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❚♦♦❧s ❢♦r ❘✐❣♦r♦✉s ❈♦♠♣✉t✐♥❣ ✉s✐♥❣ ❈❤❡❜②s❤❡✈ ❙❡r✐❡s ❆♣♣r♦①✐♠❛t✐♦♥s ◆✐❝♦❧❛s ❇r✐s❡❜❛rr❡ ▼✐♦❛r❛ ❏♦❧❞❡s ✶ ✴ ✷✾

✸✳ ❈❤❡❜②s❤❡✈ ♠♦❞❡❧s ✭❈▼✮ ❍♦✇❄ ❯s❡ ❋❧♦❛t✐♥❣✲P♦✐♥t ❛s s✉♣♣♦rt ❢♦r ❝♦♠♣✉t❛t✐♦♥s ✭❢❛st✮ ❇♦✉♥❞ r♦✉♥❞♦✛✱ ❞✐s❝r❡t✐③❛t✐♦♥✱ tr✉♥❝❛t✐♦♥ ❡rr♦rs ✐♥ ♥✉♠❡r✐❝❛❧ ❛❧❣♦r✐t❤♠s ❈♦♠♣✉t❡ ❡♥❝❧♦s✉r❡s ✐♥st❡❛❞ ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥s ❲❤❛t❄ ✶✳ ■♥t❡r✈❛❧ ❛r✐t❤♠❡t✐❝ ✭■❆✮ ✷✳ ❚❛②❧♦r ♠♦❞❡❧s ✭❚▼✮ ❲❤❡r❡❄ ❇❡❛♠ P❤②s✐❝s ✭▼✳ ❇❡r③✱ ❑✳ ▼❛❦✐♥♦✮✱ ▲♦r❡♥t③ ❛ttr❛❝t♦r ✭❲✳ ❚✉❝❦❡r✮✱ ❋❧②s♣❡❝❦ ♣r♦❥❡❝t ✭❘✳ ❩✉♠❦❡❧❧❡r✮ ❘✐❣♦r♦✉s ❝♦♠♣✉t✐♥❣ t♦♦❧s ❲❤②❄ ●❡t t❤❡ ❝♦rr❡❝t ❛♥s✇❡r✱ ♥♦t ❛♥ ✑❛❧♠♦st✑ ❝♦rr❡❝t ♦♥❡ ❇r✐❞❣❡ t❤❡ ❣❛♣ ❜❡t✇❡❡♥ s❝✐❡♥t✐✜❝ ❝♦♠♣✉t✐♥❣ ❛♥❞ ♣✉r❡ ♠❛t❤❡♠❛t✐❝s ✲ s♣❡❡❞ ❛♥❞ r❡❧✐❛❜✐❧✐t② ✷ ✴ ✷✾

✸✳ ❈❤❡❜②s❤❡✈ ♠♦❞❡❧s ✭❈▼✮ ❲❤❛t❄ ✶✳ ■♥t❡r✈❛❧ ❛r✐t❤♠❡t✐❝ ✭■❆✮ ✷✳ ❚❛②❧♦r ♠♦❞❡❧s ✭❚▼✮ ❲❤❡r❡❄ ❇❡❛♠ P❤②s✐❝s ✭▼✳ ❇❡r③✱ ❑✳ ▼❛❦✐♥♦✮✱ ▲♦r❡♥t③ ❛ttr❛❝t♦r ✭❲✳ ❚✉❝❦❡r✮✱ ❋❧②s♣❡❝❦ ♣r♦❥❡❝t ✭❘✳ ❩✉♠❦❡❧❧❡r✮ ❘✐❣♦r♦✉s ❝♦♠♣✉t✐♥❣ t♦♦❧s ❲❤②❄ ●❡t t❤❡ ❝♦rr❡❝t ❛♥s✇❡r✱ ♥♦t ❛♥ ✑❛❧♠♦st✑ ❝♦rr❡❝t ♦♥❡ ❇r✐❞❣❡ t❤❡ ❣❛♣ ❜❡t✇❡❡♥ s❝✐❡♥t✐✜❝ ❝♦♠♣✉t✐♥❣ ❛♥❞ ♣✉r❡ ♠❛t❤❡♠❛t✐❝s ✲ s♣❡❡❞ ❛♥❞ r❡❧✐❛❜✐❧✐t② ❍♦✇❄ ❯s❡ ❋❧♦❛t✐♥❣✲P♦✐♥t ❛s s✉♣♣♦rt ❢♦r ❝♦♠♣✉t❛t✐♦♥s ✭❢❛st✮ ❇♦✉♥❞ r♦✉♥❞♦✛✱ ❞✐s❝r❡t✐③❛t✐♦♥✱ tr✉♥❝❛t✐♦♥ ❡rr♦rs ✐♥ ♥✉♠❡r✐❝❛❧ ❛❧❣♦r✐t❤♠s ❈♦♠♣✉t❡ ❡♥❝❧♦s✉r❡s ✐♥st❡❛❞ ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥s ✷ ✴ ✷✾

