❆♣❧✐❝❛❝✐♦♥❡s ❞❡ ❧❛ ❞❡r✐✈❛❞❛
▼✳❙❝✳ ❆❧❡❥❛♥❞r♦ ❯❣❛❧❞❡ ▲❡ó♥
✯ ❆❝❛❞é♠✐❝♦
❊s❝✉❡❧❛ ❞❡ ▼❛t❡♠át✐❝❛✱ ❯◆❆
✶
✯
❮♥❞✐❝❡
✶✳ ❘❛③♦♥❡s ❞❡ ❝❛♠❜✐♦ r❡❧❛❝✐♦♥❛❞❛s
✸
✷✳ ❱❛❧♦r❡s ❡①tr❡♠♦s ② ❝♦♥❝❛✈✐❞❛❞ ❞❡ ✉♥❛ ❢✉♥❝✐ó♥
✹
✸✳ Pr♦❜❧❡♠❛s ❞❡ ♦♣t✐♠✐③❛❝✐ó♥
✼
✹✳ ❈✉❛❞r♦ ❞❡ ✈❛r✐❛❝✐ó♥ ② tr❛③❛❞♦ ❞❡ ❣rá✜❝❛s
✾
✺✳ ❘❡❣❧❛ ❞❡ ▲✬ ❍♦♣✐t❛❧✲❇❡r♥♦✉❧❧✐
✺✳✶✳
❋♦r♠❛s ✐♥❞❡t❡r♠✐♥❛❞❛s
✺✳✷✳
❋♦r♠❛s ✐♥❞❡t❡r♠✐♥❛❞❛s
∞−∞
1 ∞ ✱ ∞0
✶✶
②
②
0·∞
00 ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✷
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✷
✻✳ ❊❥❡r❝✐❝✐♦s
✶✸
✼✳ ❇✐❜❧✐♦❣r❛❢í❛
✶✽
✷
✶✳
❘❛③♦♥❡s ❞❡ ❝❛♠❜✐♦ r❡❧❛❝✐♦♥❛❞❛s
❈♦♠♦ s❡ ✈✐ó ❡♥ ❡❧ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r✱ ❧❛ ❞❡r✐✈❛❞❛ s❡ ♣✉❡❞❡ ✐♥t❡r♣r❡t❛r ❝♦♠♦ ✉♥❛ r❛③ó♥
❞❡ ❝❛♠❜✐♦✳ P♦r ❡st♦✱ ② ❝♦♠♦ ✉♥❛ ❛♣❧✐❝❛❝✐ó♥ ❞✐r❡❝t❛ ❞❡ ❧❛ r❡❣❧❛ ❞❡ ❧❛ ❝❛❞❡♥❛✱ s❡ ♣✉❡❞❡♥
r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s q✉❡ ✐♥✈♦❧✉❝r❡♥ ❞♦s ♦ ♠ás r❛③♦♥❡s ❞❡ ❝❛♠❜✐♦ r❡❧❛❝✐♦♥❛❞❛s ♣♦r ❛❧❣✉♥❛
❡❝✉❛❝✐ó♥ ❡♥ ♣❛rt✐❝✉❧❛r✳
❯♥❛ ❡str❛t❡❣✐❛ ♣❛r❛ r❡s♦❧✈❡r ❡st❡ t✐♣♦ ❞❡ ♣r♦❜❧❡♠❛s ❡s ❧❛ s✐❣✉✐❡♥t❡✿
✶✳ ▲❡❡r ❡❧ ♣r♦❜❧❡♠❛ ②✱ ❞❡ s❡r ♣♦s✐❜❧❡✱ ❤❛❝❡r ✉♥ ❞✐❜✉❥♦ q✉❡ ✐❧✉str❡ ❧❛ s✐t✉❛❝✐ó♥✳
✷✳ ❆s✐❣♥❛r ❧❡tr❛s ❛ t♦❞❛s ❧❛s ❝❛♥t✐❞❛❞❡s q✉❡ ✈❛rí❛♥ ❝♦♥ r❡s♣❡❝t♦ ❛❧ t✐❡♠♣♦✳
✸✳ ❉❡t❡r♠✐♥❛r ❧❛ r❡❧❛❝✐ó♥ ❡①✐st❡♥t❡ ❡♥tr❡ ❧❛s ✈❛r✐❛❜❧❡s ✐♥✈♦❧✉❝r❛❞❛s✳
