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s r rs t s s rs rts r s

SLIDE 1

❚♦♦❧s ❢♦r ❘✐❣♦r♦✉s ❈♦♠♣✉t✐♥❣ ✉s✐♥❣ ❈❤❡❜②s❤❡✈ ❙❡r✐❡s ❆♣♣r♦①✐♠❛t✐♦♥s

▼✐♦❛r❛ ❏♦❧❞❡s ❘❆■▼ ✷✵✶✶

✲✶✲

SLIDE 2

▼♦t✐✈❛t✐♦♥

✲✷✲

SLIDE 3

▼♦t✐✈❛t✐♦♥

✲✷✲

SLIDE 4

▼♦t✐✈❛t✐♦♥

✲✷✲

SLIDE 5

▼♦t✐✈❛t✐♦♥

✲✷✲

SLIDE 6

❘✐❣♦r♦✉s ❝♦♠♣✉t✐♥❣ t♦♦❧s

❲❤②❄

Pr♦✈✐❞❡s r✐❣♦r♦✉s❧② ✈❡r✐✜❡❞ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t s♦❧✉t✐♦♥s ❯s❡❞ t♦ ❝♦♠♣❧❡t❡ ♣r♦♦❢s

❍♦✇❄

❯s❡ ❋❧♦❛t✐♥❣✲P♦✐♥t ❛s s✉♣♣♦rt ❢♦r ❝♦♠♣✉t❛t✐♦♥s ✭❢❛st✮ ❇♦✉♥❞ r♦✉♥❞♦✛✱ 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✮

✲✸✲

SLIDE 7

❘✐❣♦r♦✉s ❝♦♠♣✉t✐♥❣ t♦♦❧s

❲❤②❄

Pr♦✈✐❞❡s r✐❣♦r♦✉s❧② ✈❡r✐✜❡❞ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t s♦❧✉t✐♦♥s ❯s❡❞ t♦ ❝♦♠♣❧❡t❡ ♣r♦♦❢s

❍♦✇❄

❯s❡ ❋❧♦❛t✐♥❣✲P♦✐♥t ❛s s✉♣♣♦rt ❢♦r ❝♦♠♣✉t❛t✐♦♥s ✭❢❛st✮ ❇♦✉♥❞ r♦✉♥❞♦✛✱ 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✮

✲✸✲

SLIDE 8

❘✐❣♦r♦✉s ❝♦♠♣✉t✐♥❣ t♦♦❧s

❲❤②❄

Pr♦✈✐❞❡s r✐❣♦r♦✉s❧② ✈❡r✐✜❡❞ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t s♦❧✉t✐♦♥s ❯s❡❞ t♦ ❝♦♠♣❧❡t❡ ♣r♦♦❢s

❍♦✇❄

❯s❡ ❋❧♦❛t✐♥❣✲P♦✐♥t ❛s s✉♣♣♦rt ❢♦r ❝♦♠♣✉t❛t✐♦♥s ✭❢❛st✮ ❇♦✉♥❞ r♦✉♥❞♦✛✱ 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✮

✲✸✲

SLIDE 9

❘✐❣♦r♦✉s ❝♦♠♣✉t✐♥❣ t♦♦❧s

❲❤②❄

Pr♦✈✐❞❡s r✐❣♦r♦✉s❧② ✈❡r✐✜❡❞ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t s♦❧✉t✐♦♥s ❯s❡❞ t♦ ❝♦♠♣❧❡t❡ ♣r♦♦❢s

❍♦✇❄

❯s❡ ❋❧♦❛t✐♥❣✲P♦✐♥t ❛s s✉♣♣♦rt ❢♦r ❝♦♠♣✉t❛t✐♦♥s ✭❢❛st✮ ❇♦✉♥❞ r♦✉♥❞♦✛✱ 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✮

✲✸✲

SLIDE 10

❲❤❛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✿ ✇❤❡r❡ ✐s ❛ ✈❡r② ❣♦♦❞ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ✳ ❘✐❣♦r♦✉s q✉❛❞r❛t✉r❡✿

✲✹✲

SLIDE 11

❲❤❛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

x∈[a, b]

{|f(x) − g(x)|}, ✇❤❡r❡ g ✐s ❛ ✈❡r② ❣♦♦❞ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ f✳ ❘✐❣♦r♦✉s q✉❛❞r❛t✉r❡✿ π =

