DIFERENCIAIS E O CÁLCULO APROXIMADO

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✶ ❉✐❢❡r❡♥❝✐❛✐s ❡ ♦ ❈á❧❝✉❧♦ ❆♣r♦①✐♠❛❞♦ ✹

✶✳✶ ❆❝rés❝✐♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ ❉✐❢❡r❡♥❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✸ ■♥t❡r♣r❡t❛çã♦ ●❡♦♠étr✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✹ ❆♣❧✐❝❛çã♦ ❞❡ ❉✐❢❡r❡♥❝✐❛✐s ♥❛ ❋ís✐❝❛ ✲ P❡rí♦❞♦ ❞❡ ✉♠ Pê♥❞✉❧♦ ❙✐♠♣❧❡s ✶✸ ✶✳✺ ❆♣❧✐❝❛çã♦ ❞❡ ❉✐❢❡r❡♥❝✐❛✐s ✲ ❈á❧❝✉❧♦ ❞❡ ❊rr♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✺✳✶ EM ax ✲ ❊rr♦ ▼á①✐♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹

✶✳✺✳✷ ERel ✲ ❊rr♦ ❘❡❧❛t✐✈♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹

✶✳✺✳✸ EP c ✲ ❊rr♦ P❡r❝❡♥t✉❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹

✶✳✻ ▲✐st❛ ❞❡ ❊①❡r❝í❝✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼

(4)

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❈❛♣ít✉❧♦ ✶

❉✐❢❡r❡♥❝✐❛✐s ❡ ♦ ❈á❧❝✉❧♦ ❆♣r♦①✐♠❛❞♦

❉✐✈❡rs♦s ♣r♦❜❧❡♠❛s ♥❛s ár❡❛s ❞❡ ▼❛t❡♠át✐❝❛✱ ◗✉í♠✐❝❛ ♦✉ ❊♥❣❡♥❤❛r✐❛ ❡stã♦ ♣r❡♦❝✉♣❛❞♦s ❝♦♠ ❛s ✐♥t❡r✲r❡❧❛çõ❡s ❡♥tr❡ ❛s ♠✉❞❛♥ç❛s ♥❛s ♣r♦♣r✐❡❞❛❞❡s ❢ís✐❝❛s ♦✉ ❣❡♦♠étr✐❝❛s✱ ❞❡❝♦rr❡♥t❡s ❞❛s ✈❛r✐❛çõ❡s ❡♠ ✉♠ ♦✉ ♠❛✐s ♣❛râ♠❡tr♦s q✉❡ ❞❡✜♥❡♠ ♦ ❡st❛❞♦ ✐♥✐❝✐❛❧ ❞❡ ✉♠ s✐st❡♠❛✳ ❊st❛s ♠✉❞❛♥ç❛s ♣♦❞❡♠ s❡r ❣r❛♥❞❡s ♦✉ ♣❡q✉❡♥❛s✳ P❛rt✐❝✉❧❛r♠❡♥t❡✱ ♥❡st❡ ❧✐✈r♦✱ s❡rã♦ ❛❜♦r❞❛❞❛s ❛s ♣❡q✉❡♥❛s ✈❛r✐❛çõ❡s✳

❉✉r❛♥t❡ ♦ ❡st✉❞♦ ❞❛s ❞❡r✐✈❛❞❛s✱ dy

dx ❢♦✐ ✐♥t❡r♣r❡t❛❞♦ ❝♦♠♦ ✉♠❛ ú♥✐❝❛

❡♥t✐❞❛❞❡ r❡♣r❡s❡♥t❛♥❞♦ ❛ ❞❡r✐✈❛❞❛ ❞❡y❡♠ r❡❧❛çã♦ ❛ x✳ ◆❡st❡ ❧✐✈r♦ sã♦ ✐♥tr♦❞✉③✐❞♦s

s✐❣♥✐✜❝❛❞♦s ❞✐❢❡r❡♥t❡s ♣❛r❛ dy ❡ dx✱ ♦ q✉❡ ♣❡r♠✐t✐rá tr❛t❛r dy

dx ❝♦♠♦ ✉♠❛ r❛③ã♦✳

❚❛♠❜é♠ s❡rá ❞✐s❝✉t✐❞♦ ❝♦♠♦ ❛s ❞❡r✐✈❛❞❛s ♣♦❞❡♠ s❡r ✉s❛❞❛s ♣❛r❛ ❛♣r♦✲ ①✐♠❛r ❢✉♥çõ❡s ❝♦♠♣❧✐❝❛❞❛s ♣♦r ❢✉♥çõ❡s ❧✐♥❡❛r❡s ♠❛✐s s✐♠♣❧❡s✳ ❚❛✐s ❢✉♥çõ❡s sã♦ ❞❡♥♦♠✐♥❛❞❛s ❧✐♥❡❛r✐③❛çõ❡s ❡ s❡ ❜❛s❡✐❛♠ ❡♠ r❡t❛s t❛♥❣❡♥t❡s✳

✶✳✶ ❆❝rés❝✐♠♦s

❙❡❥❛y=f(x)✉♠❛ ❢✉♥çã♦✳ ❉❡✜♥❡✲s❡ ♦ ❛❝rés❝✐♠♦ ❞❡x✱ ❞❡♥♦t❛❞♦ ♣♦r∆x✱

❝♦♠♦✿

∆x=x2−x1,

♦♥❞❡ x1, x2 ∈D(f)✳

❆ ✈❛r✐❛çã♦ ❞❡ x♦r✐❣✐♥❛ ✉♠❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ✈❛r✐❛çã♦ ❞❡y✱ ❞❡♥♦t❛❞❛ ♣♦r

∆y✱ ❞❛❞❛ ♣♦r✿

∆y =f(x2)−f(x1),

(5)

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✶✳✷✳ ❉■❋❊❘❊◆❈■❆▲

♦✉

∆y=f(x1 + ∆x)−f(x1).