✹✳ ❙✉st✐t✉✐r ❧❛ ✐♥❢♦r♠❛❝✐ó♥ ❞❛❞❛ ❡♥ ❧❛ ❡❝✉❛❝✐ó♥ ❞❡❧ ♣✉♥t♦ ❛♥t❡r✐♦r✱ ♣❛r❛ ❞❡r✐✈❛r ② ❧✉❡❣♦
❞❡s♣❡❥❛r ❧❛ r❛③ó♥ ❞❡ ❝❛♠❜✐♦ ❞❡s❝♦♥♦❝✐❞❛✳
❆ ✉♥ ❣❧♦❜♦ ❡s❢ér✐❝♦ s❡ ❧❡ ❜♦♠❜❡❛ ❛✐r❡ ❞❡ ♠♦❞♦ q✉❡ s✉ ✈♦❧✉♠❡♥ ❛✉♠❡♥t❛ ❛
r❛③ó♥ ❞❡ ✶✵✵ cm3 /s✳ ➽❈♦♥ q✉é r❛♣✐❞❡③ ❛✉♠❡♥t❛ ❡❧ r❛❞✐♦ ❞❡❧ ❣❧♦❜♦ ❝✉❛♥❞♦ ❡❧ ❞✐á♠❡tr♦
♠✐❞❡ ✺✵ ❝♠❄
❊❥❡♠♣❧♦ ✶✳
❯♥❛ ❡s❝❛❧❡r❛ ❞❡ ✶✵ ♣✐❡s ❞❡ ❧♦♥❣✐t✉❞ s❡ ❛♣♦②❛ ❝♦♥tr❛ ✉♥❛ ♣❛r❡❞✳ ▲❛ ❜❛s❡ ❞❡ ❧❛
❡s❝❛❧❡r❛ s❡ ❞❡s❧✐③❛ ❛❧❡❥á♥❞♦s❡ ❞❡ ❧❛ ♣❛r❡❞ ❛ r❛③ó♥ ❞❡ ✶ ♣✐❡✴s✳ ➽❈♦♥ q✉é r❛♣✐❞❡③ s❡ ❞❡s❧✐③❛
❤❛❝✐❛ ❛❜❛❥♦ ❧❛ ♣❛rt❡ s✉♣❡r✐♦r ❞❡ ❧❛ ❡s❝❛❧❡r❛ ❡♥ ❡❧ ✐♥st❛♥t❡ ❡♥ q✉❡ ❧❛ ❜❛s❡ ❡stá ❛ ✻ ♣✐❡s ❞❡
❧❛ ♣❛r❡❞❄
❊❥❡♠♣❧♦ ✷✳
❯♥ t❛♥q✉❡ ❞❡ ❛❣✉❛ ❝ó♥✐❝♦ t✐❡♥❡ ✉♥❛ ❛❧t✉r❛ ❞❡ ✹ ♠ ② ❡❧ r❛❞✐♦ ♠✐❞❡ ✷ ♠✳ ❙✐
s❡ ❜♦♠❜❡❛ ❛❣✉❛ ❞❡♥tr♦ ❞❡❧ t❛♥q✉❡ ❛ r❛③ó♥ ❞❡ ✷ m3 /min✱ ❡♥❝✉❡♥tr❡ ❧❛ ✈❡❧♦❝✐❞❛❞ ❛ ❧❛ q✉❡
❛✉♠❡♥t❛ ❡❧ ♥✐✈❡❧ ❞❡❧ ❛❣✉❛ ❝✉❛♥❞♦ ❧❧❡✈❛ ✸ ♠ ❞❡ ♣r♦❢✉♥❞✐❞❛❞✳
❊❥❡♠♣❧♦ ✸✳
❯♥ ❛✉t♦♠ó✈✐❧ ❆ ✈✐❛❥❛ ❤❛❝✐❛ ❡❧ ♦❡st❡ ❛ ✺✵ ❦♠✴❤✳ ❖tr♦ ❛✉t♦♠ó✈✐❧ ❇ ✈✐❛❥❛
❤❛❝✐❛ ❡❧ ♥♦rt❡ ❛ ✻✵ ❦♠✴❤✳ ❆♠❜♦s ❛✉t♦♠ó✈✐❧❡s s❡ ❞✐r✐❣❡♥ ❤❛❝✐❛ ❧❛ ✐♥t❡rs❡❝❝✐ó♥ ❞❡ ❧❛s ❞♦s
❝❛rr❡t❡r❛s✳ ➽❆ q✉é ✈❡❧♦❝✐❞❛❞ s❡ ❛♣r♦①✐♠❛♥ ❧♦s ❛✉t♦♠ó✈✐❧❡s ❡♥tr❡ sí ❝✉❛♥❞♦ ❆ ❡stá ❛ ✵✳✸
❦♠ ② ❇ ❛ ✵✳✹ ❦♠ ❞❡ ❧❛ ✐♥t❡rs❡❝❝✐ó♥❄
❊❥❡♠♣❧♦ ✹✳
