Estimation with Derivatives

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• Questions?
  • “Mars” problem in problem set 5?
    • Why not plug in values for r?
      • Gives you average change, not instantaneous
    • Take derivative of r instead
      • Negative implies getting closer, positive implies receding
  • Weight on string from problem set 6
    
• Spinning Swing Problem
  • w = C / (r²m), what is dw/dr?
    
• Estimating with Derivatives
  • Example: how much more would it cost me to make dirt cone 0.5 ft bigger in radius?
    
    • Estimate cost as dC/dr times 0.5
    • Assume dirt costs $5/cubic yard
    • Cone: V = ( πr²h)/ 3
    • What is dC/dr at r = 10 ft?
      • 50π/81
    • So extra 6 inches should cost about 50π/81 × 0.5 = 25π/81 dollars
    • But oops: only a quarter cone, so divide V by 4, can divide dV/dr and dC/dr by 4 also
  • What’s 98⁴?
    • Estimate from 100⁴ = 100,000,000
    • Then subtract 2×d/dx(x⁴) evaluated at x = 100
      • = 2×4×100³
      • = 8,000,000
    • So estimate that 98⁴ is about 92,000,000
    • (Actual value is about 98,236,816, i.e., estimate was only off by around 0.25%
• Next
  • Extrema
  • Read section 4.1

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