✸✳ ❈❤❡❜②s❤❡✈ ♠♦❞❡❧s ✭❈▼✮ ❘✐❣♦r♦✉s ❝♦♠♣✉t✐♥❣ t♦♦❧s ❲❤②❄ ●❡t t❤❡ ❝♦rr❡❝t ❛♥s✇❡r✱ ♥♦t ❛♥ ✑❛❧♠♦st✑ ❝♦rr❡❝t ♦♥❡ ❇r✐❞❣❡ t❤❡ ❣❛♣ ❜❡t✇❡❡♥ s❝✐❡♥t✐✜❝ ❝♦♠♣✉t✐♥❣ ❛♥❞ ♣✉r❡ ♠❛t❤❡♠❛t✐❝s ✲ s♣❡❡❞ ❛♥❞ r❡❧✐❛❜✐❧✐t② ❍♦✇❄ ❯s❡ ❋❧♦❛t✐♥❣✲P♦✐♥t ❛s s✉♣♣♦rt ❢♦r ❝♦♠♣✉t❛t✐♦♥s ✭❢❛st✮ ❇♦✉♥❞ r♦✉♥❞♦✛✱ ❞✐s❝r❡t✐③❛t✐♦♥✱ tr✉♥❝❛t✐♦♥ ❡rr♦rs ✐♥ ♥✉♠❡r✐❝❛❧ ❛❧❣♦r✐t❤♠s ❈♦♠♣✉t❡ ❡♥❝❧♦s✉r❡s ✐♥st❡❛❞ ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥s ❲❤❛t❄ ✶✳ ■♥t❡r✈❛❧ ❛r✐t❤♠❡t✐❝ ✭■❆✮ ✷✳ ❚❛②❧♦r ♠♦❞❡❧s ✭❚▼✮ ❲❤❡r❡❄ ❇❡❛♠ P❤②s✐❝s ✭▼✳ ❇❡r③✱ ❑✳ ▼❛❦✐♥♦✮✱ ▲♦r❡♥t③ ❛ttr❛❝t♦r ✭❲✳ ❚✉❝❦❡r✮✱ ❋❧②s♣❡❝❦ ♣r♦❥❡❝t ✭❘✳ ❩✉♠❦❡❧❧❡r✮ ✷ ✴ ✷✾

