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  1. 1Limit of sin⁡(3x)/x\sin(3x)/x
  2. 2Limit by factoring
  3. 3Limit of (1+2/x)x(1 + 2/x)^x
  4. 4L'Hopital's rule, limit
  5. 5Limit at infinity, rational function
  6. 6Limit of (1−cos⁡x)/x2(1 - \cos x)/x^2
  7. 7Limit of (tan⁡x−x)/x3(\tan x - x)/x^3
  8. 8Limit of xe−xx e^{-x}
  9. 9Derivative of a cubic
  10. 10Product rule derivative
  11. 11Quotient rule derivative
  12. 12Derivative of ln⁡(x2+1)\ln(x^2 + 1)
  13. 13Chain rule on square root
  14. 14Chain rule, f(x)=(2x+1)5f(x) = (2x + 1)^5
  15. 15Implicit differentiation on circle
  16. 16Implicit differentiation, folium
  17. 17Tangent line to y=x3y = x^3
  18. 18Critical points of x3−3xx^3 - 3x
  19. 19Absolute max on a closed interval
  20. 20Inflection point of cubic
  21. 21Maximizing rectangle area
  22. 22Optimizing box volume
  23. 23Related rates, sphere volume
  24. 24Related rates, circle area
  25. 25Definite integral of polynomial
  26. 26Integral of sin⁡x\sin x from 00 to π\pi
  27. 27Integral of ln⁡x\ln x by parts
  28. 28Integration by parts
  29. 29Integral of 2x/(x2+1)2x/(x^2 + 1)
  30. 30Integral of sin⁡2x\sin^2 x
  31. 31u-substitution integral
  32. 32Improper integral of 1/x21/x^2
  33. 33Improper integral of 1/x1/\sqrt{x}
  34. 34Improper integral of e−2xe^{-2x}
  35. 35Area between x2x^2 and xx
  36. 36Area under a parabola
  37. 37Volume by disk method
  38. 38Volume by disk method
  39. 39Average value of x2x^2
  40. 40Geometric series sum
  41. 41Telescoping series sum
  42. 42Sum of a geometric series
  43. 43Maclaurin series of e2xe^{2x}
  44. 44Maclaurin series of cos⁡x\cos x
  45. 45Radius of convergence, ratio test
  46. 46Partial derivative in x
  47. 47Partial derivative of exye^{xy}
  48. 48Double integral of x+yx + y
  49. 49Double integral, inner yy first
  50. 50Separable ODE dy/dx=2ydy/dx = 2y
  51. 51Solving dy/dx=xdy/dx = x

22Optimizing box volume

An open box is made from a 12 cm by 12 cm sheet by cutting equal squares from each corner and folding up the sides. Find the maximum volume of the box.

Cutting squares of side x gives V(x) = x(12−2x)². Setting V′(x)=0 gives x=2 (the valid root), and V(2)=2(8)²=128 cm³, the maximum volume. Checking nearby x=1 and x=3 give smaller volumes, confirming the peak.

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