❙❡ ❞❡❥❛ ❝❛❡r ✉♥❛ ♣✐❡❞r❛ ❡♥ ❛❣✉❛s tr❛♥q✉✐❧❛s✱ ❧♦ ❝✉❛❧ ♣r♦✈♦❝❛ ✉♥❛ ♦♥❞❛ ❡①✲
♣❛♥s✐✈❛✳ ❙✐ ❡❧ r❛❞✐♦ ❞❡ ❧❛ ♦♥❞❛ ❝r❡❝❡ ❛ r❛③ó♥ ❞❡ ✺ ❝♠✴s✱ ❞❡t❡r♠✐♥❡ ❧❛ ✈❛r✐❛❝✐ó♥ ❞❡❧ ár❡❛
❞❡ ❧❛ ♦♥❞❛ ❡♥ ❡❧ ✐♥st❛♥t❡ ❡♥ q✉❡ ❡❧ r❛❞✐♦ ♠✐❞❡ ✽ ❝♠✳
❊❥❡♠♣❧♦ ✺✳
❯♥ ❤♦♠❜r❡ ❝❛♠✐♥❛ ❡♥ ❧í♥❡❛ r❡❝t❛ ❛ ✉♥❛ ✈❡❧♦❝✐❞❛❞ ❞❡ ✹ ♣✐❡s✴s✳ ❯♥ r❡✢❡❝t♦r
s❡ ❡♥❝✉❡♥tr❛ ✉❜✐❝❛❞♦ ❛ ✷✵ ♣✐❡s ❞❡ ❛❧t✉r❛ ② s❡ ♠❛♥t✐❡♥❡ ❡♥ ❞✐r❡❝❝✐ó♥ ❛❧ ❤♦♠❜r❡✳ ➽❆ q✉é
✈❡❧♦❝✐❞❛❞ ❡stá ❣✐r❛♥❞♦ ❡❧ r❡✢❡❝t♦r ❝✉❛♥❞♦ ❡❧ ❤♦♠❜r❡ s❡ ❡♥❝✉❡♥tr❛ ❛ ✶✺ ♣✐❡s ❞❡ ❧❛ ❜❛s❡ ❞❡❧
r❡✢❡❝t♦r❄
❊❥❡♠♣❧♦ ✻✳
✸
✷✳
❱❛❧♦r❡s ❡①tr❡♠♦s ② ❝♦♥❝❛✈✐❞❛❞ ❞❡ ✉♥❛ ❢✉♥❝✐ó♥
❉❡✜♥✐❝✐ó♥ ✶✳
❙❡❛
f
✉♥❛ ❢✉♥❝✐ó♥✳ ❯♥ ♥ú♠❡r♦
c ∈ Df
s❡ ❧❧❛♠❛
♠á①✐♠♦ ❛❜s♦❧✉t♦ ❞❡
f
s✐ ② s♦❧♦ s✐
f (c) ≥ f (x)∀x ∈ Df
♠í♥✐♠♦ ❛❜s♦❧✉t♦ ❞❡
f
s✐ ② s♦❧♦ s✐
f (c) ≤ f (x)∀x ∈ Df
❙❡ ❞❡✜♥❡ ✉♥ ♠á①✐♠♦ ✭♠í♥✐♠♦✮ r❡❧❛t✐✈♦ ❝♦♠♦ ✉♥ ✈❛❧♦r ♠á①✐♠♦ ✭♠í♥✐♠♦✮ ❡♥ ✉♥ ✐♥t❡r✈❛❧♦
I ✱ I ⊂ Df ✳ ▲♦s ♠á①✐♠♦s ② ♠í♥✐♠♦s ❞❡ ✉♥❛ ❢✉♥❝✐ó♥ s❡ ❧❧❛♠❛♥ ✈❛❧♦r❡s ❡①tr❡♠♦s ❞❡ ❧❛
❢✉♥❝✐ó♥✳
❊♥ ❧❛ ✜❣✉r❛ ✶✱ t♦♠❛♥❞♦ ❝♦♠♦ ❞♦♠✐♥✐♦ ❞❡ ❧❛ ❢✉♥❝✐ó♥ ❡❧ ✐♥t❡r✈❛❧♦ [0, x2 ]✱ s❡ ♣r❡s❡♥t❛ ✉♥
♠í♥✐♠♦ ❛❜s♦❧✉t♦ ❡♥ x = 0✱ ✉♥ ♠á①✐♠♦ r❡❧❛t✐✈♦ ❡♥ x = x0 ✱ ✉♥ ♠í♥✐♠♦ r❡❧❛t✐✈♦ ❡♥ x = x1 ②
✉♥ ♠á①✐♠♦ ❛❜s♦❧✉t♦ ❡♥ x = x2 ✳
❋✐❣✉r❛ ✶✳
❱❛❧♦r❡s ❡①tr❡♠♦s ♣❛r❛ f ❡♥ [0, x2 ]✳
y
f (x)
0
❚❡♦r❡♠❛ ✷✳✶✳
❙❡❛♥
f
x0
x1
x
x2
✉♥❛ ❢✉♥❝✐ó♥ ❝♦♥t✐♥✉❛ ②
x0
