Puzzles about calculus

Can you solve these puzzles about differentiation and integration?

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When learning A-level maths, much time is devoted to learning how to differentiate and integrate. For this week’s blog post, I have collected some puzzles based on these skills. They should be fun to solve, present a few surprises and maybe even provide a teacher or two with an extra challenge for capable students.

The answers to these puzzles will appear here from Sunday at 8am.


Let’s start with an integral:

An Integral

Source: mscroggs.co.uk
What is
$$\int_0^{\frac\pi2}\frac1{1+\tan^ax}\,dx?$$

This might look difficult to you. You might first start by substituting in some specific values for $a$, and this would be a good way to start, but solving the integral in general is difficult. However, considering the second integral $$\int_0^{\frac\pi2}\frac1{1+\cot^ax}\,dx$$ along with the integral in the question should help.

Before I say too much, I’m going to stop and leave any further discussion of this puzzle for the answers [available from Sunday 8am].

Next, let’s have a go at a few derivatives:

$x$ to the power of $x$ to the power of $x$ to the power of …

Source: mscroggs.co.uk
Let $\displaystyle y=x^{x^{x^{x^{.^{.^.}}}}}$ (with an infinite chain of powers of $x$).What is $\displaystyle \frac{dy}{dx}$?

Differentiate this

Source: Alex Bolton
Let $\displaystyle a=\frac{\ln(\ln x)}{\ln x}$. Let $b=x^a$. Let $\displaystyle y=e^b$.

What is $\displaystyle \frac{dy}{dx}$?

Finally, the last puzzle of this blog post asks you to find all functions whose integrals satisfy certain properties:

Find Them All

Source: Twenty-six Years of Problem Posing by John Mason
Find all continuous positive functions $f$ on $[0,1]$ such that:
$$\int_0^1 f(x) dx=1\\
\text{and }\int_0^1 xf(x) dx=\alpha\\
\text{and }\int_0^1 x^2f(x) dx=\alpha^2$$

The answers to these puzzles will appear here from Sunday at 8am.

Matthew is a postdoctoral researcher at the University of Cambridge. He hasn’t had time to play Klax since the noughties, but he’s pretty sure that Coke is it!

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