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❈❤❡❜②s❤❡✈ ■♥t❡r♣♦❧❛t✐♦♥ P♦❧②♥♦♠✐❛❧✲❜❛s❡❞ ❚♦♦❧s ❢♦r ❘✐❣♦r♦✉s ❈♦♠♣✉t✐♥❣ ◆✐❝♦❧❛s ❇r✐s❡❜❛rr❡ ▼✐♦❛r❛ ❏♦❧❞❡s ❆♣r✐❧ ✾✱ ✷✵✶✵ ✶ ✴ ✸✽

▼♦t✐✈❛t✐♦♥ ✷ ✴ ✸✽

▼♦t✐✈❛t✐♦♥ ✷ ✴ ✸✽

▼♦t✐✈❛t✐♦♥ ✷ ✴ ✸✽

▼♦t✐✈❛t✐♦♥ ✷ ✴ ✸✽

✸✳ ❈❤❡❜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 ■♠ ■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s ❊❣✳ ❘❛♥❣❡ ❜♦✉♥❞✐♥❣ ❢♦r ❢✉♥❝t✐♦♥s ❊❣✳ ■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ✭■❆✮ ❊❛❝❤ ✐♥t❡r✈❛❧ ❂ ♣❛✐r ♦❢ ✢♦❛t✐♥❣✲♣♦✐♥t ♥✉♠❜❡rs π ∈ [3 . 1415 , 3 . 1416] ✺ ✴ ✸✽

✱ ❜✉t ■♠ ❘❛♥❣❡ ❜♦✉♥❞✐♥❣ ❢♦r ❢✉♥❝t✐♦♥s ❊❣✳ ■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ✭■❆✮ ❊❛❝❤ ✐♥t❡r✈❛❧ ❂ ♣❛✐r ♦❢ ✢♦❛t✐♥❣✲♣♦✐♥t ♥✉♠❜❡rs π ∈ [3 . 1415 , 3 . 1416] ■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s ❊❣✳ [1 , 2] + [ − 3 , 2] = [ − 2 , 4] ✺ ✴ ✸✽

✱ ❜✉t ■♠ ■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ✭■❆✮ ❊❛❝❤ ✐♥t❡r✈❛❧ ❂ ♣❛✐r ♦❢ ✢♦❛t✐♥❣✲♣♦✐♥t ♥✉♠❜❡rs 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]) = [0 , 7] ✺ ✴ ✸✽

■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ✭■❆✮ ❊❛❝❤ ✐♥t❡r✈❛❧ ❂ ♣❛✐r ♦❢ ✢♦❛t✐♥❣✲♣♦✐♥t ♥✉♠❜❡rs 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]) = [0 , 7] x ∈ [ − 1 , 2] , f ( x ) ∈ [0 , 7] ✱ ❜✉t ■♠ ( f ) = [3 / 4 , 7] ✺ ✴ ✸✽