✉♥ ✈❛❧♦r ❡①tr❡♠♦ ❞❡
f✳
❊♥t♦♥❝❡s
′
f (x0 ) = 0✳
❊❧ t❡♦r❡♠❛ ❛♥t❡r✐♦r ♥♦s ❞✐❝❡ q✉❡ ❧❛ ❞❡r✐✈❛❞❛ ❞❡ ✉♥❛ ❢✉♥❝✐ó♥ ❡s ❝❡r♦ ❡♥ ❧♦s ✈❛❧♦r❡s ❡①tr❡✲
♠♦s✳ ▲❛ ✐♥t❡r♣r❡t❛❝✐ó♥ ❣❡♦♠étr✐❝❛ ❞❡ ❡st♦ ❡s q✉❡✱ ❡♥ ❧♦s ✈❛❧♦r❡s ❡①tr❡♠♦s✱ ❧❛ ❣rá✜❝❛ ❞❡ ❧❛
❢✉♥❝✐ó♥ t✐❡♥❡ r❡❝t❛s t❛♥❣❡♥t❡s ❤♦r✐③♦♥t❛❧❡s✳
❉❡❜❡ q✉❡❞❛r ❝❧❛r♦ q✉❡ ❡❧ r❡❝í♣r♦❝♦ ❞❡ ❡st❡ t❡♦r❡♠❛ ♥♦ ❡s ❝✐❡rt♦✳ P♦r ❡❥❡♠♣❧♦✱ s✐ f (x) = x3 ✱
s❡ ♣✉❡❞❡ ❝♦♠♣r♦❜❛r q✉❡✱ ♣❛r❛ x0 = 0✱ f ′ (x0 ) = f ′ (0) = 0✱ ♣❡r♦ x0 = 0 ♥♦ ❡s ♠á①✐♠♦ ♥✐
♠í♥✐♠♦✳ ❊♥ ❧❛ ✜❣✉r❛ ✷ s❡ ♣✉❡❞❡ ❝♦rr♦❜♦r❛r ❡st❛ ❛✜r♠❛❝✐ó♥✳
❋✐❣✉r❛ ✷✳
●rá✜❝❛ ❞❡ f (x) = x3 ✳
y
x
✹
❉❡✜♥✐❝✐ó♥ ✷
❝rít✐❝♦ ❞❡
✳
✭P✉♥t♦ ❝rít✐❝♦✮
❯♥ ♣✉♥t♦
f✳
❚❡♦r❡♠❛ ✷✳✷✳
❙❡❛
f
x 0 ∈ Df ✱
✉♥❛ ❢✉♥❝✐ó♥ ❝♦♥t✐♥✉❛ ❡♥
t❛❧ q✉❡
f ′ (x0 ) = 0✱
[a, b]✳
❛✮ ❙✐
f ′ (x) > 0 ∀x ∈]a, b[✱
❡♥t♦♥❝❡s
f
❡s ❡str✐❝t❛♠❡♥t❡ ❝r❡❝✐❡♥t❡ ❡♥
❜✮ ❙✐
f ′ (x) < 0 ∀x ∈]a, b[✱
❡♥t♦♥❝❡s
f
❡s ❡str✐❝t❛♠❡♥t❡ ❞❡❝r❡❝✐❡♥t❡ ❡♥
❊❥❡♠♣❧♦ ✼✳
s❡ ❧❧❛♠❛ ♣✉♥t♦
[a, b]✳
[a, b]✳
❉❡t❡r♠✐♥❛r ❧♦s ✐♥t❡r✈❛❧♦s ❞❡ ♠♦♥♦t♦♥í❛ ❞❡ ❧❛ ❢✉♥❝✐ó♥ ❞❛❞❛ ♣♦r
f (x) =
x3 x2
+
− 2x✳
3
2
❙♦❧✉❝✐ó♥✳ ❙❡ ❞❡r✐✈❛ ❧❛ ❢✉♥❝✐ó♥ ♣❛r❛ ❤❛❧❧❛r ❡❧ s✐❣♥♦ ❞❡ s✉ ❞❡r✐✈❛❞❛✳
f ′ (x) = x2 + x − 2 = (x + 2)(x − 1) = 0
⇒ x = −2 ∧ x = 1
✭♣✉♥t♦s
−∞
x+2
x−1
f′
f
▲✉❡❣♦
−2 1
−
−
+
ր
f
+
−
−
ց
∞
+
+
+
ր
❡s ❝r❡❝✐❡♥t❡ ❡♥
] − ∞, −2[
