Sindbad~EG File Manager

// 12.20.23  added missing comma at line 447
// 12.30.23  added a space in problem 6 of problems7
// 1.2.23  corrected j0 to J0 in problems15

/*  6.17.13,  problems read from graph.mhw */
char *problems1[] = {
"1. x^3 - ax", // shows how to use a parameter
"2. sin ax", // gets the extrema right even for large a
"3. tan x", // handles the singularities right
"4. x^n", // try with 'erase after parameter change'  off
"5. x^2 - bx + 1", // b has default increment 0.1
"6. x sin x",
"7. 1/x", // singularities handled right
"8. 1/x^a",
"9. x/(x-1)",
"10. x/(x^2-1)", // singularities handled right
"11. 1/x + 1/(x-1)",
"12. x^4 - ax^2",
"13. x sin x^3", // competition can't handle this
"14. sin(1/x)", // or this
"15. sum( (-1)^(k+1) sin(kx)/k, k, 1,n)", // Fourier series of sawtooth function
"16. sum( x^k, k, 0, n)", // MacLaurin series of 1/(1-x)
"17. sin(ax+b)", // show use of two parameters
"18. x^2 + ax + b", // another example of use of two parameters
"19. sum(1/(x-k),k,-n,n)", // handles singularities right
"20. x^8 - 5x^7 + 9x^6 - 9x^5 + 8x^4 - 4x^3", // example from Miller p. 264
"21. x ln x", // computes that there are no singularities */
"22. x^x",
"23. x tan x", // singularities come out right
"24. x cot x", // no singularity at origin
"25. sqrt x tan x",
"26. sqrt x cot x", // sqrt x doesn't kill the singularity at 0
"27. sqrt( x tan x)",
"28. 1/x^2 - 1/sin^2 x", // singularities are x=n \pi x!=0
""
};

/*  6.17.13,  problems read from grquad.mhw */
char *problems2[] = {
"1. x^2",
"2. -x^2",
"3. ax^2",
"4. (x-a)^2",
"5. -(x-a)^2",
"6. x^2 + a",
"7. x^2 - ax",
"8. x^2 - ax + c",
"9. t^2 - 2t + 1",
"10. t^2 - 1",
"11. (t-1)(t+1)",
"12. ax^2 + cx + d",
"13. c(x-a)^2",
"14. (x-a)^2 + c",
"15. x^2 - 2ax + a^2",
"16. (x + 1)^2 - 3",
"17. (1/2)(x - 1)^2 + 2",
"18. -(1/2)(x + 2)^2 - 3",
"19. 6(x - (1/4))^2 + 15",    // | -32 < y < 32",
"20. -(4/11)(x - 7)^2 - 10",  // | -32 < y < 32",
"21. (3x - 1)^2 - 16",
"22. (x - 2)^2 -9",
"23. x^2 - 3",
"24. 7x^2 - 4x",
"25. x(x - 1) - 2",
"26. x^2 + 7x + 10",
"27. 2(x^2 - 1) - 3x",
"28. x^2 - 2x - 3",
"29. 15 x^2 + 34 x + 15",
"30. x^2 - 2x - 8",
"31. x^2 - 4x - 4",
"32. 6x^2 + x - 12",
"33. 5x^2 - 11x + 2",
"34. (x - 1/2)^2 - 7/2",
"35. (x + 3)^2 - 6",
"36. x^2 - 4x",
"37. 2x^2 - 9x - 18",
"38. x^2 - 4",
"39. -x^2 + 3",
"40. x^2 - 2x - 2",
"41. 2x^2 - 5x - 3",
"42. -x^2 - 4x - 5",
"43. 2x^2 - x - 1",
"44. 4x^2 + 11x + 6",
"45. x^2 - 2",
"46. 2x^2 + 3",
"47. -2x^2 + 3",
"48. x^2 + 4x + 4",
"49. x^2 - 2x - 3",
"50. (x - 1)^2 -4",
""
};

