Introduction to Optimization

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• Misc
• Hour exam 2
  • • Nov 11? Nov 13? Nov 18?
    • Decision:
      • Nov 13 after break is review time
      • Nov 18 is test
    • Material on derivatives since first exam (e.g., chain rule, implicit differentiation, related rates, extreme values, Mean Value Theorem, optimization, etc.)
    • Rules and format similar to first hour exam
      • Including open book, notes, references
      • But probably slightly more questions (7 - 9), and slightly harder questions e.g., more word problems
• Questions?
• Graph Sketching
  • g(x) = x² + sinx
    • g′(x) = 2x + cosx
      • = 0 at x = -0.4502, at most one minimum/maximum
    • g′′(x) = 2 - sinx
      • > 0, so graph is concave up, x = -0.4502 is location of minimum
    • Function values?
      • g(0) = 0
      • Minimum g = -0.23....
    • Asymptotic behavior?
      • As x gets very positive or very negative -1 ≤ sinx ≤ 1 but x² grows without bound
    • Estimated shape of curve:
      
    • What interval to plot over?
      • Should include minimum, and wide enough range of x values to see at least one cycle of sine
      • Look at actual graph with R
        

        > g = function(x) { x^2 + sin(x) }
        > g(-0.4502)
        [1] -0.2324656
        > curve( g(x), -5, 5 )

        
        
        

        > curve( g(x), -5, 5, ylim=c(-1,1) )

        
        
• Optimization
  • Section 4.5
    • Optimization problems basically involve finding maxima or minima
    • Steps to solve optimization problems
      • Read problem
      • Draw picture
      • Introduce variables
      • Write equation for unknown(s)
      • Test critical points and endpoints
    • Examples
      • Physics
      • Economics
  • Examples
    • Radios, televisions, etc. are tuned to a particular station by an electrical circuit whose resistance (R) depends on the frequency (f) of the radio wave according to the formula
      • R = (Lf) / (1 + LCf²)
      • L and C are parameters of the tuner (and hide some constants of proportionality in some versions of the formula).
      • What is the frequency at which resistance is highest for fixed L and C?
      • Find dR/df (remembering that L and C are constant)
      • Find critical points (dR/df = 0)
        
    • Do manufacturers of buckets actually use dimensions that come close to minimizing the amount of material needed to make the bucket?
      • Bucket = cylinder
      • Surface area proportional to amount of plastic and cost
        • = 2πrh + πr²
      • This has 2 variables, we want only 1
      • So rewrite h in terms of V (a constant for each bucket) and r
        • Volume V = πr²h
        • h = V/(πr²)
      • Now start setting up expression to differentiate
        
  • Problem set
• Next
  • One more optimization example
  • Introduction to integrals

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