’t Approached Tackling Such Questions Hi, WHATSQ, and thanks for your message! I am absolutely not a number theorist, although I must admit to getting a bit curious about

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Dear Uncle Colin, I’m ok with my basic power laws, but I don’t understand why $x^0$ is always 1, and I get mixed up when it’s a fraction or a negative power. Can you help? Running Out Of Time Hi, ROOT, and thanks for your message! If it’s any consolation,

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Integration by substitution, rigorously Dear Uncle Colin, Can you explain why integration by substitution works? I get that you’re not allowed to ‘cancel’ the $dx$s, but can’t see how it works otherwise. – Reasonable Interpretation Got Our Understanding Ridiculed Hi, RIGOUR, and thanks for your message. First up, confession time:

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Posted in ask uncle colin.

Dear Uncle Colin, In a recent test, I was asked to differentiate $\frac{x^2+4}{\sqrt{x^2+4}}$. Obviously, my first thought was to simplify it to $\br{x^2+4}^{-\frac{1}{2}}$, but I’m not allowed to do that: only to use the quotient rule and the fact that $\diff {\sqrt{f(x)}}{x} = \frac{f'(x)}{2f(x)}$. When Evaluating, Inappropriate Rules Demanded Hi,

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