/*  6.17.13,  problems read from grpoly.mhw */
char *problems3[] = {
"1. y = ax",
"2. y = x + a",
"3. y = ax + b",
"4. y = ax^2",
"5. y = ax^2 + b",
"6. y = (x-a)^2",
"7. y = x^3",
"8. y = x^3 + a",
"9. y = x^3 - ax",
"10. y = ax^3",
"11. y = x^3 + ax^2",
"12. y = (x-a)^3",
"13. y = x^4",
"14. y = ax^4",
"15. y = x^4 + a",
"16. y = x^4 - ax^2",
"17. y = x^4 - x^3",
"18. y = x^4 - ax",
"19. y = (x-a)^4",
"20. y = x^n",
"21. y = ax^3 + cx + d",
"22. y = x^4 + ax^3 + cx^2 + d",
"23. y = x^8",
"24. y = x^12",
"25. y = x^7",
"26. y = x^9",
"27. y = x^13",
"28. y = (x-2)^4",
"29. y = (x+3)^7",
"30. y = -x^4 + 2",
"31. y = -(x-5)^7 - 3",
"32. y = x^4 + x^2 - 5x",
"33. y = -(x^5) + 3x^4 - 9x + 3",
"34. y = 3x^3 - 6x^2 + 12",
"35. y = -5 x^6 - 3 x^4 + 8",
"36. y = -(x^3)",
"37. y =  x^4 - 2",
"38. y = -(x - 3)^3",
"39. y = x^5 + 1",
"40. y = (x - 4)^3",
"41. y = 2x^5 - x + 6",
"42. y = (1/2) x^3 + 1",
"43. y = -2 x^6 + 3x + 7",
"44. y = 10 x^3",
"45. y = (1/2) x^5 + 4 x^2",
"46. y = 3 x^6 - x - 11",
"47. y = -(x^6) +2x^3 - x",
"48. y = x^4 + 9x - 1",
"49. y = -5 x^4",
"50. y = x^4 - x^3  + 12 x",
""
};

/*  6.17.13,  problems read from grlin.mhw */
char *problems4[] = {
"1. y = x           | -8 <= x <= 8, -8 <= y <= 8",
"2. y = -x          | -8 <= x <= 8, -8 <= y <= 8",
"3. y = 3x          | -8 <= x <= 8, -8 <= y <= 8",
"4. y = -(1/4)x     | -8 <= x <= 8, -8 <= y <= 8",
"5. y = (2/3)x      | -8 <= x <= 8, -8 <= y <= 8",
"6. y = ax          | -8 <= x <= 8, -8 <= y <= 8",
"7. y = x + a       | -8 <= x <= 8, -8 <= y <= 8",
"8. y = a - x       | -8 <= x <= 8, -8 <= y <= 8",
"9. y = 2x + a      | -8 <= x <= 8, -8 <= y <= 8",
"10. y = 3x + a      | -8 <= x <= 8, -8 <= y <= 8",
"11. y = ax + c      | -8 <= x <= 8, -8 <= y <= 8",
"12. y = -3x + 3     | -8 <= x <= 8, -8 <= y <= 8",
"13. y = -ax - 6     | -8 <= x <= 8, -8 <= y <= 8",
"14. y = 2x - 13     | -8 <= x <= 8, -8 <= y <= 8",
"15. y = (3/5)x + 3  | -8 <= x <= 8, -8 <= y <= 8",
"16. y = x + 2       | -8 <= x <= 8, -8 <= y <= 8",
"17. y = 3x - 4      | -8 <= x <= 8, -8 <= y <= 8",
"18. y = (1/a)x      | -8 <= x <= 8, -8 <= y <= 8",
"19. y = 2           | -8 <= x <= 8, -8 <= y <= 8",
"20. y = -3          | -8 <= x <= 8, -8 <= y <= 8",
"21. y = 4x + 6      | -8 <= x <= 8, -8 <= y <= 8",
"22. y = -3x + 8     | -8 <= x <= 8, -8 <= y <= 8",
"23. y = x - 5       | -8 <= x <= 8, -8 <= y <= 8",
"24. y = 2x          | -8 <= x <= 8, -8 <= y <= 8",
"25. y = pi x        | -8 <= x <= 8, -8 <= y <= 8",
"26. y = x/2         | -8 <= x <= 8, -8 <= y <= 8",
"27. y = -3x         | -8 <= x <= 8, -8 <= y <= 8",
"28. y = -x          | -8 <= x <= 8, -8 <= y <= 8",
"29. y = -x/a        | -8 <= x <= 8, -8 <= y <= 8",
"30. y = (1/3)x +5   | -8 <= x <= 8, -8 <= y <= 8",
"31. y = -x - 3      | -8 <= x <= 8, -8 <= y <= 8",
"32. y = 4x - 3      | -8 <= x <= 8, -8 <= y <= 8",
"33. y = -ax -c      | -8 <= x <= 8, -8 <= y <= 8",
"34. y = -(1/2)x - 3 | -8 <= x <= 8, -8 <= y <= 8",
"35. y = -9x + 3     | -8 <= x <= 8, -8 <= y <= 8",
"36. y = -(1/a)x - (1/2)  | -8 <= x <= 8, -8 <= y <= 8",
"37. y = 6x - 4      | -8 <= x <= 8, -8 <= y <= 8",
"38. y = -x + 8      | -8 <= x <= 8, -8 <= y <= 8",
"39. y = -(1/7)x - 2 | -8 <= x <= 8, -8 <= y <= 8",
"40. y = x/3 - 2     | -8 <= x <= 8, -8 <= y <= 8",
"41. y = -4x - 24    | -8 <= x <= 8, -8 <= y <= 8",
"42. y = x/2 - 5     | -8 <= x <= 8, -8 <= y <= 8",
"43. y = (1/2)x - 12 | -8 <= x <= 8, -8 <= y <= 8",
"44. y = -3x + 6     | -8 <= x <= 8, -8 <= y <= 8",
"45. y = (1/2)x - 2  | -8 <= x <= 8, -8 <= y <= 8",
"46. y = 2 -(1/2)(x-1) | -8 <= x <= 8, -8 <= y <= 8",
"47. y = -3x - 8     | -8 <= x <= 8, -8 <= y <= 8",
"48. y = x + 10      | -8 <= x <= 8, -8 <= y <= 8",
"49. y = 3x + 6      | -8 <= x <= 8, -8 <= y <= 8",
"50. y = (a/c)x + (1/2)  | -8 <= x <= 8, -8 <= y <= 8",
""
};

