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Category: Coordinate Geometry

117 minutes

117 minutes

Input interpretation
117 minutes
Comparison as half‐life
~~ 23 × half-life of uranium-241 (~~ 300 s )
Offsets from current time
117 minutes from now | 6:08 am HKT | Friday, July 21, 2017
117 minutes before now | 2:14 am HKT | Friday, July 21, 2017
Interpretations
time
Area of polar curves

Area of polar curves

Input interpretation
polar curves | area enclosed
Result
bifoliate | A = (pi a^2)/(2 sqrt(2))
bifolium | A = (pi a^2)/2
cardioid | A = (3 pi a^2)/2
cycloid of Ceva | A = 3 pi a^2
circle | A = pi a^2
circle parallel curve | A = pi (a + k)^2
cranioid | A = 1/2 pi (a^2 + 4 b c F_1(1/2;-1/2, -1/2;1;p, q) – b^2 (p – 2) – c^2 (q – 2))
folium of Descartes | A = (3 a^2)/2
fourth heart curve | A = 1/420 (-196 sqrt(5) 6^(3/4) tan^(-1)(root of 6 x^4 – 25 near x = -1.42872 + 1) + 2400 2F1(1, 1, 5/4, 24/49) + 2555 – 1176 log(6) + 98 sqrt(5) 6^(3/4) log(12 + 5 sqrt(6) – 2 sqrt(5) 6^(3/4)) – 98 sqrt(5) 6^(3/4) log(12 + 5 sqrt(6) + 2 sqrt(5) 6^(3/4)) + 196 sqrt(5) 6^(3/4) tan^(-1)(1 + sqrt(5)/6^(1/4))) – (5 pi^(3/2) (25 2F1(3/4, 1, 1/2, 49/24) + 38 2F1(3/4, 1, 9/4, -25/24)))/(96 sqrt(2) Gamma(5/4) Gamma(9/4)) + (6 – (21 – 7 i)/(sqrt(5) 6^(1/4))) pi
hippopede curve | A = 2 pi (2 a – b)
Area enclosed by parametric curve

Area enclosed by parametric curve

Input interpretation
parametric plane curves | area enclosed
Result
astroid | A = (3 pi a^2)/8
astroid pedal curve | A = -1/8 pi (a^2 + 4 (x_0^2 + y_0^2))
bifoliate | A = (pi a^2)/(2 sqrt(2))
bifolium | A = (pi a^2)/2
cardioid | A = (3 pi a^2)/2
cycloid of Ceva | A = 3 pi a^2
circle | A = pi a^2
circle parallel curve | A = pi (a + k)^2
cranioid | A = 1/2 pi (a^2 + 4 b c F_1(1/2;-1/2, -1/2;1;p, q) – b^2 (p – 2) – c^2 (q – 2))
cycloid | A = 3 pi a^2
Y=x^(1/2)

Y=x^(1/2)

Input
y = sqrt(x)
Geometric figure
parabola
Plots
Root
x = 0
Properties as a real function
Derivative
(d)/(dx)(sqrt(x)) = 1/(2 sqrt(x))
Implicit derivatives
Global minimum
min{sqrt(x)} = 0 at x = 0