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ELF	>�y@@1. x^3 - ax2. sin ax3. tan x4. x^n5. x^2 - bx + 16. x sin x7. 1/x8. 1/x^a9. x/(x-1)10. x/(x^2-1)11. 1/x + 1/(x-1)12. x^4 - ax^213. x sin x^314. sin(1/x)15. sum( (-1)^(k+1) sin(kx)/k, k, 1,n)16. sum( x^k, k, 0, n)17. sin(ax+b)18. x^2 + ax + b19. sum(1/(x-k),k,-n,n)20. x^8 - 5x^7 + 9x^6 - 9x^5 + 8x^4 - 4x^321. x ln x22. x^x23. x tan x24. x cot x25. sqrt x tan x26. sqrt x cot x27. sqrt( x tan x)28. 1/x^2 - 1/sin^2 x1. x^22. -x^23. ax^24. (x-a)^25. -(x-a)^26. x^2 + a7. x^2 - ax8. x^2 - ax + c9. t^2 - 2t + 110. t^2 - 111. (t-1)(t+1)12. ax^2 + cx + d13. c(x-a)^214. (x-a)^2 + c15. x^2 - 2ax + a^216. (x + 1)^2 - 317. (1/2)(x - 1)^2 + 218. -(1/2)(x + 2)^2 - 319. 6(x - (1/4))^2 + 1520. -(4/11)(x - 7)^2 - 1021. (3x - 1)^2 - 1622. (x - 2)^2 -923. x^2 - 324. 7x^2 - 4x25. x(x - 1) - 226. x^2 + 7x + 1027. 2(x^2 - 1) - 3x28. x^2 - 2x - 329. 15 x^2 + 34 x + 1530. x^2 - 2x - 831. x^2 - 4x - 432. 6x^2 + x - 1233. 5x^2 - 11x + 234. (x - 1/2)^2 - 7/235. (x + 3)^2 - 636. x^2 - 4x37. 2x^2 - 9x - 1838. x^2 - 439. -x^2 + 340. x^2 - 2x - 241. 2x^2 - 5x - 342. -x^2 - 4x - 543. 2x^2 - x - 144. 4x^2 + 11x + 645. x^2 - 246. 2x^2 + 347. -2x^2 + 348. x^2 + 4x + 449. x^2 - 2x - 350. (x - 1)^2 -41. y = ax2. y = x + a3. y = ax + b4. y = ax^25. y = ax^2 + b6. y = (x-a)^27. y = x^38. y = x^3 + a9. y = x^3 - ax10. y = ax^311. y = x^3 + ax^212. y = (x-a)^313. y = x^414. y = ax^415. y = x^4 + a16. y = x^4 - ax^217. y = x^4 - x^318. y = x^4 - ax19. y = (x-a)^420. y = x^n21. y = ax^3 + cx + d22. y = x^4 + ax^3 + cx^2 + d23. y = x^824. y = x^1225. y = x^726. y = x^927. y = x^1328. y = (x-2)^429. y = (x+3)^730. y = -x^4 + 231. y = -(x-5)^7 - 332. y = x^4 + x^2 - 5x33. y = -(x^5) + 3x^4 - 9x + 334. y = 3x^3 - 6x^2 + 1235. y = -5 x^6 - 3 x^4 + 836. y = -(x^3)37. y =  x^4 - 238. y = -(x - 3)^339. y = x^5 + 140. y = (x - 4)^341. y = 2x^5 - x + 642. y = (1/2) x^3 + 143. y = -2 x^6 + 3x + 744. y = 10 x^345. y = (1/2) x^5 + 4 x^246. y = 3 x^6 - x - 1147. y = -(x^6) +2x^3 - x48. y = x^4 + 9x - 149. y = -5 x^450. y = x^4 - x^3  + 12 x1. y = x           | -8 <= x <= 8, -8 <= y <= 82. y = -x          | -8 <= x <= 8, -8 <= y <= 83. y = 3x          | -8 <= x <= 8, -8 <= y <= 84. y = -(1/4)x     | -8 <= x <= 8, -8 <= y <= 85. y = (2/3)x      | -8 <= x <= 8, -8 <= y <= 86. y = ax          | -8 <= x <= 8, -8 <= y <= 87. y = x + a       | -8 <= x <= 8, -8 <= y <= 88. y = a - x       | -8 <= x <= 8, -8 <= y <= 89. y = 2x + a      | -8 <= x <= 8, -8 <= y <= 810. y = 3x + a      | -8 <= x <= 8, -8 <= y <= 811. y = ax + c      | -8 <= x <= 8, -8 <= y <= 812. y = -3x + 3     | -8 <= x <= 8, -8 <= y <= 813. y = -ax - 6     | -8 <= x <= 8, -8 <= y <= 814. y = 2x - 13     | -8 <= x <= 8, -8 <= y <= 815. y = (3/5)x + 3  | -8 <= x <= 8, -8 <= y <= 816. y = x + 2       | -8 <= x <= 8, -8 <= y <= 817. y = 3x - 4      | -8 <= x <= 8, -8 <= y <= 818. y = (1/a)x      | -8 <= x <= 8, -8 <= y <= 819. y = 2           | -8 <= x <= 8, -8 <= y <= 820. y = -3          | -8 <= x <= 8, -8 <= y <= 821. y = 4x + 6      | -8 <= x <= 8, -8 <= y <= 822. y = -3x + 8     | -8 <= x <= 8, -8 <= y <= 823. y = x - 5       | -8 <= x <= 8, -8 <= y <= 824. y = 2x          | -8 <= x <= 8, -8 <= y <= 825. y = pi x        | -8 <= x <= 8, -8 <= y <= 826. y = x/2         | -8 <= x <= 8, -8 <= y <= 827. y = -3x         | -8 <= x <= 8, -8 <= y <= 828. y = -x          | -8 <= x <= 8, -8 <= y <= 829. y = -x/a        | -8 <= x <= 8, -8 <= y <= 830. y = (1/3)x +5   | -8 <= x <= 8, -8 <= y <= 831. y = -x - 3      | -8 <= x <= 8, -8 <= y <= 832. y = 4x - 3      | -8 <= x <= 8, -8 <= y <= 833. y = -ax -c      | -8 <= x <= 8, -8 <= y <= 834. y = -(1/2)x - 3 | -8 <= x <= 8, -8 <= y <= 835. y = -9x + 3     | -8 <= x <= 8, -8 <= y <= 836. y = -(1/a)x - (1/2)  | -8 <= x <= 8, -8 <= y <= 837. y = 6x - 4      | -8 <= x <= 8, -8 <= y <= 838. y = -x + 8      | -8 <= x <= 8, -8 <= y <= 839. y = -(1/7)x - 2 | -8 <= x <= 8, -8 <= y <= 840. y = x/3 - 2     | -8 <= x <= 8, -8 <= y <= 841. y = -4x - 24    | -8 <= x <= 8, -8 <= y <= 842. y = x/2 - 5     | -8 <= x <= 8, -8 <= y <= 843. y = (1/2)x - 12 | -8 <= x <= 8, -8 <= y <= 844. y = -3x + 6     | -8 <= x <= 8, -8 <= y <= 845. y = (1/2)x - 2  | -8 <= x <= 8, -8 <= y <= 846. y = 2 -(1/2)(x-1) | -8 <= x <= 8, -8 <= y <= 847. y = -3x - 8     | -8 <= x <= 8, -8 <= y <= 848. y = x + 10      | -8 <= x <= 8, -8 <= y <= 849. y = 3x + 6      | -8 <= x <= 8, -8 <= y <= 850. y = (a/c)x + (1/2)  | -8 <= x <= 8, -8 <= y <= 81. y = 1/x2. y = 1/(x-a)3. y = x/(x-a)4. y = x + 1/x5. y = x^2 + 1/x6. y = x^3 + 1/x7. 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)16. y = x/(x-1)17. y = x(x^2-1)18. y = 1/x^a19. y =  (x^2 - 1)/(x-1)20. y =  (x^2 - a)/x1. y = x^(1/2)2. y = x^(1/3)3. y = x^(1/a)4. y = x^b5. 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)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^x7. 2^x - x8. 2^x - x^29. 2^x - x^n10. 2^x + a11. (1/2)^x       @This is a decay, not a growth, because the base is less than 1.12. (99/100)^x13. (101/100)^x14. (1+b)^x15. (1+b)^(ax)16. 2^x + 2^(-x)17. e^x18. ae^x19. e^(ax)20. e^x - ax21. e^x - ax^222. xe^x23. ce^(ax)24. ce^(-ax)25. e^(x^2)26. e^(sqrt x)27. e^x + e^(-x)28. e^x - e^(-x)1. log x2. log(2,x)3. log abs(x)4. log(x-a)5. - log x6. a log x7. log(a,x)       @Try to find the value of a that makes the slope 1 when $y=0$.8. ln x9. ln(x) + a10. ln(x-a)11. a ln x12. 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)/x23. x ln x24. x - ln x25. x^2 - ln x26. 1/x - ln x1. sqrt x2. root(n, x)3. sqrt(1-x) + sqrt(x-1)4. x - sqrt x5. sqrt(x)/x6. 