◆❛ ❋✐❣✉r❛ ✶✳✶✱ ♣♦❞❡✲s❡ ♦❜s❡r✈❛r ❣r❛✜❝❛♠❡♥t❡ ♦ s✐❣♥✐✜❝❛❞♦ ❞❡∆x ❡∆y✳

❋✐❣✉r❛ ✶✳✶✿ ❘❡♣r❡s❡♥t❛çã♦ ❣rá✜❝❛ ❞♦s ✐♥❝r❡♠❡♥t♦s∆x❡ ∆y✳

✶✳✷ ❉✐❢❡r❡♥❝✐❛❧

❉❡✜♥✐çã♦ ✶✳✷✳✶✳ ❉✐❢❡r❡♥❝✐❛❧ ❞❡ ✉♠❛ ❢✉♥çã♦ é ♦ ❛❝rés❝✐♠♦ s♦❢r✐❞♦ ♣❡❧❛ ♦r❞❡♥❛❞❛ ❞❛

r❡t❛ t❛♥❣❡♥t❡ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛ ✉♠ ❛❝rés❝✐♠♦ ∆xs♦❢r✐❞♦ ♣♦r x✳

❉❡✜♥✐çã♦ ✶✳✷✳✷✳ ❙❡❥❛♠ y = f(x) ✉♠❛ ❢✉♥çã♦ ❞❡r✐✈á✈❡❧ ❡ ∆x ✉♠ ❛❝rés❝✐♠♦ ❞❡ x✳

❉❡✜♥❡✲s❡✿

❛✮ ❛ ❞✐❢❡r❡♥❝✐❛❧ ❞❛ ✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡ x✱ ❞❡♥♦t❛❞❛ ♣♦rdx✱ ❝♦♠♦ ∆x=dx✳

❜✮ ❛ ❞✐❢❡r❡♥❝✐❛❧ ❞❛ ✈❛r✐á✈❡❧ ❞❡♣❡♥❞❡♥t❡ y✱ ❞❡♥♦t❛❞❛ ♣♦r dy✱ ❝♦♠♦ dy=f′₍_x₎_·_∆_x✳

❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❉❡✜♥✐çã♦ ✶✳✷✳✷✱ é ♣♦ssí✈❡❧ ❡s❝r❡✈❡r dy = f′₍_x₎_·_dx ♦✉ dy

dx =f ′₍_x₎✳

❆ ♥♦t❛çã♦ dy

dx✱ ❥á ✉s❛❞❛ ♣❛r❛ f

′₍_x₎✱ ♣♦❞❡ ❛❣♦r❛ s❡r ❝♦♥s✐❞❡r❛❞❛ ✉♠ q✉♦✲

❝✐❡♥t❡ ❡♥tr❡ ❞✉❛s ❞✐❢❡r❡♥❝✐❛✐s✳

❖❜s❡r✈❛çã♦ ✶✳✷✳✶✳ ◗✉❛♥❞♦ ◆❡✇t♦♥ ❡ ▲❡✐❜♥✐③ ♣✉❜❧✐❝❛r❛♠ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡ s❡✉s

❡st✉❞♦s r❡❧❛❝✐♦♥❛❞♦s ❛♦ ❈á❧❝✉❧♦✱ ❝❛❞❛ ✉♠ ✉s♦✉ ✉♠❛ ♥♦t❛çã♦ ♣❛r❛ ❛ ❞❡r✐✈❛❞❛✳ ◆❡st❡

(6)

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✶✳✸✳ ■◆❚❊❘P❘❊❚❆➬➹❖ ●❊❖▼➱❚❘■❈❆

❝✉rs♦✱ ❛❞♦t❛♠✲s❡ ❛s ♥♦t❛çõ❡s ❧✐♥❤❛ ✭❞❡ ▲❛❣r❛♥❣❡✮✱ y′✱ ❡ ❞❡ ♦♣❡r❛❞♦r ❞✐❢❡r❡♥❝✐❛❧ ✭❞❡

▲❡✐❜♥✐③✮✱ dy

dx✳ ❆ ♥♦t❛çã♦ ❞♦ ♣♦♥t♦✱ ❞❡s❡♥✈♦❧✈✐❞❛ ♣♦r ◆❡✇t♦♥✱ ♥ã♦ s❡rá ❡①♣❧♦r❛❞❛

♥❡st❡ ♠❛t❡r✐❛❧✳

❊①❡♠♣❧♦ ✶✳✷✳✶✳ ❙❡ f(x) = 3x2

−2x+ 1✱ ❞❡t❡r♠✐♥❡ dy✳

❙♦❧✉çã♦✿

❚❡♠✲s❡ q✉❡ dy =f′₍_x₎_dx✳ P❛r❛ _f₍_x_{) = 3}_x2

−2x+ 1✱ ❛♣❧✐❝❛♥❞♦ ❛s r❡❣r❛s

❞❡ ❞❡r✐✈❛❞❛ ❞❛ s✉❜tr❛çã♦✱ ❞❡r✐✈❛❞❛ ❞❛ ♣♦tê♥❝✐❛ ❞❡x❡ ❞❛ ❞❡r✐✈❛❞❛ ❞❡ ✉♠❛ ❝♦♥st❛♥t❡✱ f′₍_x_{) = 6}_x₋₂✳ ▲♦❣♦✱_dy _{= (6}_x₋₂₎_dx✳

❊①❡♠♣❧♦ ✶✳✷✳✷✳ ❈❛❧❝✉❧❛r ❛ ❞✐❢❡r❡♥❝✐❛❧ ❞❛s ❢✉♥çõ❡s✿

❛✮ y=√1 +x2

❜✮ y= 1 3tg

3

(x) + tg(x)✳

❙♦❧✉çã♦✿

❛✮ ❈♦♠♦ dy =f′₍_x₎_dx ❡_y₌_f₍_x_{) =}√_{1 +}_x2✱ t❡♠✲s❡✱ ❛♣❧✐❝❛♥❞♦ ❛ r❡❣r❛ ❞❛ ❝❛❞❡✐❛✿

f′₍_x_{) =} _√ x

1 +x2✳ ▲♦❣♦✱dy =f

′₍_x₎_dx₌ _√ x

1 +x2dx✳

❜✮ ❆♣❧✐❝❛♥❞♦ ❛s r❡❣r❛s ❞❡ ❞❡r✐✈❛❞❛ ❞❛ ♣♦tê♥❝✐❛✱ ❞❛ ❝❛❞❡✐❛ ❡ ❞❛ ❞❡r✐✈❛❞❛ ❞❛ t❛♥❣❡♥t❡✱ t❡♠✲s❡✿

f′₍_x_{) =} 1

3 ·3tg

2

(x)_·sec2

(x) + sec2

(x)

= sec2

(x)[tg2

(x) + 1]

f′₍_x_{) = sec}4

(x).