/*  6.17.13,  problems read from grrat.mhw */
char *problems5[] = {
"1. y = 1/x",
"2. y = 1/(x-a)",
"3. y = x/(x-a)",
"4. y = x + 1/x",
"5. y = x^2 + 1/x",
"6. y = x^3 + 1/x",
"7. y = x + 1/(x-a)",
"8. y = x^2 + 1/(x-a)",
"9. y = 1/(x-1) + 1/(x+1)",
"10. y = 1/(x-1) - 1/(x+1)",
"11. y = x + 1/(x-1) + 1/(x+1)",
"12. y = x + 1/(x-1) - 1/(x+1)",
"13. y = x^2 + 1/(x-1) + 1/(x+1)",
"14. y = x^2 + 1/(x-1) - 1/(x+1)",
"15. y = sum(1/(x-k),k,-n,n)", // handles singularities right
"16. y = x/(x-1)",
"17. y = x(x^2-1)",
"18. y = 1/x^a",
"19. y =  (x^2 - 1)/(x-1)",
"20. y =  (x^2 - a)/x",
""
};

/*  6.17.13,  problems read from grfract.mhw */
char *problems6[] = {
"1. y = x^(1/2)",
"2. y = x^(1/3)",
"3. y = x^(1/a)",
"4. y = x^b",
"5. y = (x-a)^(1/2)",
"6. y = x^(a/2)",
"7. y = ax^(1/2)",
"8. y = -x^(1/2)",
"9. y = x^(-1/2)",
"10. y = x^(-1/3)",
"11. y = x^(-1/a)",
"12. y = x^(1/2) + (1-x)^(1/2)",
"13. y = x^(1/2) - (1-x)^(1/2)",
"14. y = x^(1/3) + (1-x)^(1/3)",
"15. y = x^(1/3) - (1-x)^(1/4)",
"16. y = x^(1/2) - x^(1/3)",
""
};

/*  6.17.13,  problems read from grexp.mhw */
char *problems7[] = {
"1. 2^x       @Notice the slope at 0 is less than one.",
"2. 3^x       @Notice the slope at 0 is more than one.",
"3. a^x       @Try to determine the value of $a$ for which the slope is 1 when $x=0$.",
"3. 2^(-x)       @This shows exponential decay, not growth, because of the minus sign.",
"4. a^(-x)       @Try to determine the value of $a$ for which the slope is -1 when $x=0$.",
"5. 2^(ax)",
"6. a 2^x",
"7. 2^x - x",
"8. 2^x - x^2",
"9. 2^x - x^n",
"10. 2^x + a",
"11. (1/2)^x       @This is a decay, not a growth, because the base is less than 1.",
"12. (99/100)^x",
"13. (101/100)^x",
"14. (1+b)^x",
"15. (1+b)^(ax)",
"16. 2^x + 2^(-x)",
"17. e^x",
"18. ae^x",
"19. e^(ax)",
"20. e^x - ax",
"21. e^x - ax^2",
"22. xe^x",
"23. ce^(ax)",
"24. ce^(-ax)",
"25. e^(x^2)",
"26. e^(sqrt x)",
"27. e^x + e^(-x)",
"28. e^x - e^(-x)",
""
};