❘✐❣♦r♦✉s ❝♦♠♣✉t✐♥❣ t♦♦❧s ❲❤②❄ ●❡t t❤❡ ❝♦rr❡❝t ❛♥s✇❡r✱ ♥♦t ❛♥ ✑❛❧♠♦st✑ ❝♦rr❡❝t ♦♥❡ ❇r✐❞❣❡ t❤❡ ❣❛♣ ❜❡t✇❡❡♥ s❝✐❡♥t✐✜❝ ❝♦♠♣✉t✐♥❣ ❛♥❞ ♣✉r❡ ♠❛t❤❡♠❛t✐❝s ✲ s♣❡❡❞ ❛♥❞ r❡❧✐❛❜✐❧✐t② ❍♦✇❄ ❯s❡ ❋❧♦❛t✐♥❣✲P♦✐♥t ❛s s✉♣♣♦rt ❢♦r ❝♦♠♣✉t❛t✐♦♥s ✭❢❛st✮ ❇♦✉♥❞ r♦✉♥❞♦✛✱ ❞✐s❝r❡t✐③❛t✐♦♥✱ tr✉♥❝❛t✐♦♥ ❡rr♦rs ✐♥ ♥✉♠❡r✐❝❛❧ ❛❧❣♦r✐t❤♠s ❈♦♠♣✉t❡ ❡♥❝❧♦s✉r❡s ✐♥st❡❛❞ ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥s ❲❤❛t❄ ✶✳ ■♥t❡r✈❛❧ ❛r✐t❤♠❡t✐❝ ✭■❆✮ ✷✳ ❚❛②❧♦r ♠♦❞❡❧s ✭❚▼✮ ✸✳ ❈❤❡❜②s❤❡✈ ♠♦❞❡❧s ✭❈▼✮ ❲❤❡r❡❄ ❇❡❛♠ P❤②s✐❝s ✭▼✳ ❇❡r③✱ ❑✳ ▼❛❦✐♥♦✮✱ ▲♦r❡♥t③ ❛ttr❛❝t♦r ✭❲✳ ❚✉❝❦❡r✮✱ ❋❧②s♣❡❝❦ ♣r♦❥❡❝t ✭❘✳ ❩✉♠❦❡❧❧❡r✮ ✷ ✴ ✷✾

Pr❛❝t✐❝❛❧ ❊①❛♠♣❧❡s✿ ❈♦♠♣✉t✐♥❣ s✉♣r❡♠✉♠ ♥♦r♠s ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ❢✉♥❝t✐♦♥s✿ ✇❤❡r❡ ✐s ❛ ✈❡r② ❣♦♦❞ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ✳ ❘✐❣♦r♦✉s q✉❛❞r❛t✉r❡✿ ❲❤❛t ❦✐♥❞ ♦❢ ♣r♦❜❧❡♠s ❝❛♥ ✇❡ ✭❈▼✮ ❛❞❞r❡ss ❄ ❈✉rr❡♥t❧② ✇❡ ❝♦♥s✐❞❡r ✉♥✐✈❛r✐❛t❡ ❢✉♥❝t✐♦♥s f ✱ ✏s✉✣❝✐❡♥t❧② s♠♦♦t❤✑ ♦✈❡r [ a, b ] ✳ ✸ ✴ ✷✾

❲❤❛t ❦✐♥❞ ♦❢ ♣r♦❜❧❡♠s ❝❛♥ ✇❡ ✭❈▼✮ ❛❞❞r❡ss ❄ ❈✉rr❡♥t❧② ✇❡ ❝♦♥s✐❞❡r ✉♥✐✈❛r✐❛t❡ ❢✉♥❝t✐♦♥s f ✱ ✏s✉✣❝✐❡♥t❧② s♠♦♦t❤✑ ♦✈❡r [ a, b ] ✳ Pr❛❝t✐❝❛❧ ❊①❛♠♣❧❡s✿ ❈♦♠♣✉t✐♥❣ s✉♣r❡♠✉♠ ♥♦r♠s ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ❢✉♥❝t✐♦♥s✿ sup {| f ( x ) − g ( x ) |} , x ∈ [ a, b ] ✇❤❡r❡ g ✐s ❛ ✈❡r② ❣♦♦❞ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ f ✳ ❘✐❣♦r♦✉s q✉❛❞r❛t✉r❡✿ 1 � 4 π = 1 + x 2 d x 0 ✸ ✴ ✷✾