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 x13. root(3, x)/sqrt x14. x - sqrt(x^2+1)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.1. sin x2. cos x4. cot x5. sec x6. csc x7. tan(ax)8. cot(ax)9. sec(ax)10. csc(ax)11. tan(x-b)1. arctan x2. arcsin x3. arccos x4. arcsec x5. arccsc x6. arccot x7. arctan(tan x)1. sinh x2. cosh x3. tanh x4. sech x5. cosh x6. coth x1. sin(x^2)2. sin(sqrt x)3. sin(sqrt(x^2 -1))4. sin(1/x)5. sin^2 x6. sqrt(1 - sin^2 x)7. cos(x) - sin(x)8. a sin x + c cos x9. a sin^2 x + c cos^2 x10. (sin x)/x11. (sin x)/x^212. cos(x)/x13. (1-cos x)/x14. (1-cos x)/x^215. 1/x + sin x16. 1/x^2 + sin x17. x + sin x18. x^2 + sin x19. 1/(x-a) + sin x20. x^3 - ax + b sin x21. x sin x22. x^2 sin x23. x sin x^324. 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 1807.1. J0(x)  | -10 < x < 10, -1 < y < 12. J1(x) | -10 < x < 10, -1 < y < 13. J(n,x) | -10 < x < 10, -1 < y < 14. K0(x) | -10 < x < 10, -1 < y < 15. K1(x) | -10 < x < 10, -1 < y < 16. I0(x) | -10 < x < 10, -1 < y < 17. I1(x) | -10 < x < 10, -1 < y < 18. I(n,x) | -10 < x < 10, -1 < y < 11. 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)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.1. 2x + 2, -2x + 22. ax + b, x/a + b3. ax^2 - b, ax^2 + b4. x^2, x^35. x^3, x^56. x^5, x^77. x^n, x^(n+1)8. sqrt x, root(3, x) | -2 < x < 2, -1.5 < y < 1.59. x^(1/3), x^(1/4) | -3 < x < 3, -1.5 < y < 1.510. root(3, x), root(5, x) | -2 < x < 2, -1.5 < y < 1.511. x^n, 2^x12. 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 x1. ax + b, x/a + b2. ax^2 + b, ax^2 - b3. x^a, x^b4. sqrt x, root(n+2, x) | -2 < x < 2, -1.5 < y < 1.55. x^(1/m), x^(1/n) | -3 < x < 3, -1.5 < y < 1.56. root(n+2,x), root(m+2, x) | -2 < x < 2, -1.5 < y < 1.57. x^n, 2^x8. sin x, cos x, tan x9. tanh x, atan x1. (2-x)/5 <= y <= 0 | -1 < x < 3, -1 < y < 12. (3x+2)/(-3) <= y <= (5x+1)/4  | -2 < x < 2, -2 < y < 23. 2x^2 + x <= y <= 1 | -2 < x < 2,  -2 < y < 24. (x-sqrt 2)(x+sqrt 2) < y < 0 | -2 < x < 2, -2.5 < y < 25. x^3 + 2x^2 <= y <= 8x | -4 < x < 4, -30 < y < 306. x^3 + x^2 - 9x -9 < y < 0 | -4 < x < 4, -30 < y < 307. x^4 < y < x^2 + 12 | -4 < x < 4, -25 < y < 258. (x^2-9)^2 -2 < y < -3(x^2-9) |  -4 < x < 4 , -20 < y < 909. (x-4)/(x^2+2x) <= y <= 0 | -10 < x < 10, -20 < y < 1510. (2x+1)/(x-1) <= y <= (x-1)/(x+1) | -8 < x < 8, -10 < y < 1011. sqrt(2sqrt(x+1)) < y < sqrt (3x-5) | -2 < x < 5, -2 < y < 512. x/(2 sqrt(x+1)) + sqrt(x+1) < y < 0  | -2 < x < 2, -3 < y < 113. x < y < abs x | -2 <= y <= 214. x < y < sqrt x | -0.5 <=x <= 2, -0.5 <= y <= 215. x < y < abs(x)/x | -2 <= y <= 216. y < tan x17. sin x < y < x18. sin x < y < ln x19. ln x < y < tan x20. sin ((n+1)x) < y < x21. sin (nx) < y < sin ((m+1)x)�5..;	5f.2'V	^V	�V	5�.̢	5�.��	5�.��	5.	5C.;3	5h.NX	5�._}	5�.o�	5�.{�	5�.��	5!.�	��	5[.�K	5�.�p	5�.	��	5�.�	I!I/4:!;9!I?4:!;9!I?$>%I6.�
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