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Category: Algebra

2sin 2x

2sin 2x

Input 2 sin(2 x) Roots x = (pi n)/2, n element Z Series expansion at x=0 4 x – (8 x^3)/3 + (8 x^5)/15 + O(x^6) (Taylor series) Derivative d/dx(2 sin(2 x)) = 4 cos(2 x) Indefinite integral integral 2 sin(2 x) dx = -cos(2 x) + constant Global maxima max{2 sin(2 x)} = 2 at x = pi n + pi/4 for integer n Definite integral over a half-period integral_0^(pi/2) 2 sin(2 x) dx = 2 Definite integral mean…

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2t 3t

2t 3t

Input 2 t (3 t) Result 6 t^2 Geometric figure parabola Root t = 0 Polynomial discriminant Delta = 0 Derivative d/dt(2 t (3 t)) = 12 t Indefinite integral integral 6 t^2 dt = 2 t^3 + constant Global minimum min{2 t (3 t)} = 0 at t = 0

2qr

2qr

Input interpretation 2 long quarterweights Comparison ~~ 2 × mass of a Good Delivery gold bar ( 400 oz t ) Interpretation mass

214/7

214/7

Input 214/7 Exact result 214/7 (irreducible) Decimal approximation 30.57142857142857142857142857142857142857142857142857142857… Repeating decimal 30.571428^_ (period 6) Mixed fraction 30 4/7 Quotient and remainder 30×7 + 4 Prime factorization 2×7^(-1)×107 Continued fraction [30; 1, 1, 3] Egyptian fraction expansion 30 + 1/2 + 1/14 Occurrence in convergents (45 e)/4~~30, 31, 61/2, 153/5, 214/7, 367/12, 948/31, … (simple continued fraction convergent sequence)

3 8 7 8

3 8 7 8

Input {3, 8, 7, 8} Total 3 + 8 + 7 + 8 = 26 Statistics mean | 6.5 median | 7.5 sample standard deviation | 2.38 Vector length sqrt(186)~~13.6382 Normalized vector (sqrt(3/62), 4 sqrt(2/93), 7/sqrt(186), 4 sqrt(2/93)) Possible closed form (no form found in terms of holonomic sequences)

2nc to c

2nc to c

Input interpretation convert 2 nanocycles to cycles Result 2×10^-9 cycles Corresponding quantity Phase angle change Deltaphi from Deltaphi = 2pic: | 13 nrad (nanoradians) | 7.2×10^-7° (degrees)