P♦rt❛♥t♦✱ dy= sec4

(x)dx✳

✶✳✸ ■♥t❡r♣r❡t❛çã♦ ●❡♦♠étr✐❝❛

❙❡❥❛ y = f(x) ✉♠❛ ❢✉♥çã♦ ❞❡r✐✈á✈❡❧✱ ❝✉❥♦ ❣rá✜❝♦ é ✐❧✉str❛❞♦ ♥❛ ❋✐❣✉r❛

✶✳✸✳

❈♦♥s✐❞❡r❡ ♦s ♣♦♥t♦sP(x1, f(x1))✱M(x2, f(x1))❡Q(x2, f(x2))✳ ❖ ❛❝rés✲

❝✐♠♦∆xq✉❡ ❞❡✜♥❡ ❛ ❞✐❢❡r❡♥❝✐❛❧dx❡stá ❣❡♦♠❡tr✐❝❛♠❡♥t❡ r❡♣r❡s❡♥t❛❞♦ ♣❡❧❛ ♠❡❞✐❞❛

❞♦ s❡❣♠❡♥t♦ P M✳

(7)

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✶✳✸✳ ■◆❚❊❘P❘❊❚❆➬➹❖ ●❊❖▼➱❚❘■❈❆

❋✐❣✉r❛ ✶✳✷✿ ■♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❛ ❞✐❢❡r❡♥❝✐❛❧✳

❖ ❛❝rés❝✐♠♦ ∆y ❡stá r❡♣r❡s❡♥t❛❞♦ ♣❡❧❛ ♠❡❞✐❞❛ ❞♦ s❡❣♠❡♥t♦ M Q✳

❙❡❥❛ t ❛ r❡t❛ t❛♥❣❡♥t❡ à ❝✉r✈❛ y =f(x) ♥♦ ♣♦♥t♦ P✳ ❊st❛ r❡t❛ ❝♦rt❛ ❛

r❡t❛ ✈❡rt✐❝❛❧x=x2♥♦ ♣♦♥t♦R✱ ❢♦r♠❛♥❞♦ ♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦P M R✳ ❆ ✐♥❝❧✐♥❛çã♦

✭❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r✮ ❞❡st❛ r❡t❛ é ❞❛❞❛ ♣♦r f′₍_x₁_{) = tg(}_α₎✳

❖❜s❡r✈❛♥❞♦ ♦ tr✐â♥❣✉❧♦ P M R✱ é ♣♦ssí✈❡❧ ❡s❝r❡✈❡r✿

f′₍_x

1) = tg(α) =

M R P M.

♦♥❞❡M R❡P M sã♦ r❡s♣❡❝t✐✈❛♠❡♥t❡ ❛s ♠❡❞✐❞❛s ❞♦s s❡❣♠❡♥t♦sM R❡P M✳ ❯s❛♥❞♦

♦ ❢❛t♦ ❞❡ q✉❡ f′₍_x_{) =} dy

dx ❝♦♥❝❧✉✐✲s❡ q✉❡ dy=M R✱ ❥á q✉❡ dx=P M✳

P❡❧❛ s✉❜st✐t✉✐çã♦ ❞❡ ∆y ♣♦r dy ❝♦♠❡t❡✲s❡ ✉♠ ❡rr♦ q✉❡ é ❝❛❧❝✉❧❛❞♦ ♣❡❧♦

♠ó❞✉❧♦ ❞❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦ ✈❛❧♦r ❡①❛t♦ ❞❛ ✈❛r✐❛çã♦ ❞❡ y✭∆y✮ ❡ ♦ ✈❛❧♦r ❛♣r♦①✐♠❛❞♦

❞❛ ✈❛r✐❛çã♦ ❞❡ y ✭dy✮✱ ✐st♦ é✱

e=_|∆y₋dy_|.

❖ q✉❡ ❛❝♦♥t❡❝❡ ❝♦♠ ∆y₋dy q✉❛♥❞♦ ∆x t♦r♥❛✲s❡ ♠✉✐t♦ ♣❡q✉❡♥♦❄

❖❜s❡r✈❛✲s❡ q✉❡✱ q✉❛♥❞♦ ∆x t♦r♥❛✲s❡ ♠✉✐t♦ ♣❡q✉❡♥♦✱ ♦ ♠❡s♠♦ ♦❝♦rr❡

❝♦♠ ❛ ❞✐❢❡r❡♥ç❛ ∆y₋dy✳

❊♠ ❡①❡♠♣❧♦s ♣rát✐❝♦s✱ ❝♦♥s✐❞❡r❛✲s❡ ∆y _≈ dy ✭❧ê✲s❡✿ ∆y ❛♣r♦①✐♠❛❞❛✲

♠❡♥t❡ ✐❣✉❛❧ ❛ dy✮✱ ❞❡s❞❡ q✉❡ ♦ ∆x ❝♦♥s✐❞❡r❛❞♦ s❡❥❛ ✉♠ ✈❛❧♦r ♣❡q✉❡♥♦✳

❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ♣❛r❛ ✈❛❧♦r❡s ♣❡q✉❡♥♦s ❞❡ ∆x✱

∆y_≈dy.

(8)

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✶✳✸✳ ■◆❚❊❘P❘❊❚❆➬➹❖ ●❊❖▼➱❚❘■❈❆

❙❛❜❡♥❞♦✲s❡ q✉❡ dy =f′₍_x_)∆_x✱ t❡♠✲s❡✿

∆y_≈f′₍_x_)∆_x.

