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Read the problem Formul Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema What is the largest topless trapezoid that can be made by folding up the ends of a 30 in length of wire? Read the problem

  1. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema What is the largest ‘topless’ trapezoid that can be made by folding up the ends of a 30 in length of wire?

  2. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema What is the largest ‘topless’ trapezoid that can be made by folding up the ends of a 30 in length of wire? Trying to maximise area

  3. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema What is the largest ‘topless’ trapezoid that can be made by folding up the ends of a 30 in length of wire? Trying to maximise area Why not like this?

  4. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema Area of a trapezoid: A = 1 2( t + b ) h t and b are length of top and bottom, and h is the height (in inches)

  5. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema Three input variables, t , b , h

  6. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema Three input variables, t , b , h , plus two more, s and e : b + 2 s = 30 ⇒ b = 30 − 2 s By Pythagorean theorem, s 2 = h 2 + e 2 ⇒ e = � s 2 − h 2

  7. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema Three input variables, t , b , h , plus two more, s and e : b + 2 s = 30 ⇒ b = 30 − 2 s By Pythagorean theorem, s 2 = h 2 + e 2 ⇒ e = � s 2 − h 2 and � s 2 − h 2 . t = b + 2 e = 30 − 2 s + 2

  8. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema Three input variables, t , b , h , plus two more, s and e : � s 2 − h 2 b = 30 − 2 s and t = 30 − 2 s + 2

  9. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema Three input variables, t , b , h , plus two more, s and e : � s 2 − h 2 b = 30 − 2 s and t = 30 − 2 s + 2 New formula: A = 1 2( t + b ) h

  10. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema Three input variables, t , b , h , plus two more, s and e : � s 2 − h 2 b = 30 − 2 s and t = 30 − 2 s + 2 New formula: A = 1 � s 2 − h 2 ) + (30 − 2 s ) � � (30 − 2 s + 2 h 2

  11. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema Three input variables, t , b , h , plus two more, s and e : � s 2 − h 2 b = 30 − 2 s and t = 30 − 2 s + 2 New formula: A = 1 � s 2 − h 2 ) + (30 − 2 s ) � � (30 − 2 s + 2 h 2 � s 2 − h 2 ) h = (30 − 2 s +

  12. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema t , b , h , s , and e are lengths, so all are ≥ 0

  13. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema t , b , h , s , and e are lengths, so all are ≥ 0 b = 30 − 2 s ≥ 0 ⇒ s ≤ 15

  14. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema t , b , h , s , and e are lengths, so all are ≥ 0 b = 30 − 2 s ≥ 0 ⇒ s ≤ 15 s 2 = h 2 + e 2 ⇒ s ≥ h

  15. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema t , b , h , s , and e are lengths, so all are ≥ 0 b = 30 − 2 s ≥ 0 ⇒ s ≤ 15 s 2 = h 2 + e 2 ⇒ s ≥ h Domain is ( s , h ) with 0 ≤ s ≤ 15 and 0 ≤ h ≤ s

  16. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema t , b , h , s , and e are lengths, so all are ≥ 0 b = 30 − 2 s ≥ 0 ⇒ s ≤ 15 s 2 = h 2 + e 2 ⇒ s ≥ h Domain is ( s , h ) with 0 ≤ s ≤ 15 and 0 ≤ h ≤ s

  17. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema Interior of the domain is where all inequalities are strict: 0 < s < 15 and 0 < h < s .

  18. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema Interior of the domain is where all inequalities are strict: 0 < s < 15 and 0 < h < s . Find where � ∂ A ∂ s , ∂ A � = � ∇ A = 0 ∂ h (or ∇ A is undefined)

  19. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema Interior of the domain is where all inequalities are strict: 0 < s < 15 and 0 < h < s . Find where � ∂ A ∂ s , ∂ A � = � ∇ A = 0 ∂ h (or ∇ A is undefined) Be careful! There are solutions not in the interior of the domain

  20. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema Boundary is where we have equality: s = 0 and 0 ≤ h ≤ s

  21. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema Boundary is where we have equality: s = 0 and 0 ≤ h ≤ 0

  22. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema Boundary is where we have equality: s = 0 and h = 0

  23. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema Boundary is where we have equality: s = 0 and h = 0 s = 15 and 0 ≤ h ≤ s

  24. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema Boundary is where we have equality: s = 0 and h = 0 s = 15 and 0 ≤ h ≤ 15

  25. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema Boundary is where we have equality: s = 0 and h = 0 s = 15 and 0 ≤ h ≤ 15 0 ≤ s ≤ 15 and h = 0

  26. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema Boundary is where we have equality: s = 0 and h = 0 s = 15 and 0 ≤ h ≤ 15 0 ≤ s ≤ 15 and h = 0 0 ≤ s ≤ 15 and h = s

  27. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema Treat each case separately When s = 0, the only possibility is h = 0, so add (0 , 0) to the list

  28. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema Treat each case separately When s = 15:

  29. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema Treat each case separately When s = 15: � s 2 − h 2 ) h A = (30 − 2 s +

  30. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema Treat each case separately When s = 15: � (15) 2 − h 2 ) h A = (30 − 2(15) +

  31. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema Treat each case separately When s = 15: � (15) 2 − h 2 ) h A = (30 − 2(15) + � 225 − h 2 · h . =

  32. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema Treat each case separately When s = 15: � (15) 2 − h 2 ) h A = (30 − 2(15) + � 225 − h 2 · h . = Domain is 0 ≤ h ≤ s .

  33. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema Treat each case separately When s = 15: � (15) 2 − h 2 ) h A = (30 − 2(15) + � 225 − h 2 · h . = Domain is 0 ≤ h ≤ 15. One critical point ( s , h ) = (15 , h ) with 0 < h < 15 (where?)

  34. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema Treat each case separately When s = 15: � (15) 2 − h 2 ) h A = (30 − 2(15) + � 225 − h 2 · h . = Domain is 0 ≤ h ≤ 15. One critical point ( s , h ) = (15 , h ) with 0 < h < 15 (where?) Endpoints ( s , h ) = (15 , 0) and ( s , h ) = (15 , 15)

  35. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema List so far: Interior critical point

  36. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema List so far: Interior critical point (0 , 0)

  37. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema List so far: Interior critical point (0 , 0) (15 , 0) and (15 , 15), and another critical point where s = 15

  38. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema List so far: Interior critical point (0 , 0) (15 , 0) and (15 , 15), and another critical point where s = 15 . . .

  39. Read the problem Formulæ Cut down independent variables Domain Calculus (interior) Calculus (boundary) Extrema Plug all points ( s , h ) into A : s h A 0 0 0 15 0 0 15 15 0 . . . . . . . . .

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