silemus.
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  1. 4L'Hopital's rule, limit
  2. 6Limit of (1−cos⁡x)/x2(1 - \cos x)/x^2
  3. 16Implicit differentiation, folium
  4. 21Maximizing rectangle area
  5. 22Optimizing box volume
  6. 23Related rates, sphere volume
  7. 24Related rates, circle area
  8. 27Integral of ln⁡x\ln x by parts
  9. 28Integration by parts
  10. 30Integral of sin⁡2x\sin^2 x
  11. 35Area between x2x^2 and xx
  12. 37Volume by disk method
  13. 41Telescoping series sum
  14. 44Maclaurin series of cos⁡x\cos x
  15. 45Radius of convergence, ratio test
  16. 49Double integral, inner yy first
  17. 50Separable ODE dy/dx=2ydy/dx = 2y

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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