❊❥❡♠♣❧♦ ✽ ✭❈♦♠♦ ❡❥❡r❝✐❝✐♦✮✳
♣♦r
x+1
f (x) =
✳
x−1
❚❡♦r❡♠❛ ✷✳✸
❝rít✐❝♦s✮
②
]1, ∞[❀
② ❡s ❞❡❝r❡❝✐❡♥t❡ ❡♥ ❡♥
] − 2, 1[✳
❉❡t❡r♠✐♥❛r ❧♦s ✐♥t❡r✈❛❧♦s ❞❡ ♠♦♥♦t♦♥í❛ ❞❡ ❧❛ ❢✉♥❝✐ó♥ ❞❛❞❛
✳
✭❈r✐t❡r✐♦ ❞❡ ❝♦♥❝❛✈✐❞❛❞✮ ❙❡❛
f
✉♥❛ ❢✉♥❝✐ó♥ ❝♦♥t✐♥✉❛ ❡♥
❛✮ ❙✐
f ′′ (x) > 0✱ ∀x ∈]a, b[✱
❡♥t♦♥❝❡s
f
❡s ❝ó♥❝❛✈❛ ❤❛❝✐❛ ❛rr✐❜❛ ❡♥
❜✮ ❙✐
f ′′ (c) < 0✱ ∀x ∈]a, b[✱
❡♥t♦♥❝❡s
f
❡s ❝ó♥❝❛✈❛ ❤❛❝✐❛ ❛❜❛❥♦ ❡♥
❊❥❡♠♣❧♦ ✾✳
❉❡t❡r♠✐♥❛r ❧❛ ❝♦♥❝❛✈✐❞❛❞ ❞❡ ❧❛ ❢✉♥❝✐ó♥ ❞❛❞❛ ♣♦r
❙♦❧✉❝✐ó♥✳ ❈❛❧❝✉❧❛♥❞♦ ❧❛s ❞❡r✐✈❛❞❛s ❞❡
f
[a, b]✳
]a, b[✳
]a, b[✳
f (x) =
x3 x2
+
− 2x✳
3
2
s❡ t✐❡♥❡✿
f ′ (x) = x2 + x − 2
f ′′ (x) = 2x + 1
❊♥t♦♥❝❡s✱ ♣♦r ❡❧ ❝r✐t❡r✐♦ ❞❡ ❝♦♥❝❛✈✐❞❛❞✱ s✐
2x + 1 > 0✱
❝ó♥❝❛✈❛ ❤❛❝✐❛ ❛rr✐❜❛✳
▲✉❡❣♦✱
f
❡s ❝ó♥❝❛✈❛ ❤❛❝✐❛ ❛❜❛❥♦ ❡♥
1
−∞, −
2
✺
♦ s❡❛✱ s✐
x>−
1
✱
2
② ❝ó♥❝❛✈❛ ❤❛❝✐❛ ❛rr✐❜❛ ❡♥
❧❛ ❢✉♥❝✐ó♥ ❡s
1
− ,∞
2
❉❡✜♥✐❝✐ó♥ ✸ ✭P✉♥t♦ ❞❡ ✐♥✢❡①✐ó♥✮✳ ❯♥ ♣✉♥t♦ c ∈ Df s❡ ❧❧❛♠❛ ♣✉♥t♦ ❞❡ ✐♥✢❡①✐ó♥ ❞❡ f ✱
s✐ ❡♥ é❧ ❤❛② ✉♥ ❝❛♠❜✐♦ ❞❡ ❝♦♥❝❛✈✐❞❛❞✳
❊❥❡♠♣❧♦ ✶✵ ✭❈♦♠♦ ❡❥❡r❝✐❝✐♦✮✳ ❉❡t❡r♠✐♥❛r ❧♦s ♣✉♥t♦s ❞❡ ✐♥✢❡①✐ó♥ ② ❧❛ ❝♦♥❝❛✈✐❞❛❞ ❞❡ ❧❛
x4
❢✉♥❝✐ó♥ ❞❛❞❛ ♣♦r f (x) = − + x2 + 1✳
2
❚❡♦r❡♠❛ ✷✳✹ ✭❈r✐t❡r✐♦ ❞❡ ❧❛ s❡❣✉♥❞❛ ❞❡r✐✈❛❞❛✮✳ ❙❡❛ f ✉♥❛ ❢✉♥❝✐ó♥ t❛❧ q✉❡ f ′′ ❡s ❝♦♥t✐♥✉❛
② x = c ✉♥ ♣✉♥t♦ ❝rít✐❝♦ ❞❡ f ✳
❛✮ ❙✐ f ′′ (c) < 0✱ ❡♥t♦♥❝❡s x = c ❡s ✉♥ ♠á①✐♠♦✳