P❡❧❛ ❉❡✜♥✐çã♦ ✶✳✷✳✷✿

∆y =f(x+ ∆x)₋f(x),

❧♦❣♦✱

f(x+ ∆x)₋f(x)_≈f′₍_x_)∆_x,

♦✉ ❛✐♥❞❛✱

f(x+ ∆x)_≈f′₍_x_)∆_x₊_f₍_x₎_. ✭✶✳✸✳✶✮

❆ ❡st❡ ♣r♦❝❡ss♦ ❝❤❛♠❛✲s❡ ❞❡ ❧✐♥❡❛r✐③❛çã♦ ❞❡ ❢✱ ❡♠ t♦r♥♦ ❞❡ x✳

❊①❡♠♣❧♦ ✶✳✸✳✶✳ ❈♦♠♣❛r❡ ♦s ✈❛❧♦r❡s ❞❡∆y ❡dy s❡y=x3

+x2

−2x+ 1 ❡x ✈❛r✐❛r

✭❛✮ ❞❡ 2 ♣❛r❛ 2,05 ❡ ✭❜✮ ❞❡ 2 ♣❛r❛ 2,01✳

❙♦❧✉çã♦✿

✭❛✮ ❚❡♠✲s❡ q✉❡ ∆x= 0,05✱ ❛ss✐♠ ♣♦r ❞❡✜♥✐çã♦✿

∆y =f(x+ ∆x)₋f(x)

=f(2 + 0,05)₋f(2)

= [(2,05)3

+ (2,05)2

−2(2,05) + 1]₋[23

+ 22

−2(2) + 1] ∆y = 0,717625.

P❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ dy✱ q✉❛♥❞♦ x= 2 ❡∆x= 0,05 t❡♠✲s❡✿

dy =f′₍_x₎_dx_{= (3}_x2

+ 2x₋2)dx

= [3(2)2

+ 2(2)₋2]0,05

dy = 0,7.

❖❜s❡r✈❡ q✉❡ ♦ ❡rr♦ ❝♦♠❡t✐❞♦ ❛♦ ✉s❛r ❞✐❢❡r❡♥❝✐❛✐s e = _|∆y₋dy_| é ❞❡

✵✱✵✶✼✻✷✺✳

✭❜✮ ❚❡♠✲s❡ q✉❡ ∆x= 0,01✱ ❛ss✐♠ ♣♦r ❞❡✜♥✐çã♦✿

∆y =f(x+ ∆x)₋f(x)

=f(2 + 0,01)₋f(2)

= [(2,01)3

+ (2,01)2

−2(2,01) + 1]₋[23

+ 22

−2(2) + 1] ∆y = 0,140701.

(9)

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P❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ dy✱ q✉❛♥❞♦ x= 2 ❡∆x= 0,01 t❡♠✲s❡✿

dy =f′₍_x₎_dx_{= (3}_x2

+ 2x₋2)dx

= [3(2)2

+ 2(2)₋2]0,01

dy = 0,14.

❖❜s❡r✈❛çã♦ ✶✳✸✳✶✳ ❖❜s❡r✈❡ ❛ ❢✉♥çã♦ ❞♦ ❊①❡♠♣❧♦ ✶✳✸✳✶✳ ❈♦♠♣❛r❛♥❞♦ ∆y ❝♦♠

dy ♣❡r❝❡❜❡✲s❡ q✉❡ ❛ ❛♣r♦①✐♠❛çã♦ ♣♦r ❞✐❢❡r❡♥❝✐❛✐s dy t♦r♥❛✲s❡ ♠❡❧❤♦r à ♠❡❞✐❞❛ q✉❡

∆x ✜❝❛ ♠❡♥♦r✳

❉❡ ❢❛t♦✱ ♣❛r❛ ♦ ✐t❡♠ ✭❛✮✱ ♦ ❡rr♦ ❝♦♠❡t✐❞♦ ❛♦ ❡♠♣r❡❣❛r ❞✐❢❡r❡♥❝✐❛✐s é

e = _|∆y ₋dy_| = _|0,717625₋0,7_| = 0,017625✳ ❊♥q✉❛♥t♦ q✉❡ ♣❛r❛ ♦ ✐t❡♠ ✭❜✮✱

♦♥❞❡ ∆x é ♠❡♥♦r✱ ♦ ❡rr♦ ❝♦♠❡t✐❞♦ ❛♦ ❡♠♣r❡❣❛r ❞✐❢❡r❡♥❝✐❛✐s é e = _|∆y ₋dy_| =

|0,140701₋0,14_|= 0,000701✳

❖❜s❡r✈❛çã♦ ✶✳✸✳✷✳ ❊♠ ❛❧❣✉♥s ❝❛s♦s é ♠❛✐s ❢á❝✐❧ ❝❛❧❝✉❧❛r ♦ dy✱ ♣♦✐s ♣❛r❛ ❢✉♥çõ❡s

♠❛✐s ❝♦♠♣❧✐❝❛❞❛s ♣♦❞❡ s❡r ✐♠♣♦ssí✈❡❧ ❝❛❧❝✉❧❛r ❡①❛t❛♠❡♥t❡ ♦ ✈❛❧♦r ❞❡ ∆y✳ ◆❡ss❡s

❝❛s♦s✱ ❛ ❛♣r♦①✐♠❛çã♦ ♣♦r ❞✐❢❡r❡♥❝✐❛✐s t♦r♥❛✲s❡ ♠✉✐t♦ út✐❧✳

❊①❡♠♣❧♦ ✶✳✸✳✷✳ ❙❡ y = 2x2

−6x+ 5✱ ❝❛❧❝✉❧❡ ♦ ❛❝rés❝✐♠♦ ∆y ♣❛r❛ x = 3 ❡ ∆x= 0,001✳

❙♦❧✉çã♦✿

P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ∆y✱ ❡s❝r❡✈❡✲s❡✿

∆y = f(x1 + ∆x)−f(x1)

= f(3 + 0,001)₋f(3)

= [2(3,001)2

−6(3,001) + 5]₋[2(32

)₋6(3) + 5]

= 5,006002₋5

∆y = 0,006002.

