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

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

Archive

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