❜✮ ❙✐ f ′′ (c) > 0✱ ❡♥t♦♥❝❡s x = c ❡s ✉♥ ♠í♥✐♠♦✳
❚❡♦r❡♠❛ ✷✳✺ ✭❈r✐t❡r✐♦ ❞❡ ❧❛ ♥✲és✐♠❛ ❞❡r✐✈❛❞❛✮✳ ❙❡❛ c ✉♥ ♣✉♥t♦ ❝rít✐❝♦ ❞❡ f t❛❧ q✉❡
f ′ (c) = f ′′ (c) = · · · = f n (c) = 0 ② f n+1 (c) 6= 0✳
❛✮ ❙✐ n + 1 ❡s ♣❛r✱ ❡♥t♦♥❝❡s
• x = c ❡s ✉♥ ♠á①✐♠♦ r❡❧❛t✐✈♦ s✐ f n+1 (c) < 0✳
• x = c ❡s ✉♥ ♠í♥✐♠♦ r❡❧❛t✐✈♦ s✐ f n+1 (c) > 0✳
❜✮ ❙✐ n + 1 ❡s ✐♠♣❛r✱ ❡♥t♦♥❝❡s x = c ♥♦ ❡s ♠á①✐♠♦ ♥✐ ♠í♥✐♠♦✳
❊❥❡♠♣❧♦ ✶✶✳ ❉❡t❡r♠✐♥❛r✱ s✐ ❡①✐st❡♥✱ ❧♦s ✈❛❧♦r❡s ❡①tr❡♠♦s ❞❡ ❧❛ ❢✉♥❝✐ó♥ f (x) = 31 x3 −
x2 − 3x + 3✳
❙♦❧✉❝✐ó♥✳ ❆♣❧✐❝❛♥❞♦ ❡❧ ❝r✐t❡r✐♦ ❞❡ ❧❛ s❡❣✉♥❞❛ ❞❡r✐✈❛❞❛ s❡ t✐❡♥❡✿
f ′ (x) = x2 − 2x − 3 = 0
(x − 3)(x + 1) = 0
⇒ x = 3 ② x = −1
✭♣✉♥t♦s ❝rít✐❝♦s✮
f ′′ (x) = 2x − 2✳ ▲✉❡❣♦✱
f (3) = 4 > 0 ⇒ (3, f (3)) ❡s ✉♥ ♣✉♥t♦ ♠í♥✐♠♦✳
f (−1) = −4 < 0 ⇒ (−1, f (−1)) ❡s ✉♥ ♣✉♥t♦ ♠á①✐♠♦✳
❊❥❡♠♣❧♦ ✶✷✳ ❉❡t❡r♠✐♥❡ s✐ ❧❛ ❢✉♥❝✐ó♥ f (x) = −4x3 ♣♦s❡❡ ✈❛❧♦r❡s ❡①tr❡♠♦s✳
❙♦❧✉❝✐ó♥✳ ❈❛❧❝✉❧❛♥❞♦ ❧❛s ❞❡r✐✈❛❞❛s s❡ t✐❡♥❡✿
f ′ (x) = −12x2 = 0 ⇒ x = 0 ✭♣✉♥t♦ ❝rít✐❝♦✮
f ′′ (x) = −24x
✻
❈♦♠♦ f ′′ (0) = 0✱ ♥♦ ♣✉❡❞❡ ✉s❛rs❡ ❡❧ ❝r✐t❡r✐♦ ❞❡ s❡❣✉♥❞❛ ❞❡r✐✈❛❞❛✳ ❊♥t♦♥❝❡s s❡ ✉t✐❧✐③❛ ❡❧
❝r✐t❡r✐♦ ❞❡ ❧❛ ♥✲és✐♠❛ ❞❡r✐✈❛❞❛✿
f ′′′ (x) = −24 ② f ′′′ (0) = −24 6= 0✳ ▲✉❡❣♦✱ ❝♦♠♦ ✸ ❡s ✐♠♣❛r ✭s❡ ❧❧❡❣ó ❛ ❧❛ t❡r❝❡r❛ ❞❡r✐✈❛❞❛✮✱