P♦rt❛♥t♦ ♦ ❛❝rés❝✐♠♦ ∆y= 0,006002✳

❊①❡♠♣❧♦ ✶✳✸✳✸✳ ❙❡ y = 6x2

−4✱ ❝❛❧❝✉❧❡ ♦ ❛❝rés❝✐♠♦ ∆y ❡ dy ♣❛r❛ x = 2 ❡ ∆x= 0,001✳

❙♦❧✉çã♦✿

(10)

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❯t✐❧✐③❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ ∆y✱ t❡♠✲s❡

∆y = f(x1+ ∆x)−f(x1)

= f(2 + 0,001)₋f(2)

= [6_·(2,001)2

−4]₋[6(22

)₋4]

= 20,024006₋20

∆y = 0,024006.

P♦rt❛♥t♦✱ ∆y= 0,024006✳

P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ dy ❡ s❛❜❡♥❞♦ q✉❡ f(x) = 6x2

−4✱ t❡♠✲s❡✿

dy = f′₍_x₎_dx

= 12x∆x

= 12(2)(0,001)

dy = 0,024.

▲♦❣♦✱ dy = 0,024✳

❖❜s❡r✈❛✲s❡ q✉❡ ❛ ❞✐❢❡r❡♥ç❛|∆y₋dy_|= 0,000006s❡r✐❛ ♠❡♥♦r✱ ❝❛s♦ ❢♦ss❡

✉s❛❞♦ ✉♠ ✈❛❧♦r ♠❡♥♦r q✉❡ ✵✱✵✵✶ ♣❛r❛ ∆x✳

❊①❡♠♣❧♦ ✶✳✸✳✹✳ ❈❛❧❝✉❧❡ ✉♠ ✈❛❧♦r ❛♣r♦①✐♠❛❞♦ ♣❛r❛ p3

65,5✉s❛♥❞♦ ❞✐❢❡r❡♥❝✐❛✐s✳

❙♦❧✉çã♦✿

❙❡❥❛y=f(x)❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ♣♦rf(x) = √3

x✳ ❆♣❧✐❝❛♥❞♦ ❛ ❧✐♥❡❛r✐③❛çã♦

❞❛ ❢✉♥çã♦ f✱ r❡♣r❡s❡♥t❛❞❛ ♣❡❧❛ ❡q✉❛çã♦ ✭✶✳✸✳✶✮✱ ❡s❝r❡✈❡✲s❡✿

y+dy=√3

x+ ∆x

❡

dy= 1

3x23

dx.

❚❡♠✲s❡ x = 64 ❡ ∆x = 1,5✱ ♣♦✐s ✻✹ é ♦ ❝✉❜♦ ♣❡r❢❡✐t♦ ♠❛✐s ♣ró①✐♠♦ ❞❡

✻✺✱✺✳

P♦rt❛♥t♦✱

x+ ∆x= 65,5

dx= ∆x= 1,5

y=√3

64 = 4

(11)

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✶✳✸✳ ■◆❚❊❘P❘❊❚❆➬➹❖ ●❊❖▼➱❚❘■❈❆

❡

dy= 1

3(64)23 ·

1,5 = 1,5

3_·16 = 0,03125.

▲♦❣♦✱

3

p

65,5 = p3

64 + 1,5_≈y+dy.

❋✐♥❛❧♠❡♥t❡✱

3

p

65,5_≈y+dy= 4 + 0,03125_≈4,03125.

❖❜s❡r✈❡ q✉❡✱ ❛♦ ✉t✐❧✐③❛r ✉♠❛ ❝❛❧❝✉❧❛❞♦r❛✱ ♦❜té♠✲s❡ q✉❡

3

p

65,5_≈4,0310089894.

P♦rt❛♥t♦ ♦ ❡rr♦ ❝♦♠❡t✐❞♦ ❛♦ ✉t✐❧✐③❛r ❞✐❢❡r❡♥❝✐❛✐s é ❞❡ ✵✱✵✵✵✷✹✶✵✶✵✻✱ ♦✉ s❡❥❛✱ ❞❡ ✷✱✹✶✵✶✵✻×10−4_✳

❊①❡♠♣❧♦ ✶✳✸✳✺✳ ❖❜t❡♥❤❛ ✉♠ ✈❛❧♦r ❛♣r♦①✐♠❛❞♦ ♣❛r❛ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ✜♥❛ ❝♦r♦❛

❝✐❧í♥❞r✐❝❛ ❞❡ ❛❧t✉r❛ ✶✷ ♠✱ r❛✐♦ ✐♥t❡r✐♦r ✼ ♠ ❡ ❡s♣❡ss✉r❛ ✵✱✵✺ ♠✳ ❉❡t❡r♠✐♥❡ ♦ ❡rr♦ ❞❡❝♦rr❡♥t❡ ❞❛ ✉t✐❧✐③❛çã♦ ❞❡ ❞✐❢❡r❡♥❝✐❛✐s✳

❙♦❧✉çã♦✿

❋✐❣✉r❛ ✶✳✸✿ ❈♦r♦❛ ❝✐❧í♥❞r✐❝❛

❆ ❋✐❣✉r❛ ✶✳✸ r❡♣r❡s❡♥t❛ ♦ só❧✐❞♦ ❞❡ ❛❧t✉r❛ h✱ r❛✐♦ ✐♥t❡r✐♦r r ❡ ❡s♣❡ss✉r❛

∆r✳ ❖ ✈♦❧✉♠❡ ❞♦ ❝✐❧✐♥❞r♦ ✐♥t❡r✐♦r é ❞❛❞♦ ♣♦r✿

V =πr2_·h=π(7)2

·12 = 588π♠3.

❍❛✈❡♥❞♦ ✉♠ ❛❝rés❝✐♠♦ ∆r✱ ♦ ✈♦❧✉♠❡ ❞❛ ❝♦r♦❛ s❡rá ✐❣✉❛❧ à ✈❛r✐❛çã♦∆V

❡♠ V✳ ❯s❛♥❞♦ ❞✐❢❡r❡♥❝✐❛✐s✱ t❡♠✲s❡✿

∆V _≈dV

(12)

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✶✳✸✳ ■◆❚❊❘P❘❊❚❆➬➹❖ ●❊❖▼➱❚❘■❈❆

❡

dV = 2πr_·h_·∆r

dV = 2π(7)(12)(0,05) = 8,4π♠3

.