/*  6.17.13,  problems read from grlog.mhw */
char *problems8[] = {
"1. log x",
"2. log(2,x)",
"3. log abs(x)",
"4. log(x-a)",
"5. - log x",
"6. a log x",
"7. log(a,x)       @Try to find the value of a that makes the slope 1 when $y=0$.",
"8. ln x",
"9. ln(x) + a",
"10. ln(x-a)",
"11. a ln x",
"12. a ln(x-c)",
"13. ln(1/x)",
"14. ln(abs(1/x))",
"15. ln(x^2)",
"16. ln(x^3)",
"17. ln(x^n)",
"18. ln(1-x)",
"19. ln(x^3-ax)",
"20. ln(x - sqrt(x))",
"21. ln(sqrt(x) - x)",
"22. (ln x)/x",
"23. x ln x",
"24. x - ln x",
"25. x^2 - ln x",
"26. 1/x - ln x",
""
};

/*  6.17.13,  problems read from gralg.mhw */
char *problems9[] = {
"1. sqrt x",
"2. root(n, x)",
"3. sqrt(1-x) + sqrt(x-1)",
"4. x - sqrt x",
"5. sqrt(x)/x",
"6. sqrt(x^2+1)",
"7. x/(1-x)",
"8. x/(1-x) + sqrt(x^2 + 1)",
"9. sqrt(x^2-a^2)",
"10. sqrt(a^2-x^2)",
"11. sqrt(x^2 + a^2)",
"12. root(3, x) - sqrt x",
"13. root(3, x)/sqrt x",
"14. x - sqrt(x^2+1)",
""
};

/*  6.17.13,  problems read from grsincos.mhw */
char *problems10[] = {
"1. sin(ax)       @$a$ determines the frequency of the wave.",
"2. cos(ax)       @$a$ determines the frequency of the wave.",
"3. sin(x-b)       @$b$ determines the phase of the wave.",
"4. cos(x-b)       @$b$ determines the phase of the wave.",
"5. a sin x       @$a$ determines the amplitude of the wave.",
"6. a cos x       @$a$ determines the amplitude of the wave.",
"7. a sin(x-b)       @$a$ controls the amplitude, $b$ controls the phase.",
"8. a cos(x-b)       @$a$ controls the amplitude, $b$ controls the phase.",
"9. sin(x-n pi/ 2)       @A phase shift of $pi/2$ changes a sine to a cosine.",
"10. cos(x-n pi/ 2)       @A phase shift of $pi/2$ changes a cosine to a sine.",
"11. sin(x + n pi)       @Shifting the phase by a multiple of $2 pi$ doesn't change the graph.",
"12. cos(x + n pi)       @Shifting the phase by a multiple of $2 pi$ doesn't change the graph.",
""
};

/*  6.17.13,  problems read from grtrig.mhw */
char *problems11[] = {
"1. sin x",
"2. cos x",
"3. tan x",
"4. cot x",
"5. sec x",
"6. csc x",
"7. tan(ax)",
"8. cot(ax)",
"9. sec(ax)",
"10. csc(ax)",
"11. tan(x-b)",
""
};

/*  6.17.13,  problems read from grarctrig.mhw */
char *problems12[] = {
"1. arctan x",
"2. arcsin x",
"3. arccos x",
"4. arcsec x",
"5. arccsc x",
"6. arccot x",
"7. arctan(tan x)",
""
};

/*  6.17.13,  problems read from grhtrig.mhw */
char *problems13[] = {
"1. sinh x",
"2. cosh x",
"3. tanh x",
"4. sech x",
"5. cosh x",
"6. coth x",
""
};

