Questions asked by A. S. Adikesavan

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• How do you prove that the 8-sd approximation to the value of the infinite continued fraction #0.0123456789/(1+0.0123456789/(1+0.0123456789/(1+...))) =0.012196914# ?
  
• How do you make the graph for #y=ln(1+x/(ln(1-x)))#?
  
• How do you find the interval of existence for the real function #ln(1+x/(ln(1-x)))#?
  
• In the first Mean Value Theorem #f(b)=f(a)+(b-a)f'(c), a<c<b, f(x) =log_2 x, a=1 and f'(c)=1. How do you find b and c?
  
• #a<b<c<d#. How do you find the solution(s) of |x-a|+|x-c|=|x-b|+|x-d|? It is verifiable that, for #(a, b, c, d)=(1, 2, 3, 5), x=7/2# is a solution.
  
• From a unit sphere, the part between two parallel planes equidistant from the center, and with spacing 1 unit in-between, is removed. The remaining parts are joined together face-to-face, precisely. How do you find the volume of this new solid?
  
• How do you find the volume of the central part of the unit sphere that is bounded by the planes #x=+-1/5, y=+-1/5 and z=+-1/5#?
  
• How do you find the inverse of # y = e^x-1/x#?
  
• How do you evaluate the continued fraction #f(e)=e-1+1/(e-1+1/(e-1+1/(e-1+1/(e-1+1/...# ?
  
• How do you find the value of #sin^(-1)(sin (cos^(-1) (sin (pi/12))))#?
  
• Peshawar in Pakistan and Columbia in SC, USA are at the same latitude #54^o N#. Their longitudes are #71.7^o E and 81.1^o W#. The radius of the Earth at this latitude is nearly 6371 km. How do you find the latitude-circle distance between the two cities?
  
• The periodic function #x = 2 sin ( t/10 )# satisfies #x^2-3x+2=0#. How do you find all the values of t?
  
• How do you find 4-sd rounded approximation(s) to the solution(s) of #e^x-1/x=pi#?
  
• Vectors #A = ( L, 1, 0 ), B = ( 0, M, 1 ) and C = ( 1, 0, N ). A X B and B X C# are parallel. How do you prove that #L M N + 1 = 0?#
  
• How do you prove that the 4-sd approximation to the value of #log_2(2+1/log_2(2+1/log_2(2+...)))# is 1.428?
  
• Here, about 10 h ago, Cesaero R presented a super 28-sd y(2) as a zero of #2^y=2+1/y#. How do you find super strings y(b) as zeros of #b^y=b+1/y, b = 10, e and pi#? How do you prove uniqueness or otherwise of y(b)?
  
• The parameter a for the closed curve #r = f(theta;a)# is rational. For example, a is rational in #r=a(1+cos theta)#. How do you prove that its length, area enclosed and the volume of the solid of revolution, about an axis, are all transcendental?
  
• How do you find A, B and C, given that #A sin ( B x + C ) = cos ( cos^(-1) sin x + sin^(-1) cos x ) + sin (cos^(-1) sin x + sin^(-1) cos x )#?
  
• #F( x ; b) = log_b(x+1/log_b(x+1/log_b(x+...))), x>=b>1.#How do you approximate F(15, 10)=1.19958, nearly?
  
• For #n > 1#, I have designated the value of #log_n(pi+1/log_n(pi+1/log_n(pi+...)))# as the function #pin(n)#. Inversely, given #pin (8)=0.72544666#, how do you approximate # pi#?
  
• How do you prove graphically that there are no/two solutions to #x^2 + y^2 = 1 and x+ y = sqrt 2 +-1/2#?
  
• The Functional Continued Fraction (F C F) of exponential class is defined by #a_(cf) (x;b) = a^(x+b/(a^(x+b/a^(x +...)))), a > 0#. Upon setting a = e = 2.718281828.., how do you prove that #e_(cf) ( 0.1; 1 ) = 1.880789470#, nearly?
  