✱ ❜✉t ■♠ ■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s ❊❣✳ ❘❛♥❣❡ ❜♦✉♥❞✐♥❣ ❢♦r ❢✉♥❝t✐♦♥s ❊❣✳ ■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ✭■❆✮ ❊❛❝❤ ✐♥t❡r✈❛❧ ❂ ♣❛✐r ♦❢ ✢♦❛t✐♥❣✲♣♦✐♥t ♥✉♠❜❡rs ✭♠✉❧t✐♣❧❡ ♣r❡❝✐s✐♦♥ ■❆ ❧✐❜r❛r✐❡s ❡①✐st✱ ❡✳❣✳ ▼P❋■ ✶ ✮ ✶ ❤tt♣✿✴✴❣❢♦r❣❡✳✐♥r✐❛✳❢r✴♣r♦❥❡❝ts✴♠♣❢✐✴ ✹ ✴ ✷✾

✱ ❜✉t ■♠ ■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s ❊❣✳ ❘❛♥❣❡ ❜♦✉♥❞✐♥❣ ❢♦r ❢✉♥❝t✐♦♥s ❊❣✳ ■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ✭■❆✮ ❊❛❝❤ ✐♥t❡r✈❛❧ ❂ ♣❛✐r ♦❢ ✢♦❛t✐♥❣✲♣♦✐♥t ♥✉♠❜❡rs ✭♠✉❧t✐♣❧❡ ♣r❡❝✐s✐♦♥ ■❆ ❧✐❜r❛r✐❡s ❡①✐st✱ ❡✳❣✳ ▼P❋■ ✶ ✮ π ∈ [3 . 1415 , 3 . 1416] ✶ ❤tt♣✿✴✴❣❢♦r❣❡✳✐♥r✐❛✳❢r✴♣r♦❥❡❝ts✴♠♣❢✐✴ ✹ ✴ ✷✾

✱ ❜✉t ■♠ ❘❛♥❣❡ ❜♦✉♥❞✐♥❣ ❢♦r ❢✉♥❝t✐♦♥s ❊❣✳ ■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ✭■❆✮ ❊❛❝❤ ✐♥t❡r✈❛❧ ❂ ♣❛✐r ♦❢ ✢♦❛t✐♥❣✲♣♦✐♥t ♥✉♠❜❡rs ✭♠✉❧t✐♣❧❡ ♣r❡❝✐s✐♦♥ ■❆ ❧✐❜r❛r✐❡s ❡①✐st✱ ❡✳❣✳ ▼P❋■ ✶ ✮ π ∈ [3 . 1415 , 3 . 1416] ■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s ❊❣✳ [1 , 2] + [ − 3 , 2] = [ − 2 , 4] ✶ ❤tt♣✿✴✴❣❢♦r❣❡✳✐♥r✐❛✳❢r✴♣r♦❥❡❝ts✴♠♣❢✐✴ ✹ ✴ ✷✾

✱ ❜✉t ■♠ ■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ✭■❆✮ ❊❛❝❤ ✐♥t❡r✈❛❧ ❂ ♣❛✐r ♦❢ ✢♦❛t✐♥❣✲♣♦✐♥t ♥✉♠❜❡rs ✭♠✉❧t✐♣❧❡ ♣r❡❝✐s✐♦♥ ■❆ ❧✐❜r❛r✐❡s ❡①✐st✱ ❡✳❣✳ ▼P❋■ ✶ ✮ π ∈ [3 . 1415 , 3 . 1416] ■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s ❊❣✳ [1 , 2] + [ − 3 , 2] = [ − 2 , 4] ❘❛♥❣❡ ❜♦✉♥❞✐♥❣ ❢♦r ❢✉♥❝t✐♦♥s ❊❣✳ x ∈ [ − 1 , 2] , f ( x ) = x 2 − x + 1 F ( X ) = X 2 − X + 1 F ([ − 1 , 2]) = [ − 1 , 2] 2 − [ − 1 , 2] + [1 , 1] F ([ − 1 , 2]) = [0 , 4] − [ − 1 , 2] + [1 , 1] F ([ − 1 , 2]) = [ − 1 , 6] ✶ ❤tt♣✿✴✴❣❢♦r❣❡✳✐♥r✐❛✳❢r✴♣r♦❥❡❝ts✴♠♣❢✐✴ ✹ ✴ ✷✾