1

1 + x2 dx

✲✹✲

SLIDE 12

■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ✭■❆✮

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

✶❤tt♣✿✴✴❣❢♦r❣❡✳✐♥r✐❛✳❢r✴♣r♦❥❡❝ts✴♠♣❢✐✴ ✲✺✲

SLIDE 13

■♥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 ❊❣✳ ❘❛♥❣❡ ❜♦✉♥❞✐♥❣ ❢♦r ❢✉♥❝t✐♦♥s ❊❣✳ ✱ ❜✉t ■♠

✶❤tt♣✿✴✴❣❢♦r❣❡✳✐♥r✐❛✳❢r✴♣r♦❥❡❝ts✴♠♣❢✐✴ ✲✺✲

SLIDE 14

■♥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 ❊❣✳ ✱ ❜✉t ■♠

✶❤tt♣✿✴✴❣❢♦r❣❡✳✐♥r✐❛✳❢r✴♣r♦❥❡❝ts✴♠♣❢✐✴ ✲✺✲

SLIDE 15

■♥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) = x2 − x + 1 F(X) = X2 − 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] ✱ ❜✉t ■♠

✶❤tt♣✿✴✴❣❢♦r❣❡✳✐♥r✐❛✳❢r✴♣r♦❥❡❝ts✴♠♣❢✐✴ ✲✺✲

SLIDE 16

■♥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) = x2 − x + 1 F(X) = X2 − 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] x ∈ [−1, 2], f(x) ∈ [−1, 6]✱ ❜✉t ■♠(f) = [3/4, 3]

✶❤tt♣✿✴✴❣❢♦r❣❡✳✐♥r✐❛✳❢r✴♣r♦❥❡❝ts✴♠♣❢✐✴ ✲✺✲

SLIDE 17

❲❤❡♥ ■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ❞♦❡s ♥♦t s✉✣❝❡✿ ❈♦♠♣✉t✐♥❣ s✉♣r❡♠✉♠ ♥♦r♠s ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦rs

f(x) = e1/ cos(x), x ∈ [0, 1]✱ p(x) = 10

i=0 cixi✱

s✳t✳ ✐s ❛s s♠❛❧❧ ❛s ♣♦ss✐❜❧❡ ✭❘❡♠❡③ ❛❧❣♦r✐t❤♠✮

✲✻✲

SLIDE 18

❲❤❡♥ ■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ❞♦❡s ♥♦t s✉✣❝❡✿ ❈♦♠♣✉t✐♥❣ s✉♣r❡♠✉♠ ♥♦r♠s ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦rs

f(x) = e1/ cos(x), x ∈ [0, 1]✱ p(x) = 10

i=0 cixi✱

ε(x) = f(x) − p(x) s✳t✳ ✐s ❛s s♠❛❧❧ ❛s ♣♦ss✐❜❧❡ ✭❘❡♠❡③ ❛❧❣♦r✐t❤♠✮

✲✻✲

SLIDE 19

❲❤❡♥ ■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ❞♦❡s ♥♦t s✉✣❝❡✿ ❈♦♠♣✉t✐♥❣ s✉♣r❡♠✉♠ ♥♦r♠s ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦rs

f(x) = e1/ cos(x), x ∈ [0, 1]✱ p(x) = 10

i=0 cixi✱

ε(x) = f(x) − p(x) s✳t✳ ε∞ = supx∈[a, b]{|ε(x)|} ✐s ❛s s♠❛❧❧ ❛s ♣♦ss✐❜❧❡ ✭❘❡♠❡③ ❛❧❣♦r✐t❤♠✮

✲✻✲

SLIDE 20

❲❤❡♥ ■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ❞♦❡s ♥♦t s✉✣❝❡✿ ❈♦♠♣✉t✐♥❣ s✉♣r❡♠✉♠ ♥♦r♠s ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦rs

f(x) = e1/ cos(x), x ∈ [0, 1]✱ p(x) = 10

i=0 cixi✱

ε(x) = f(x) − p(x) s✳t✳ ε∞ = supx∈[a, b]{|ε(x)|} ✐s ❛s s♠❛❧❧ ❛s ♣♦ss✐❜❧❡ ✭❘❡♠❡③ ❛❧❣♦r✐t❤♠✮ ❯s✐♥❣ ■❆✱ ✱ ❜✉t