• On the scaling power of logarithmic FCF: #log_(cf) (x;a;b)=log_b (x+a/log_b(x+a/log_b (x+...))), b in (1, oo), x in (0, oo) and a in (0, oo)#. How do you prove that #log_(cf) ( "trillion"; "trillion"; "trillion" )=1.204647904#, nearly?
  
• How do you find the equation of the parabola having its focus at #( 1/2, 1/2)# and the directrix along #x - y = 1#?
  
• FCF (Functional Continued Fraction) #cosh_(cf) (x; a)=cosh(x+a/cosh(x+a/cosh(x+...)))# How do you prove that #y = cosh_(cf) (x; x)# is asymptotic to #y = cosh x#, as #x -> 0# or the graphs touch each other, at #x = 0#?
  
• The graph of #y=e^x/2# is the bisector-graph of the graphs for #y = cosh x and y = sinh x#. How do you use these graphs to show that the limit for the indeterminate form #oo-oo# could be 0?.
  
• The FCF (Functional Continued Fraction) #cosh_(cf) (x;a)=cosh (x+a/cosh(x+a/cosh(x+...)))#. How do you prove that this FCF is an even function with respect to both x and a, together?and #cosh_(cf) (x; a) and cosh_(cf) (-x;a)# are different?
  
• How do you find the range of the scalar triple product of the vectors #<cos alpha, cos beta, 0>, <0, cos beta, cos gamma> and <cos alpha, 0, cos gamma>?#
  
• How do you find the maximum volume #V(theta, phi)# of the tetrahedron, with vertices at #(0, 0, 0), (cos theta, 0, 0), (0, sin theta, 0) and (cos theta, sin theta, cos phi)?#
  
• The FCF (Functional Continued Fraction) #cosh_(cf) (x;a) = cosh(x+a/cosh(x+a/cosh(x+...)))#. How do you prove that #cosh_(cf) (0;1) = 1.3071725#, nearly and the derivative #(cosh_(cf) (x;1))'=0.56398085#, at x = 0?
  
• Using Chebyshev Polynomial #T_n (x)=cosh( n( arc cosh(x))), x > = 1# and the recurrence relation #T_(n+2)(x)=2xT_(n+1)(x) - T_n (x)#, with #T_0(x)=1 and T_1(x)=x#, how do you porve that #cosh(7 arc cosh(1.5))=421.5?#
  
• #T_n(x)# is the Chebyshev polynomial of degree n. The FCF #cosh_(cf )(T_n (x); T_n (x))=cosh(T_n(x)+(T_n(x))/cosh(T_n (x) + ...)), x >= 1#. How do you prove that the 18-sd value of this FCF for #n=2, x =1.25# is #6.00560689395441650?
  
• Using Chebyshev nth degree polynomial #T_n (x) = cos ( n ( arc cos ( x ) ) ), |x | < = 1,# how do you prove that #cos ( 10 ( arc cos ( 0.25) ) ) = 0.816894531#, nearly?
  
• A Functional Continued Fraction ( FCF ) is #exp_(cf)(a;a;a)=a^(a+a/a^(a+a/a^(a+...))), a > 1#. Choosing #a=pi#, how do you prove that the 17-sd value of the FCF is 39.90130307286401?
  
• Is there any point #(x, y)# on the curve #y=x^(x(1+1/y)), x > 0,# at which the tangent is parallel to the x-axis?
  
• A cube and a regular octahedron are carved out of of unit-radius wooden spheres . If the vertices are on the spheres, how do you prove that their volumes compare with that of the sphere, in the proportions #3/sqrt 2 : 2 : pi?#
  
• Epsilon is as significant as infinity. Is the bracketing #[in, oo]# equivalent to #(0, oo)?#
  
• How do you prove that #(x+pi/4)-(x+pi/4)^3/(3!)+(x+pi/4)^5/(5!)-... = 1-(x-pi/4)^2/(2!)+(x-pi/4)^4/(4!)=...= 1+x-x^2/(2!)-x^3/(3!)+x^4/(4!)+x^5/(5!)-..., x in (-oo, oo)?#
  
• How do you prove that the triad of graphs of #y = cosh x , y = sinh x# and ( their bisector in the y-direction ) #y =(1/2) e^x# becomes asymptotic, as #x to oo#?
  
