mscroggs.co.uk
mscroggs.co.uk

subscribe

Puzzles

An integral

Source: Alex Bolton (inspired by Book Proofs blog)
What is
$$\int_0^{\frac\pi2}\frac1{1+\tan^a(x)}\,dx?$$

Show hint


Show answer & extension

Find them all

Find all continuous positive functions, \(f\) on \([0,1]\) such that:
$$\int_0^1 f(x) dx=1\\ \mathrm{and }\int_0^1 xf(x) dx=\alpha\\ \mathrm{and }\int_0^1 x^2f(x) dx=\alpha^2$$

Show answer & extension

Integrals

$$\int_0^1 1 dx = 1$$
Find \(a_1\) such that:
$$\int_0^{a_1} x dx = 1$$
Find \(a_2\) such that:
$$\int_0^{a_2} x^2 dx = 1$$
Find \(a_n\) such that (for \(n>0\)):
$$\int_0^{a_n} x^n dx = 1$$

Show answer & extension

Double derivative

What is
$$\frac{d}{dy}\left(\frac{dy}{dx}\right)$$
when:
(i) \(y=x\)
(ii) \(y=x^2\)
(iii) \(y=x^3\)
(iv) \(y=x^n\)
(v) \(y=e^x\)
(vi) \(y=\sin(x)\)?

Show answer & extension

Differentiate this

$$f(x)=e^{x^{ \frac{\ln{\left(\ln{x}\right)}}{ \ln{x}}} }$$
Find \(f'(x)\).

Show answer

x to the power of x again

Let \(y=x^{x^{x^{x^{...}}}}\) [\(x\) to the power of (\(x\) to the power of (\(x\) to the power of (\(x\) to the power of ...))) with an infinite number of \(x\)s]. What is \(\frac{dy}{dx}\)?

Show answer & extension

Archive

Show me a random puzzle
 Most recent collections 

Sunday Afternoon Maths LXVII

Coloured weights
Not Roman numerals

Advent calendar 2018

Sunday Afternoon Maths LXVI

Cryptic crossnumber #2

Sunday Afternoon Maths LXV

Cryptic crossnumber #1
Breaking Chocolate
Square and cube endings

List of all puzzles

Tags

factors rectangles integration geometry menace coordinates taxicab geometry games perimeter speed triangle numbers 3d shapes grids remainders square roots complex numbers division pascal's triangle star numbers symmetry angles functions cryptic crossnumbers planes lines percentages quadratics shapes dodecagons sums squares ave addition trigonometry unit fractions folding tube maps sport polygons chess dice cards numbers sum to infinity rugby probabilty 2d shapes bases proportion partitions chocolate surds advent money parabolas square numbers averages probability cube numbers palindromes wordplay multiplication sequences circles volume doubling calculus coins triangles number people maths logic factorials graphs indices colouring time clocks perfect numbers irreducible numbers chalkdust crossnumber scales digits integers arrows means shape multiples balancing odd numbers regular shapes algebra dates differentiation ellipses christmas spheres routes area cryptic clues books prime numbers floors mean fractions crossnumbers crosswords hexagons

Archive

Show me a random puzzle
▼ show ▼
© Matthew Scroggs 2019