/*  6.17.13,  problems read from grcomptr.mhw */
char *problems14[] = {
"1. sin(x^2)",
"2. sin(sqrt x)",
"3. sin(sqrt(x^2 -1))",
"4. sin(1/x)",
"5. sin^2 x",
"6. sqrt(1 - sin^2 x)",
"7. cos(x) - sin(x)",
"8. a sin x + c cos x",
"9. a sin^2 x + c cos^2 x",
"10. (sin x)/x",
"11. (sin x)/x^2",
"12. cos(x)/x",
"13. (1-cos x)/x",
"14. (1-cos x)/x^2",
"15. 1/x + sin x",
"16. 1/x^2 + sin x",
"17. x + sin x",
"18. x^2 + sin x",
"19. 1/(x-a) + sin x",
"20. x^3 - ax + b sin x",
"21. x sin x",
"22. x^2 sin x",
"23. x sin x^3",
"24. sum( (-1)^(k+1) sin(kx)/k, k, 1,n)       @Fourier series of sawtooth function."
  " If Euler had seen this graph in 1753, he would not have made a"
  " famous mistake: he denied Bernoulli's claim that every function"
  " could be written as an (infinite) sum of trigonometric functions."
  " Euler believed that the sum could not be discontinuous.  The matter"
  " wasn't straightened out until Fourier published his treatise on"
  " Fourier series in 1807.",  //need comma so it ends in an empty string.
""
};

/*  6.17.13,  problems read from grbessel.mhw */
char *problems15[] = {
"1. J0(x)  | -10 < x < 10, -1 < y < 1",
"2. J1(x) | -10 < x < 10, -1 < y < 1",
"3. J(n,x) | -10 < x < 10, -1 < y < 1",
"4. K0(x) | -10 < x < 10, -1 < y < 1",
"5. K1(x) | -10 < x < 10, -1 < y < 1",
"6. I0(x) | -10 < x < 10, -1 < y < 1",
"7. I1(x) | -10 < x < 10, -1 < y < 1",
"8. I(n,x) | -10 < x < 10, -1 < y < 1",
""
};

/*  6.17.13,  problems read from grseries.mhw */
char *problems16[] = {
"1. 1/(1-x), sum( x^k, k, 0, n)",
"2. sin x, sum((-1)^k x^(2k+1)/(2k+1)!, k,0,n)",
"3. cos x, sum((-1)^(k+1) x^(2k)/(2k)!,k,0,n)",
"4. e^x, sum( x^k / k!, k, 0, n)",
"5. ln(1+x), sum((-1)^(k+1) x^k / k, k, 1, n)",
"6. arctan(x), sum((-1)^k x^(2k+1) / (2k+1), k, 0, n)",
""
};

/*  6.17.13,  problems for graph_fourier */
char *problems17[] = {
 "1. y = pi *((x+pi)/(2 pi)-floor((x+pi)/(2 pi)))-pi/2, y = sum( (-1)^(k+1) sin(kx)/k, k, 1,n)  | -12 < x < 12, -2 < y < 2 @Fourier series of sawtooth function."
 " Use the parameter button to increase n and see how the series converges."
 " If Euler had seen this graph in 1753, he would not have made a"
 " famous mistake: he denied Bernoulli's claim that every function"
 " could be written as an (infinite) sum of trigonometric functions."
 " Euler believed that the sum could not be discontinuous.  The matter"
 " wasn't straightened out until Fourier published his treatise on"
 " Fourier series in 1807.  The wiggles at the ends are called the \"Gibbs phenomenon\".",

 "2. y = sgn(x)  , y = (4/pi) * sum( sin((2k-1)x)/(2k-1), k, 1, n )  | -pi < x < pi, -1.5 < y < 1.5 @ Square wave Fourier series."
 " The limit function is the sign function, which jumps from -1 to 1 at x = 0.",

"3. y = abs(x), y = (pi/2) - (4/pi) * sum( cos((2k-1)x) / (2k-1)^2, k, 1, n )  | -pi < x < pi, -0.5 < y < pi @ Triangular wave Fourier series."
" The limit function is the absolute value function, symmetric about x = 0.",

"4. y = abs(sin(x)), y = (2/pi) - (4/pi) * sum( cos(2kx) / (4k^2 - 1), k, 1, n )  | -pi < x < pi, -0.5 < y < 1 @ Full-wave rectified sine."
" The limit function is the absolute value of sin(x), which is periodic with period 2π.",
 ""
};