❖ ✈♦❧✉♠❡ ❡①❛t♦ ❞❛ ❝♦r♦❛ ❝✐❧í♥❞r✐❝❛ é✿

∆V = π(r+ ∆r)2

·h₋πr2

·h

= π(7,05)2

·12₋π_·(72

)_·12

= 596,43π₋588π

∆V = 8,43π♠3

.

P♦rt❛♥t♦✱ ♦ ❡rr♦ ❝♦♠❡t✐❞♦ ♥❛ ❛♣r♦①✐♠❛çã♦ ♣♦r ❞✐❢❡r❡♥❝✐❛✐s é e=_|∆V ₋ dV_|= 0,03π♠3_✳

❊①❡♠♣❧♦ ✶✳✸✳✻✳ ❯♠❛ ♣❧❛❝❛ q✉❛❞r❛❞❛ ❞❡ ❧❛❞♦x✱ ❞❡ ❡s♣❡ss✉r❛ ❞❡s♣r❡③í✈❡❧✱ é ❛q✉❡✲

❝✐❞❛✳ ❖ ♣r♦❝❡ss♦ ❞❡ ❛q✉❡❝✐♠❡♥t♦ ♣r♦✈♦❝❛ ✉♠❛ ❞✐❧❛t❛çã♦ ♥❛ ♣❧❛❝❛✳ ❆❞♠✐t✐♥❞♦ ✉♠❛ ❞✐❧❛t❛çã♦ ✉♥✐❢♦r♠❡✱ ♠❛♥t❡♥❞♦ ❛ ❢♦r♠❛ q✉❛❞r❛❞❛✱ ❝❛❧❝✉❧❡ ❛ ✈❛r✐❛çã♦ ❞❛ ár❡❛ ❡♠ ❢✉♥çã♦ ❞❛ ✈❛r✐❛çã♦ ❞❡ s❡✉ ❧❛❞♦✱ ∆x✱ ♦❝♦rr✐❞❛ ❞❡✈✐❞♦ ❛♦ ♣r♦❝❡ss♦ ❞❡ ❛q✉❡❝✐♠❡♥t♦✳

❊st✐♠❡ ❛ ✈❛r✐❛çã♦ ❞❛ ár❡❛ ✉t✐❧✐③❛♥❞♦ ❞✐❢❡r❡♥❝✐❛✐s✳ ■♥t❡r♣r❡t❡ ♦ r❡s✉❧t❛❞♦✳

❙♦❧✉çã♦✿ ❆ ♣❧❛❝❛ q✉❛❞r❛❞❛ ✐♥✐❝✐❛❧♠❡♥t❡ t❡♠ ❧❛❞♦ ❞❡ ♠❡❞✐❞❛ x✳ P♦rt❛♥t♦✱ s✉❛ ár❡❛

✈❛❧❡ A(x) =x2_✳

❆♣ós ♦ ❛q✉❡❝✐♠❡♥t♦✱ ❛ ♠❡❞✐❞❛ ❞♦ s❡✉ ❧❛❞♦ ❛✉♠❡♥t❛ ❡♠ ∆x ✉♥✐❞❛❞❡s✳

P♦rt❛♥t♦✱ ❡ss❛ ♠❡❞✐❞❛ ✈❛r✐❛ ❞❡x♣❛r❛x+ ∆x❡ ❛ ár❡❛ ❞❛ ♣❧❛❝❛✱ ❛♣ós ♦ ❛q✉❡❝✐♠❡♥t♦

é✱

A(x+ ∆x) = (x+ ∆x)2

=x2+ 2x∆x+ (∆x)2

.

❆ ✈❛r✐❛çã♦ ❞❛ ár❡❛ ❞♦ q✉❛❞r❛❞♦ q✉❛♥❞♦ ❛ ♠❡❞✐❞❛ ✈❛r✐❛ ❞❡x♣❛r❛x+ ∆x

é✱

∆A=A(x+ ∆x)₋A(x) =x2

+ 2x∆x+ (∆x)2

−x2

= 2x∆x+ (∆x)2

.

P♦r ♦✉tr♦ ❧❛❞♦✱ ❡♠ t❡r♠♦s ❞❡ ❞✐❢❡r❡♥❝✐❛✐s✱ ❛ ✈❛r✐❛çã♦ ❞❛ ár❡❛ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❝♦♠♦✱

dA =A′₍_x₎_dx_{= 2}_xdx_{= 2}_x_∆_x.

❖❜s❡r✈❡ q✉❡ ♦ ❡rr♦ ❝♦♠❡t✐❞♦ ❛♦ ✉t✐❧✐③❛r ❞✐❢❡r❡♥❝✐❛✐s é e=_|∆A₋dA_|= (∆x)2_{✱ ♣♦rt❛♥t♦ q✉❛♥❞♦ ♦ ❛❝rés❝✐♠♦}

∆x é ♠✉✐t♦ ♣❡q✉❡♥♦✱ ♦ ✈❛❧♦r ❞♦ ❞✐❢❡r❡♥❝✐❛❧ dA

♣♦❞❡ s❡r ❞✐t❛ ❝♦♠♦ ✉♠❛ ❜♦❛ ❛♣r♦①✐♠❛çã♦ ♣❛r❛ ❛ ✈❛r✐❛çã♦ ❞❛ ár❡❛✳

(13)

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✶✳✹✳ ❆P▲■❈❆➬➹❖ ❉❊ ❉■❋❊❘❊◆❈■❆■❙ ◆❆ ❋❮❙■❈❆ ✲ P❊❘❮❖❉❖ ❉❊ ❯▼ P✃◆❉❯▲❖ ❙■▼P▲❊❙