• The function #y = x^x# is differentiable at x = 1/e, What is the Taylor series ( if any ) for y, about x = 1/e?
  
• How do you prove that the curves of #y=x^x# and y = the Functional Continued Fraction (FCF) generated by #y=x^(x(1+1/y))# touch #y=x#, at their common point ( 1, 1 )?
  
• #y=x^x# and y = the Functional Continued Fraction (FCF) developed by #y=x^(x(1+1/y)# are growth functions, for #x>=1#. How do you prove that at #x = 0.26938353091863307 X10^2#, y-ratio is 1 in 100 and curve-direction is vertical, nearly, for both?
  
• The graphs of the triad #y = cosh x, y = sinh x and y = e^x# meet x = 1 at (1, cosh 1), (1, sinh 1) and (1, e). How do you find these points for the FCF-triad #y=cosh(x(1+1/y)), y=sinh(x(1+1/y)) and y=e^(x(1+1/y))#?
  
• #f(0)=n and f(x)=sinh (nx)/sinh x#, for #|x|>0#. How do you make two graphs, for n = 2 and n = 3?
  
• A cylindrical pillar, with a regular nonagonal cross-section, has to be carved out of a right circular cylindrical sandal wood. If the height is 10' and diameter of the base is 1', how do you prove that the minimum possible scrap is 0.623 cft, nearly?
  
• A particle is constrained to move in #Q_1#, in a parabolic groove # y = x^2#. It was at O, at t = 0 s. At time t s, its speeds in the directions Ox and Oy are 5 and 12 cm/s, respectively. How do you prove its direction makes #67^o22'49''# with Ox?
  
• Either of two unit circles passes through the center of the other. How do you prove that the common area is #2/3pi-sqrt 3/2# areal units?
  
• Either of two unit spheres passes through the center of the other. Without using integration, how do you prove that the the common volume is nearly 1.633 cubic units?
  
• #2^N# unit spheres are conjoined such that each passes through the center of the opposite sphere. Without using integration, how do you find the common volume?and its limit, as # N to oo#?
  
• #2^N# unit circles are conjoined such that each circle passes through the center of the opposite circle. How do you find the common area? and the limit of this area, as #N to oo?#
  
• Assuming that the Earth is a sphere and its orbit around the Sun is a circle, how do you find the volume of the torus that is just sufficient to accommodate the Earth?
  
• Sans the lone satellite Luna, our planet Earth had cleared debris and other close by space orbiters. How do you find the volume of this cleared neighborhood around the Earth's orbit?
  
• Along the equator, it was no-shadow-noon on September 22 ( Autumnal Equinox). When do you expect this right-above-head-Sun noon, at Bangalore, India? The latitude of Bangalore is #12.98^o N#.
  
• How do you find in decimals the ratio ( binary 1.01)/(octal 2.4)?
  
• A ring torus is made by joining the circular ends of 1 meter long thin and elastic tube. If the radius of the cross section is 3.18 cm, how do you prove that the volume of the torus is #19961# cc?
  
• A ring torus has an ellipse of semi axes a and b as cross section. The radius to the axis of the torus from its center is c. Without integration for solid of revolution, how do you prove that the volume is #2pi^2abc#?
  
• How do you find the area ( if any ) common to the four cardioids #r=1+-cos theta and r = 1+-sin theta#?
  
• Introducing 1 AAU (Areal Astronomical Unit) for the area of a circle of radius 1 AU, how do you find 4-sd values in AAU, fot the orbital areas of solar planets?
  