✲✼✲

SLIDE 21

❲❤❡♥ ■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ❞♦❡s ♥♦t s✉✣❝❡✿ ❈♦♠♣✉t✐♥❣ s✉♣r❡♠✉♠ ♥♦r♠s ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦rs

f(x) = e1/ cos(x), x ∈ [0, 1]✱ p(x) = 10

i=0 cixi✱

ε(x) = f(x) − p(x) s✳t✳ ε∞ = supx∈[a, b]{|ε(x)|} ✐s ❛s s♠❛❧❧ ❛s ♣♦ss✐❜❧❡ ✭❘❡♠❡③ ❛❧❣♦r✐t❤♠✮ ❯s✐♥❣ ■❆✱ ✱ ❜✉t

✲✼✲

SLIDE 22

❲❤❡♥ ■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ❞♦❡s ♥♦t s✉✣❝❡✿ ❈♦♠♣✉t✐♥❣ s✉♣r❡♠✉♠ ♥♦r♠s ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦rs

f(x) = e1/ cos(x), x ∈ [0, 1]✱ p(x) = 10

i=0 cixi✱

ε(x) = f(x) − p(x) s✳t✳ ε∞ = supx∈[a, b]{|ε(x)|} ✐s ❛s s♠❛❧❧ ❛s ♣♦ss✐❜❧❡ ✭❘❡♠❡③ ❛❧❣♦r✐t❤♠✮ ❯s✐♥❣ ■❆✱ ε(x) ∈ [−233, 298]✱ ❜✉t ε(x)∞ ≃ 3.8325 · 10−5

✲✼✲

SLIDE 23

❲❤② ■❆ ❞♦❡s ♥♦t s✉✣❝❡✿ ❖✈❡r❡st✐♠❛t✐♦♥

❖✈❡r❡st✐♠❛t✐♦♥ ❝❛♥ ❜❡ r❡❞✉❝❡❞ ❜② ✉s✐♥❣ ✐♥t❡r✈❛❧s ♦❢ s♠❛❧❧❡r ✇✐❞t❤✳ ■♥ t❤✐s ❝❛s❡✱ ♦✈❡r [0, 1] ✇❡ ♥❡❡❞ 107 ✐♥t❡r✈❛❧s✦

✲✽✲

SLIDE 24

❲❤② ■❆ ❞♦❡s ♥♦t s✉✣❝❡✿ ❖✈❡r❡st✐♠❛t✐♦♥

❖✈❡r❡st✐♠❛t✐♦♥ ❝❛♥ ❜❡ r❡❞✉❝❡❞ ❜② ✉s✐♥❣ ✐♥t❡r✈❛❧s ♦❢ s♠❛❧❧❡r ✇✐❞t❤✳ ❯s❡ t❤❡ ❛♣♣r♦♣r✐❛t❡ t♦♦❧s✦

✲✽✲

SLIDE 25

❘✐❣♦r♦✉s ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥s

r❡♣❧❛❝❡❞ ✇✐t❤ ❛ r✐❣♦r♦✉s ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ✿ ✲ ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❞❡❣r❡❡ ✲ ✐♥t❡r✈❛❧ s✳ t✳ ▼❛✐♥ ♣♦✐♥t ♦❢ t❤✐s t❛❧❦✿ ❍♦✇ t♦ ❝♦♠♣✉t❡ ❄

✲✾✲

SLIDE 26

❘✐❣♦r♦✉s ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥s

f r❡♣❧❛❝❡❞ ✇✐t❤ ❛ r✐❣♦r♦✉s ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ✿ ✲ ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ T ♦❢ ❞❡❣r❡❡ n ✲ ✐♥t❡r✈❛❧ s✳ t✳ ▼❛✐♥ ♣♦✐♥t ♦❢ t❤✐s t❛❧❦✿ ❍♦✇ t♦ ❝♦♠♣✉t❡ ❄