✶✳✹ ❆♣❧✐❝❛çã♦ ❞❡ ❉✐❢❡r❡♥❝✐❛✐s ♥❛ ❋ís✐❝❛ ✲ P❡rí♦❞♦ ❞❡

✉♠ Pê♥❞✉❧♦ ❙✐♠♣❧❡s

❯♠❛ ✐♠♣♦rt❛♥t❡ ❛♣❧✐❝❛çã♦ ❞❡ ❞✐❢❡r❡♥❝✐❛✐s ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ♥❛ ❋ís✐❝❛✳ ❯♠ ❢ís✐❝♦ ❛♦ ❛♥❛❧✐s❛r ❛s ❝♦♥s❡q✉ê♥❝✐❛s ❞❡ ✉♠❛ ❡q✉❛çã♦✱ ♠✉✐t❛s ✈❡③❡s ♥❡❝❡ss✐t❛ r❡❛✲ ❧✐③❛r s✐♠♣❧✐✜❝❛çõ❡s ♥❛s ❡q✉❛çõ❡s ♠❛t❡♠át✐❝❛s q✉❡ r❡♣r❡s❡♥t❛♠ ♦s ❢❡♥ô♠❡♥♦s ❢ís✐❝♦s✳ ❊♠ ❛❧❣✉♥s ❝❛s♦s✱ ❛♣❧✐❝❛✲s❡ ✉♠❛ ❛♣r♦①✐♠❛çã♦ ❝❤❛♠❛❞❛ ❞❡ ❛♣r♦①✐♠❛çã♦ ❧✐♥❡❛r✳

❯s✉❛❧♠❡♥t❡ ♦s ♣r♦❜❧❡♠❛s ❡♥✈♦❧✈❡♥❞♦ ✉♠ ♣ê♥❞✉❧♦ s✐♠♣❧❡s sã♦ ❛♥❛❧✐s❛❞♦s ❝♦♠♦ s❡♥❞♦ ✉♠ ♦s❝✐❧❛❞♦r ❤❛r♠ô♥✐❝♦ s✐♠♣❧❡s✱ ♦s q✉❛✐s ❣❡r❛♠ ✉♠ ♠♦❞❡❧♦ ♠❛t❡♠át✐❝♦ ♣❛r❛ s✐st❡♠❛s r❡❧❛❝✐♦♥❛❞♦s ❛♦ ♠♦✈✐♠❡♥t♦ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❡①♣♦st❛ ❛ ✉♠❛ ❢♦rç❛ ❞❡ ❛tr❛çã♦ ❝♦♠ ♠❛❣♥✐t✉❞❡ ♣r♦♣♦r❝✐♦♥❛❧ ❛ ❞✐stâ♥❝✐❛ ❞❡st❛ ♣❛rtí❝✉❧❛ ❡♠ r❡❧❛çã♦ à ♦r✐❣❡♠ ❞♦ s✐st❡♠❛✳ ❉❡ss❛ ❢♦r♠❛ ♦ ♠♦✈✐♠❡♥t♦ ❞❡ ✉♠ ♣ê♥❞✉❧♦ s✐♠♣❧❡s ♣♦❞❡ s❡r ❞❡s❝r✐t♦ ♣♦r ✉♠❛ ❡q✉❛çã♦✱ ❝❤❛♠❛❞❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ♦r❞✐♥ár✐❛✱

d2

θ dt2 +

g

lsen(θ) = 0, ✭✶✳✹✳✶✮

♦♥❞❡gé ❛ ❛❝❡❧❡r❛çã♦ ❞❛ ❣r❛✈✐❞❛❞❡ ❡l♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ♣ê♥❞✉❧♦✳ P❛r❛ ✈❛❧♦r❡s ♠✉✐t♦

♣❡q✉❡♥♦s ❞♦ â♥❣✉❧♦ θ✱ ✉t✐❧✐③❛✲s❡ ❛ ❛♣r♦①✐♠❛çã♦ ❧✐♥❡❛r sen(θ) _≈ θ ❡ r❡❡s❝r❡✈❡✲s❡ ❛

❡q✉❛çã♦ ✭✶✳✹✳✶✮ ❝♦♠♦✱

d2

θ dt2 +

g lθ= 0.

❊st❛ ❡q✉❛çã♦ ✐♥❞✐❝❛ q✉❡✱ ❞❡♥tr♦ ❞❛ ❛♣r♦①✐♠❛çã♦ ❞❡ â♥❣✉❧♦s ♣❡q✉❡♥♦s✱ ♦ ♠♦✈✐♠❡♥t♦ ❞♦ ♣ê♥❞✉❧♦ s✐♠♣❧❡s é ❤❛r♠ô♥✐❝♦ s✐♠♣❧❡s✱ ❡ ♦ ♣❡rí♦❞♦ ❞❡ ♦s❝✐❧❛çã♦ ❞♦ ♣ê♥❞✉❧♦ é ❝❛❧❝✉❧❛❞♦ ❝♦♠♦✱

T = 2π

s

l g.

❊ss❡ ❢❛t♦ ♣♦❞❡ ♦❜s❡r✈❛❞♦ ♥❛ ❋✐❣✉r❛ ✶✳✹ ♦♥❞❡ s❡ ❛♣r❡s❡♥t❛♠ ♦s ❣rá✜❝♦s ❞❡ x ❡ sen(x) ♥❛s ♣r♦①✐♠✐❞❛❞❡s ❞❛ ♦r✐❣❡♠✳ ◆♦t❡ q✉❡ ❛ ❢✉♥çã♦ sen(x) ❡stá ♠✉✐t♦

♣ró①✐♠❛ ❞❛ ❢✉♥çã♦x q✉❛♥❞♦xé s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦✱ ♦✉ s❡❥❛✱ ♣ró①✐♠♦ ❞❡ ③❡r♦✳

✶✳✺ ❆♣❧✐❝❛çã♦ ❞❡ ❉✐❢❡r❡♥❝✐❛✐s ✲ ❈á❧❝✉❧♦ ❞❡ ❊rr♦s

P♦❞❡✲s❡ ❡st✐♠❛r ♦ ✈❛❧♦r ❞♦ ❡rr♦ ♣r♦♣❛❣❛❞♦ ✲ ❡rr♦ q✉❡ s❡ ❝♦♠❡t❡ q✉❛♥❞♦ s❡ ✉s❛ ✉♠❛ ❡st✐♠❛t✐✈❛ ♣❛r❛ ♦ ❛r❣✉♠❡♥t♦ ❞❛ ❢✉♥çã♦✳