/*  6.17.13,  problems read for compare_same */
char *problems18[] = {
"1. 2x + 2, -2x + 2",
"2. ax + b, x/a + b",
"3. ax^2 - b, ax^2 + b",
"4. x^2, x^3",
"5. x^3, x^5",
"6. x^5, x^7",
"7. x^n, x^(n+1)",
"8. sqrt x, root(3, x) | -2 < x < 2, -1.5 < y < 1.5",
"9. x^(1/3), x^(1/4) | -3 < x < 3, -1.5 < y < 1.5",
"10. root(3, x), root(5, x) | -2 < x < 2, -1.5 < y < 1.5",
"11. x^n, 2^x", // show relation between exponential and polynomial growth
"12. x,cos x       @ Kepler first studied the equation x = cos x in connection"
  "  with his studies of the orbits of the planets."
  "  Use the point-slope tool to find the solution.",
"13. sin x, sum((-1)^k x^(2k+1)/(2k+1)!, k,0,n)       @ MacLaurin series of sin x."
  "  Change the number of terms (using the parameter button) and see how it converges.",
"14. (pi *((x+pi)/(2 pi)-floor((x+pi)/(2 pi)))-pi/2, sum( (-1)^(k+1) sin(kx)/k, k, 1,n))  | -12 < x < 12, -2 < y < 2       @Fourier series of sawtooth function."
  " Use the parameter button to increase n and see how the series converges."
  " If Euler had seen this graph in 1753, he would not have made a"
  " famous mistake: he denied Bernoulli's claim that every function"
  " could be written as an (infinite) sum of trigonometric functions."
  " Euler believed that the sum could not be discontinuous.  The matter"
  " wasn't straightened out until Fourier published his treatise on"
  " Fourier series in 1807.  The wiggles at the ends are called the \"Gibbs phenomenon\".  ",
"15. sin x, cos x, tan x",  // one example with three functions
  ""
};

/*  6.17.13,  problems for compare_different */
char *problems19[] = {
"1. ax + b, x/a + b",
"2. ax^2 + b, ax^2 - b",
"3. x^a, x^b",
"4. sqrt x, root(n+2, x) | -2 < x < 2, -1.5 < y < 1.5",
"5. x^(1/m), x^(1/n) | -3 < x < 3, -1.5 < y < 1.5",
"6. root(n+2,x), root(m+2, x) | -2 < x < 2, -1.5 < y < 1.5",
"7. x^n, 2^x",
"8. sin x, cos x, tan x", // compare the three most important trig functions
"9. tanh x, atan x", // they have a similar form
""
};

/*  6.17.13,  problems read from grafineq.mhw */
char *problems20[] = {
"1. (2-x)/5 <= y <= 0 | -1 < x < 3, -1 < y < 1",
"2. (3x+2)/(-3) <= y <= (5x+1)/4  | -2 < x < 2, -2 < y < 2",
"3. 2x^2 + x <= y <= 1 | -2 < x < 2,  -2 < y < 2",
"4. (x-sqrt 2)(x+sqrt 2) < y < 0 | -2 < x < 2, -2.5 < y < 2",
"5. x^3 + 2x^2 <= y <= 8x | -4 < x < 4, -30 < y < 30",
"6. x^3 + x^2 - 9x -9 < y < 0 | -4 < x < 4, -30 < y < 30",
"7. x^4 < y < x^2 + 12 | -4 < x < 4, -25 < y < 25",
"8. (x^2-9)^2 -2 < y < -3(x^2-9) |  -4 < x < 4 , -20 < y < 90",
"9. (x-4)/(x^2+2x) <= y <= 0 | -10 < x < 10, -20 < y < 15",
"10. (2x+1)/(x-1) <= y <= (x-1)/(x+1) | -8 < x < 8, -10 < y < 10",
"11. sqrt(2sqrt(x+1)) < y < sqrt (3x-5) | -2 < x < 5, -2 < y < 5",
"12. x/(2 sqrt(x+1)) + sqrt(x+1) < y < 0  | -2 < x < 2, -3 < y < 1",
"13. x < y < abs x | -2 <= y <= 2",
"14. x < y < sqrt x | -0.5 <=x <= 2, -0.5 <= y <= 2",
"15. x < y < abs(x)/x | -2 <= y <= 2",
"16. y < tan x",
"17. sin x < y < x",
"18. sin x < y < ln x",
"19. ln x < y < tan x",
"20. sin ((n+1)x) < y < x",
"21. sin (nx) < y < sin ((m+1)x)",
""
};