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

23Related rates, sphere volume

A spherical balloon's radius grows at 2 cm/s. How fast is its volume increasing when the radius is 5 cm? Give the answer to 3 decimal places.

Differentiate V = (4/3)πr³ with respect to time to get dV/dt = 4πr²(dr/dt), the chain rule since r depends on t. Plugging r = 5 and dr/dt = 2 gives dV/dt = 200π ≈ 628.319 cm³/s.

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