✲✾✲

SLIDE 27

❘✐❣♦r♦✉s ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥s

f r❡♣❧❛❝❡❞ ✇✐t❤ ❛ r✐❣♦r♦✉s ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ✿ ✲ ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ T ♦❢ ❞❡❣r❡❡ n ✲ ✐♥t❡r✈❛❧ ∆ s✳ t✳ f(x) − T(x) ∈ ∆, ∀x ∈ [a, b] ▼❛✐♥ ♣♦✐♥t ♦❢ t❤✐s t❛❧❦✿ ❍♦✇ t♦ ❝♦♠♣✉t❡ ❄

✲✾✲

SLIDE 28

❘✐❣♦r♦✉s ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥s

f r❡♣❧❛❝❡❞ ✇✐t❤ ❛ r✐❣♦r♦✉s ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ✿ (T, ∆) ✲ ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ T ♦❢ ❞❡❣r❡❡ n ✲ ✐♥t❡r✈❛❧ ∆ s✳ t✳ f(x) − T(x) ∈ ∆, ∀x ∈ [a, b] ▼❛✐♥ ♣♦✐♥t ♦❢ t❤✐s t❛❧❦✿ ❍♦✇ t♦ ❝♦♠♣✉t❡ (T, ∆) ❄

✲✾✲

SLIDE 29

❚❛②❧♦r ▼♦❞❡❧s

✲ ❈♦♥s✐❞❡r ❚❛②❧♦r ❛♣♣r♦①✐♠❛t✐♦♥s ▲❡t ✱ t✐♠❡s ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥ ♦✈❡r ❛r♦✉♥❞ ✳ ✲ ❋♦r ♦❜t❛✐♥✐♥❣ ✿ ❋♦r ✏❜❛s✐❝ ❢✉♥❝t✐♦♥s✑ ✭ ✱ ❡t❝✳✮ ✉s❡ ▲❛❣r❛♥❣❡ ❢♦r♠✉❧❛ ❋♦r ✏❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥s✑✉s❡ ❛ t✇♦✲st❡♣ ♣r♦❝❡❞✉r❡✿ ✲ ❝♦♠♣✉t❡ ♠♦❞❡❧s ❢♦r ❛❧❧ ❜❛s✐❝ ❢✉♥❝t✐♦♥s❀ ✲ ❛♣♣❧② ❛❧❣❡❜r❛✐❝ r✉❧❡s ✇✐t❤ t❤❡s❡ ♠♦❞❡❧s✱ ✐♥st❡❛❞ ♦❢ ♦♣❡r❛t✐♦♥s ✇✐t❤ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❢✉♥❝t✐♦♥s✳

✲✶✵✲

SLIDE 30

❚❛②❧♦r ▼♦❞❡❧s

✲ ❈♦♥s✐❞❡r ❚❛②❧♦r ❛♣♣r♦①✐♠❛t✐♦♥s ▲❡t n ∈ N✱ n+1 t✐♠❡s ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥ f ♦✈❡r [a, b] ❛r♦✉♥❞ x0✳ f(x) =

n

f(i)(x0)(x − x0)i i!

T(x)

+ ∆n(x)

∆

✲ ❋♦r ♦❜t❛✐♥✐♥❣ ✿ ❋♦r ✏❜❛s✐❝ ❢✉♥❝t✐♦♥s✑ ✭ ✱ ❡t❝✳✮ ✉s❡ ▲❛❣r❛♥❣❡ ❢♦r♠✉❧❛ ❋♦r ✏❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥s✑✉s❡ ❛ t✇♦✲st❡♣ ♣r♦❝❡❞✉r❡✿ ✲ ❝♦♠♣✉t❡ ♠♦❞❡❧s ❢♦r ❛❧❧ ❜❛s✐❝ ❢✉♥❝t✐♦♥s❀ ✲ ❛♣♣❧② ❛❧❣❡❜r❛✐❝ r✉❧❡s ✇✐t❤ t❤❡s❡ ♠♦❞❡❧s✱ ✐♥st❡❛❞ ♦❢ ♦♣❡r❛t✐♦♥s ✇✐t❤ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❢✉♥❝t✐♦♥s✳

✲✶✵✲

SLIDE 31

❚❛②❧♦r ▼♦❞❡❧s

✲ ❈♦♥s✐❞❡r ❚❛②❧♦r ❛♣♣r♦①✐♠❛t✐♦♥s ▲❡t n ∈ N✱ n+1 t✐♠❡s ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥ f ♦✈❡r [a, b] ❛r♦✉♥❞ x0✳ f(x) =

n

f(i)(x0)(x − x0)i i!