• My estimate for the distance of the farthest Sun-size star that could be focused as a single-whole-star, by a 0.001''-precision telescope, is 30.53 light years. What is your estimate? Same, or different?
  
• Assuming that the range of #sin^(-1)x # is #(-oo, oo)#, is # x sin^(-1) x # differentiable, for #sin^(-1)x in [0, 2pi]#?
  
• How do you perform inversions for #y = x^2 and y = x^4?# Is #(dx)/(dy)# from the inverse #1/((dy)/(dx))?#
  
• My calculator displays #sin^(-1)sin 45^o=45^o, sin^(-1)sin 135^o=45^o and sin^(-1)sin 405^o=45^o#.How do you convert this answer from the calculator to the specific starter value? for similar other trigonometric forms?
  
• A fraction V in decimal form is an infinite string that comprises the non-repeat string v prefixing infinitely repeating period P of n digits. If the msd (the first digit) in P is #m^(th)# decimal digit, prove that V = v + #10^(-m) P/(1-10^(-n))#?
  
• In 1/6=1.6666..., repeating 6 is called repeatend ( or reptend ) . I learn from https://en.wikipedia.org/wiki/Repeating_decimal, the reptend in the decimal form of 1/97 is a 96-digit string. Find fraction(s) having longer reptend string(s)?
  
• Using https://socratic.org/questions/in-1-6-1-6666-repeating-6-is-called-repeatend-or-reptend-i-learn-from-https-en-w, how do you design a set of rational numbers { x } that have reptend with million digits?
  
• How do you find the real factors of #49x^6-140x^4+260x^2-169?#
  
• For f(x, y)=x-y, how do you prove that the equation #f(x, y)= x f(y,x)# represents a hyperbola? find its asymptotes?
  
• What is the distance, measured along a great circle, between locations at #(34^o N, 34^o W) and (34^ S, 34^o E)#?
  
• Given #f(x, y)=x^2+y^2-2x#, how do you the volume of the solid bounded by #z=(f(x, y)+f(y,x))/2-5/2, z = +-3?#
  
• The integer part of #f(4) =[theta^(3^4)]=2521008887 #, where the Mills constant #theta=1.30637788386.. .# was expected to be the fourth prime in the sequence f(n). Is it a prime? How do you cross check?
  
• The roots #{x_i}, i=1,2,3,...,6# of #x^6+ax^3+b=0# are such that every #|x_i|=1#. How do you prove that, if #b^2-a^2>=1, a^2-3<=b^2<=a^2+5?#. Otherwise, #b^2-5<=a^2<=b^2+3 ?#
  
• How do you prove that #sin(cot^(-1)(tan(cos^(-1)(sin x))))=sin x#?
  
• How do you find exactly #sin 78.75^o#, in a closed form?
  
• How do you prove that #(1/2)sqrt(2+sqrt(2+sqrt 2))=sqrt((1+sqrt((1+1/sqrt 2)/2)) /2)#?
  
• For mathematical exactitude, how do you express #sin 89.625^o#, in a closed form?
  
• Closest Luna Looks Super. From the data on record over centuries, how do you prove that the so-far-largest Super Moon was nearly 1.14 times the so-far-smallest, in diameter, and 1.30 times, in area of the disc?
  
• Citing examples and making graphs, how do you explain that #f(r, theta;a,b,n,alpha )=r-(a + b cos n(theta- alpha) )=0# represents a family of limacons, rose curves, circles and more?
  
• How do you make a graph of the rosy limacon #r = 1 + 2 cos 3theta#?
  
• How do you prove that the limit #n to oo sum e^(-n/10) cos(n/10)#, for n from 0 to n, is 10.5053, nearly.?
  
• #(sin x)'=cos x, (cos x)'=-sin x, int sin x dx = -cos x + C, int cos x dx = sin x + C and e^(ix)=cos x + i sin x#. Are all these OK, if x is in degree mode?
  
