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 

Advent calendar 2019

Sunday Afternoon Maths LXVII

Coloured weights
Not Roman numerals

Advent calendar 2018

Sunday Afternoon Maths LXVI

Cryptic crossnumber #2

List of all puzzles

Tags

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

Archive

Show me a random puzzle
▼ show ▼
© Matthew Scroggs 2012–2020