T(x)

+ ∆n(x)

∆

✲ ❋♦r ♦❜t❛✐♥✐♥❣ ∆✿ ❋♦r ✏❜❛s✐❝ ❢✉♥❝t✐♦♥s✑ ✭sin, cos✱ ❡t❝✳✮ ✉s❡ ▲❛❣r❛♥❣❡ ❢♦r♠✉❧❛ ∀x ∈ [a, b], ∃ξ ∈ [a, b] s✳t✳ ∆n(x, ξ) = f(n+1)(ξ)(x − x0)n+1 (n + 1)! ❋♦r ✏❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥s✑✉s❡ ❛ t✇♦✲st❡♣ ♣r♦❝❡❞✉r❡✿ ✲ ❝♦♠♣✉t❡ ♠♦❞❡❧s ❢♦r ❛❧❧ ❜❛s✐❝ ❢✉♥❝t✐♦♥s❀ ✲ ❛♣♣❧② ❛❧❣❡❜r❛✐❝ r✉❧❡s ✇✐t❤ t❤❡s❡ ♠♦❞❡❧s✱ ✐♥st❡❛❞ ♦❢ ♦♣❡r❛t✐♦♥s ✇✐t❤ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❢✉♥❝t✐♦♥s✳

✲✶✵✲

SLIDE 32

❚❛②❧♦r ▼♦❞❡❧s

✲ ❈♦♥s✐❞❡r ❚❛②❧♦r ❛♣♣r♦①✐♠❛t✐♦♥s ▲❡t n ∈ N✱ n+1 t✐♠❡s ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥ f ♦✈❡r [a, b] ❛r♦✉♥❞ x0✳ f(x) =

n

f(i)(x0)(x − x0)i i!

T(x)

+ ∆n(x)

∆

✲ ❋♦r ♦❜t❛✐♥✐♥❣ ∆✿ ❋♦r ✏❜❛s✐❝ ❢✉♥❝t✐♦♥s✑ ✭sin, cos✱ ❡t❝✳✮ ✉s❡ ▲❛❣r❛♥❣❡ ❢♦r♠✉❧❛ ❋♦r ✏❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥s✑✉s❡ ❛ t✇♦✲st❡♣ ♣r♦❝❡❞✉r❡✿ ✲ ❝♦♠♣✉t❡ ♠♦❞❡❧s (T, ∆) ❢♦r ❛❧❧ ❜❛s✐❝ ❢✉♥❝t✐♦♥s❀ ✲ ❛♣♣❧② ❛❧❣❡❜r❛✐❝ r✉❧❡s ✇✐t❤ t❤❡s❡ ♠♦❞❡❧s✱ ✐♥st❡❛❞ ♦❢ ♦♣❡r❛t✐♦♥s ✇✐t❤ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❢✉♥❝t✐♦♥s✳

✲✶✵✲

SLIDE 33

❚❛②❧♦r ▼♦❞❡❧s ✲ ❚✇♦✲st❡♣ ♣r♦❝❡❞✉r❡

❊①❛♠♣❧❡✿ f❝♦♠♣(x) = exp(sin(x) + cos(x))

✲✶✶✲

SLIDE 34

❚❛②❧♦r ▼♦❞❡❧s ✲ ❚✇♦✲st❡♣ ♣r♦❝❡❞✉r❡

❊①❛♠♣❧❡✿ f❝♦♠♣(x) = exp(sin(x) + cos(x))

✲✶✶✲

SLIDE 35

❚❛②❧♦r ▼♦❞❡❧s ✲ ❚✇♦✲st❡♣ ♣r♦❝❡❞✉r❡

❊①❛♠♣❧❡✿ f❝♦♠♣(x) = exp(sin(x) + cos(x))

✲✶✶✲

SLIDE 36

❚❛②❧♦r ▼♦❞❡❧s ✲ ❚✇♦✲st❡♣ ♣r♦❝❡❞✉r❡

❊①❛♠♣❧❡✿ f❝♦♠♣(x) = exp(sin(x) + cos(x))

✲✶✶✲

SLIDE 37