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,

Read More →
Posted in ask uncle colin, integration.

Dear Uncle Colin, How would you integrate $e^x \sin(x)$ (with respect to $x$, obviously)? - Difficult Integral, Just Kan't See The Right Answer Hi, DIJKSTRA, and thanks for your message! As seems to be the way recently, there are several ways to approach this. My favourite way One of the

Read More →
Posted in ask uncle colin.

Dear Uncle Colin, What's the correct method to find the general solution of $y''+4y'+4y=(3+x)e^{-2x}$? I've got the complementary function just fine (it's $y=Ae^{-2x}+Bxe^{-2x}$), but I'm going in circles with the particular integral. - Differential Equation's Solutions Are Really Gruesome; Ugly Exponential Scariness Hi, DESARGUES, and thanks for your message! I

Read More →
Posted in ask uncle colin.

Dear Uncle Colin, How do people do decimal calculations like $80 \times 0.15$ in their heads? It seems impossible. - Doesn't Everyone Seem Clever, Answering Readily These Evil Sums Hi, DESCARTES, and thank you for your message! There are several possible strategies for a question like this - how I

Read More →
Posted in ask uncle colin.

“What are you doing here?” “Never mind that, sensei, how on earth did they catch you?” “... I don’t want to talk about it. In any case, I was only doing 5% more than the speed of light.” “Yeah, that poses more questions than it answers, in honesty.” The Powerpoint

Read More →
Posted in ask uncle colin.

Dear Uncle Colin, According to John D Cook, you can estimate the natural logarithm of a number $x$ by working out $\log_2(x)-\log_{10}(x)$. It seems to be pretty accurate - but why? Not A Problem - Interesting Exponential Riddle Hi, Napier, and thanks for your message! That is indeed a conundrum:

Read More →
Posted in ask uncle colin.

Dear Uncle Colin, When I'm 8 metres away from a flagpole, the angle of elevation to its top is exactly 40º, according to this angle-measurey-majigger I have here. What will it be when I'm only five metres away? The Hypotenuse Eludes One; Does One Look Into Trigonometric Expressions? Hi, THEODOLITE,

Read More →
Posted in ask uncle colin, proof, trigonometry.

Dear Uncle Colin, I have a trig identity I can't prove! I have to show that $\frac{\cos(x)}{1-\sin(x)} = \tan(x) + \sec(x)$. Strangely Excited Comment About Non-Euclidean Trigonometry. Hi, SECANT, and thanks for your message! This is a slightly sneaky one, but definitely a good one to practice. Let's do it

Read More →
Posted in ask uncle colin, trigonometry.

Dear Uncle Colin, I was asked to work out $\tan\br{\theta + \piby 2}$, but the formula failed because $\tan\br{\piby 2}$ is undefined. Is there another way? - Lost Inna Mess, Infinite Trigonometry Hi, LIMIT, and thanks for your message! In fact, there are several ways to approach it! Basic geometry

Read More →
Posted in ask uncle colin, trigonometry.

Dear Uncle Colin, I'm trying to solve $2\cos(3x)-3\sin(3x)=-1$ (for $0\le \theta \lt 90º$) but I keep getting stuck and/or confused! What do you recommend? - Losing Angles, Getting Ridiculous Answers, Nasty Geometric Equation Hi, LAGRANGE, and thank you for your message! There are a couple of ways to approach this:

Read More →