❡♥t♦♥❝❡s ❡❧ ♣✉♥t♦ (0, 0) ♥♦ ❡s ♠á①✐♠♦ ♥✐ ♠í♥✐♠♦✳ ❊♥ ❧❛ ✜❣✉r❛ ✸ s❡ ♣✉❡❞❡ ❛♣r❡❝✐❛r ❡st❡
r❡s✉❧t❛❞♦✿
❋✐❣✉r❛ ✸✳
●rá✜❝❛ ❞❡ f (x) = −4x3 ✳
y
◆✐ ♠á①✐♠♦ ♥✐ ♠í♥✐♠♦
x
❊❥❡♠♣❧♦ ✶✸
✭❈♦♠♦ ❡❥❡r❝✐❝✐♦✮✳
❉❡t❡r♠✐♥❡ s✐ ❧❛ ❢✉♥❝✐ó♥
❡①tr❡♠♦s✳
✸✳
f (x) = 5x4 − 7
♣♦s❡❡ ✈❛❧♦r❡s
Pr♦❜❧❡♠❛s ❞❡ ♦♣t✐♠✐③❛❝✐ó♥
▲♦s r❡s✉❧t❛❞♦s ❡①♣✉❡st♦s ❡♥ ❧❛ s❡❝❝✐ó♥ ❛♥t❡r✐♦r ♣❡r♠✐t❡♥ ❤❛❧❧❛r ✈❛❧♦r❡s ♠á①✐♠♦s ② ♠í♥✐✲
♠♦s ❞❡ ✉♥❛ ❢✉♥❝✐ó♥✳
❙✐ s❡ ❛♣❧✐❝❛♥ ❛ ♣r♦❜❧❡♠❛s ❡♥ ❧♦s ❝✉❛❧❡s ❡stá♥ ✐♥✈♦❧✉❝r❛❞❛s ❢✉♥❝✐♦♥❡s ❝♦♥♦❝✐❞❛s✱ ❡s ♣♦s✐❜❧❡
♦❜t❡♥❡r ✈❛❧♦r❡s ó♣t✐♠♦s ❞❡ ❡❧❧❛s✳
❯♥❛ ❡str❛t❡❣✐❛ ❛❞❡❝✉❛❞❛ ♣❛r❛ ❧❛ r❡s♦❧✉❝✐ó♥ ❞❡ ❡st❡ t✐♣♦ ❞❡ ♣r♦❜❧❡♠❛s ❡s ❧❛ s✐❣✉✐❡♥t❡✿
✶✳ ■❞❡♥t✐✜❝❛r ❧❛ ❢✉♥❝✐ó♥ ❛ ♦♣t✐♠✐③❛r✳
✷✳ ❙✐ ❧❛ ❢✉♥❝✐ó♥ ❞❡❧ ♣✉♥t♦ ❛♥t❡r✐♦r ♣♦s❡❡ ❞♦s ✈❛r✐❛❜❧❡s✱ ❡s ♥❡❝❡s❛r✐♦ ✐❞❡♥t✐✜❝❛r ✉♥❛
r❡❧❛❝✐ó♥ ❛❞✐❝✐♦♥❛❧ ❡♥tr❡ ❡❧❧❛s ♣❛r❛ ❞❡s♣❡❥❛r ✉♥❛ ❡♥ tér♠✐♥♦s ❞❡ ❧❛ ♦tr❛ ② s✉st✐t✉✐r❧❛
❡♥ ❧❛ ❢✉♥❝✐ó♥ ❛ ♦♣t✐♠✐③❛r✳
✸✳ ❉❡r✐✈❛r ❧❛ ❢✉♥❝✐ó♥ ❛ ♦♣t✐♠✐③❛r ♣❛r❛ ❤❛❧❧❛r ❧♦s ♣✉♥t♦s ❝rít✐❝♦s✳
✹✳ ❯t✐❧✐③❛r ❡❧ ❈r✐t❡r✐♦ ❞❡ s❡❣✉♥❞❛ ❞❡r✐✈❛❞❛✱ ♦ ❜✐❡♥✱ ❡❧ ❞❡ ❧❛ ♥✲és✐♠❛ ❞❡r✐✈❛❞❛ ♣❛r❛
❞❡t❡r♠✐♥❛r ❧♦s ♠á①✐♠♦s ♦ ♠í♥✐♠♦s✳
✼
▲❛ s✉♠❛ ❞❡ ✉♥ ♥ú♠❡r♦ ② ❡❧ tr✐♣❧❡ ❞❡ ♦tr♦ ❡s ✻✵✳ ❊♥❝♦♥tr❛r✱ ❡♥tr❡ t♦❞♦s ❧♦s
♣❛r❡s ❞❡ ♥ú♠❡r♦s q✉❡ s❛t✐s❢❛❝❡♥ ❡st♦✱ ❛q✉❡❧ ❝✉②♦ ♣r♦❞✉❝t♦ s❡❛ ❡❧ ♠á①✐♠♦ ♣♦s✐❜❧❡✳
❊❥❡♠♣❧♦ ✶✹✳
❊❥❡♠♣❧♦ ✶✺✳
♣❡rí♠❡tr♦ P ✳