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✶✳✺✳ ❆P▲■❈❆➬➹❖ ❉❊ ❉■❋❊❘❊◆❈■❆■❙ ✲ ❈➪▲❈❯▲❖ ❉❊ ❊❘❘❖❙

❋✐❣✉r❛ ✶✳✹✿ ●rá✜❝♦s ❞❛ ❢✉♥çã♦ s❡♥♦ ❡ ❞❛ ❢✉♥çã♦ ✐❞❡♥t✐❞❛❞❡

✶✳✺✳✶

E

M ax

✲ ❊rr♦ ▼á①✐♠♦

❖ EM ax = dy é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❡rr♦ ♠á①✐♠♦✱ ❡rr♦ ♣r♦♣❛❣❛❞♦ ♦✉ ❡rr♦

❛♣r♦①✐♠❛❞♦✳ P♦❞❡ t❛♠❜é♠ s❡r ❞✐t♦ ✈❛❧♦r ❛♣r♦①✐♠❛❞♦✳ ◆♦t❡ q✉❡ EM ax ♣♦ss✉✐ ✉♥✐✲

❞❛❞❡ ❞❡ ♠❡❞✐❞❛✳

✶✳✺✳✷

E

Rel

✲ ❊rr♦ ❘❡❧❛t✐✈♦

❖ ERel =

dy

y é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❡rr♦ r❡❧❛t✐✈♦ ❡ ♥ã♦ ♣♦ss✉✐ ✉♥✐❞❛❞❡ ❞❡

♠❡❞✐❞❛✳

✶✳✺✳✸

E

P c

✲ ❊rr♦ P❡r❝❡♥t✉❛❧

❖ EP c = 100

dy

y é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❡rr♦ ♣❡r❝❡♥t✉❛❧ ❡ é ❡①♣r❡ss♦ ❝♦♠♦

✉♠❛ ♣♦r❝❡♥t❛❣❡♠✳

❊①❡♠♣❧♦ ✶✳✺✳✶✳ ▼❡❞✐✉✲s❡ ♦ ❞✐â♠❡tr♦ ❞❡ ✉♠ ❝ír❝✉❧♦ ❡ s❡ ❛❝❤♦✉ ✺✱✷ ♣♦❧❡❣❛❞❛s✱ ❝♦♠

✉♠ ❡rr♦ ♠á①✐♠♦ ❞❡ ✵✱✵✺ ♣♦❧❡❣❛❞❛s✳ ❉❡t❡r♠✐♥❡ ♦ ♠á①✐♠♦ ❡rr♦ ❛♣r♦①✐♠❛❞♦ ❞❛ ár❡❛ q✉❛♥❞♦ ❝❛❧❝✉❧❛❞❛ ♣❡❧❛ ❢ór♠✉❧❛✿

A= πD

2

4 .

❉❡t❡r♠✐♥❡ t❛♠❜é♠ ♦s ❡rr♦s r❡❧❛t✐✈♦s ❡ ♣❡r❝❡♥t✉❛❧✳ ❙♦❧✉çã♦✿

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❖ ✈❛❧♦r ❡①❛t♦ ❞❛ ár❡❛ A = πD

2

4 ♣❛r❛ D = 5,2 ♣♦❧❡❣❛❞❛s é A = 21,23

♣♦❧2 ✭♣♦❧❡❣❛❞❛s2✮✳

❆ ❞❡r✐✈❛❞❛ ❞❡ A ❡♠ r❡❧❛çã♦ ❛ D é

dA

dD =

π

2D.

❆ss✐♠✱ s✉❛ ❞✐❢❡r❡♥❝✐❛❧ dA é✿

dA = π

4 ·2D·dD = π

2 ·5,2·0,05 = 0,41♣♦❧2

.

❖ ❡rr♦ r❡❧❛t✐✈♦ é ❝❛❧❝✉❧❛❞♦ ♣♦r✿

dA

A = 0,0193.

❊ ♦ ❡rr♦ ♣❡r❝❡♥t✉❛❧ é ❝❛❧❝✉❧❛❞♦ ♣♦r✿

EP c = 100ERel

EP c = 1,93%.

❊①❡♠♣❧♦ ✶✳✺✳✷✳ ❆ ♠❡❞✐❞❛ ❞♦ r❛✐♦ ❞❡ ✉♠❛ ❡s❢❡r❛ é 0,7 ❝♠✳ ❙❡ ❡st❛ ♠❡❞✐❞❛ t✐✈❡r

✉♠❛ ♠❛r❣❡♠ ❞❡ ❡rr♦ ❞❡ 0,01❝♠✱ ❡st✐♠❡ ♦ ❡rr♦ ♣r♦♣❛❣❛❞♦ ❛♦ ✈♦❧✉♠❡ V ❞❛ ❡s❢❡r❛✳

❈❛❧❝✉❧❡ ♦ ❡rr♦ r❡❧❛t✐✈♦✳ ❙♦❧✉çã♦✿

❖ ✈♦❧✉♠❡ ❞❛ ❡s❢❡r❛ é ❞❛❞♦ ♣♦r

V = 4π 3 R

3

.

❆ ❡st✐♠❛t✐✈❛ ❞♦ r❛✐♦ ❞❛ ❡s❢❡r❛ é R= 0,7 ❡ ♦ ❡rr♦ ♠á①✐♠♦ ❞❛ ❡st✐♠❛t✐✈❛

é ∆R=dR= 0,01✳

❆ ❞✐❢❡r❡♥❝✐❛❧ ❞♦ ✈♦❧✉♠❡ ❞❛ ❡s❢❡r❛ é dV = 4πR2

dR✳ ◆❡st❡ ❡①❡♠♣❧♦✱ dV = 4π(0,7)2

(0,01) _≈0,06158❝♠3_✳

❖ ❡rr♦ r❡❧❛t✐✈♦ é ❞❛❞♦ ♣♦r✿