Puzzles about calculus

Can you solve these puzzles about differentiation and integration?

post

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 University College London. He hasn’t had time to play Klax since the noughties, but he’s pretty sure that Coke is it!

More from Chalkdust

One thought on “Puzzles about calculus

  1. Pingback: 5 Maths Gems #60 - Liveindex News People Places

Comments are closed.