• How do you find the locus of the center of the hyperbola having asymptotes given by #y-x tan alpha+1=0 and y-x tan(alpha+pi/4)+2=0#, where #alpha# varies?
  
• #y_n=log x_n, n =2,3,4,...and y_n-(n-1)/n y_(n-1)=1/n log n#, with #y_2=log sqrt2#, how do you prove that #x_n=(n!)^(1/n)#?
  
• How do you express # y=|x^2-9|+|x^4-16|+|x^6-1|#, sans the symbol #|...|#?
  
• How do you find the asymptotes of #y=|x-1|/|x-2|#?
  
• How do you express the value of #cos 2^o15' #, in mathematical exactitude?
  
• ABCDEFGH is a regular convex octagon, with #A( 0, 1 ), B( sqrt2, 1 ), E( sqrt 2, -1-sqrt 2 ) and F( 0, -1-sqrt2 )#. How do you find the coordinates of the remaining vertices? .
  
• ABCD is a unit square. If CD is moved parallel to AB, and away from AB, continuously, how do you prove that the limit AC/BD is 1?
  
• How do you find the polar equation to the tangent, at #theta = pi/12#, on #r = sin 3theta#?
  
• Like FCF, a Functional Continued Sum (FCS) #F_(fcs)(x; a) = F(x+a (F(x + a (F(x + a(F...))))#. With binary x and y, #y = log_2(x; 1) = log_2(x + log_2(x+log_2(x+...)))#. How do you find the binary at x = (101)_2?
  
• #y = sin_(fcs)(x; 1) = sin (x + sin(x + sin ( x + sin( x + ...))))#. How do you find y, at #x = 1 radian#?
  
• The FCS #tan_(fcs)(pix: pi) = tan (pix+ pi (tan (pix + pi tan (pix +...)#. How do you graph this FCS to reveal x-intercepts #0, +-1, +_2, +3|-...#?
  
• FCS # y = tan_(fcs)( x; -1 ) = tan ( x -tan(x- tan ( x - ...#. How do you prove that, piecewise, at y = 1, the tangent is inclined at #33^o 41' 24'' # to the x-axis, nearly?
  
• FCS #y = exp_(fcs)(x; 1) = e^(x+e^(x+e^(x+... )))#. How do you prove that, as #y to oo, y + x to C < oo#?
  
• #x+-y = 0# are asymptotes to the family of Rectangular Hyperbolas (RH) #x^2-y^2=+-c^2#. How do you prove that the multitude of straight lines #x+-y=a# are asymptotes to the RH family?
  
• How do you create a graph of #r = sin ((4theta)/3)#?
  
• From the crests of #y = sin x, 1 = sin (pi/2) = sin ((5/2)pi) = sin ((9/2)pi) = ...# How do you evaluate #sin^(-1) sin (pi/2), sin^(-1) sin ((5/2)pi), sin^(-1) sin ((9/2)pi), ...?#
  
• How do you write polar equations of hyperbolas, from the polar equations of their asymptotes?
  
• Defining the wholesome inverse operator #(sin)^(-1)# by #Y = (sin)^(-1)(X) = k pi + (-1)^k sin^(-1)X, k= 0, +-1, +-2, +-3,...#, how do you find the points of inflexion of the FCS # y = ((sin)^(-1))_(fcs) (x; 1) = (sin)^(-1)(x+y)?#
  
• #(sin )^(-1)# is the piecewise-wholesome inverse sine operator. The FCS y = #(sin )^(-1)(x+(sin )^(-1)(x+(sin )^(-1)(x+...)))#. How do you find the amplitude and the period, of this FCS wave?
  
• With Meter Scale of mm- precision and Compass, how do you mark exactly #x = sqrt k# cm, k = 2, 3, 5, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 10, 21, 22, 23, 24, 26, ... and 99?
  
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