❍❛❧❧❛r ❧❛s ❞✐♠❡♥s✐♦♥❡s ❞❡❧ r❡❝tá♥❣✉❧♦ ❞❡ ♠❛②♦r ár❡❛ ♣♦s✐❜❧❡✱ q✉❡ t❡♥❣❛
▲❛ ❣❡♥❡r❛tr✐③ ❞❡ ✉♥ ❝♦♥♦ ❝✐r❝✉❧❛r r❡❝t♦ ❡s ✉♥❛ ❝♦♥st❛♥t❡ a✳ ❉❡t❡r♠✐♥❛r ❧❛
❛❧t✉r❛ ❝♦rr❡s♣♦♥❞✐❡♥t❡ ❛❧ ❝♦♥♦ ❞❡ ♠❛②♦r ✈♦❧✉♠❡♥✳
❊❥❡♠♣❧♦ ✶✻✳
▲❛ ❤✐♣♦t❡♥✉s❛ ❞❡ ✉♥ tr✐á♥❣✉❧♦ r❡❝tá♥❣✉❧♦ ♠✐❞❡ ✻ ✉♥✐❞❛❞❡s✳ ❍❛❧❧❛r ❧❛s ♠❡✲
❞✐❞❛s ❞❡ ❧♦s ❝❛t❡t♦s ❞❡ ♠❛♥❡r❛ t❛❧ q✉❡ ❡❧ ár❡❛ ❞❡❧ tr✐á♥❣✉❧♦ s❡❛ ❧❛ ♠á①✐♠❛ ♣♦s✐❜❧❡✳
❊❥❡♠♣❧♦ ✶✼✳
❙❡ ❞❡s❡❛ ❝♦♥str✉✐r ✉♥❛ ❝❛❥❛ ♣❛r❛❧❡❧❡♣í♣❡❞❛✱ ❞❡ ❜❛s❡ r❡❝t❛♥❣✉❧❛r ② ❝♦♥ t❛♣❛✱
❝✉②♦ ✈♦❧✉♠❡♥ s❡❛ ❞❡ ✼✷ cm3 ✳ ▲♦s ❧❛❞♦s ❞❡ ❧❛ ❜❛s❡ ❡stá♥ ❡♥ ❧❛ r❛③ó♥ 1 : 2 ✭✉♥♦ ❡s ❡❧ ❞♦❜❧❡
❞❡❧ ♦tr♦✮✳ ➽❈✉á❧❡s ❞❡❜❡♥ s❡r ❧❛s❞✐♠❡♥s✐♦♥❡s ❞❡ ❧❛ ❝❛❥❛ ♣❛r❛ q✉❡ s✉ s✉♣❡r✜❝✐❡ t♦t❛❧ s❡❛ ❧❛
♠❡♥♦r ♣♦s✐❜❧❡❄
❊❥❡♠♣❧♦ ✶✽✳
❯♥❛ ✈❡♥t❛♥❛ t✐❡♥❡ ❢♦r♠❛ ❞❡ ✉♥ r❡❝tá♥❣✉❧♦ ❝♦r♦♥❛❞♦ ♣♦r ✉♥ s❡♠✐❝ír❝✉❧♦✳
❍❛❧❧❛r s✉s ❞✐♠❡♥s✐♦♥❡s s✐ s✉ ♣❡rí♠❡tr♦ ❡s ✶✷ m ② ❡❧ ár❡❛ ❧❛ ♠❛②♦r ♣♦s✐❜❧❡✳
❊❥❡♠♣❧♦ ✶✾✳
❯♥ r❡❝tá♥❣✉❧♦ ❣✐r❛ s♦❜r❡ ✉♥♦ ❞❡ s✉s ❧❛❞♦s ② ❣❡♥❡r❛ ✉♥ ❝✐❧✐♥❞r♦ ❝✐r❝✉❧❛r r❡❝t♦✳
❙✐ ❡❧ ♣❡rí♠❡tr♦ ❞❡❧ r❡❝tá♥❣✉❧♦ ❡s ❞❡ ✷✹ ❝♠✱ ➽❝✉á❧❡s s♦♥ ❧❛s ❞✐♠❡♥s✐♦♥❡s ❞❡❧ r❡❝tá♥❣✉❧♦ q✉❡
❣❡♥❡r❛ ❡❧ ❝✐❧✐♥❞r♦ ❞❡ ♠❛②♦r ✈♦❧✉♠❡♥❄ ➽❈✉á❧ ❡s ❡s❡ ✈♦❧✉♠❡♥❄
❊❥❡♠♣❧♦ ✷✵✳
❯♥ ❝❛♠✐ó♥ ❝♦♥s✉♠❡ 0,002x ❧✐tr♦s ❞❡ ❝♦♠❜✉st✐❜❧❡ ♣♦r ❦✐❧ó♠❡tr♦ ❝✉❛♥❞♦
✈✐❛❥❛ ❛ x ❦♠✴❤✱ ♣❛r❛ 70 ≤ x ≤ 100✳ ❙✐ ❡❧ ❝♦♠❜✉st✐❜❧❡ ❝✉❡st❛ ✩ ✶✳✷ ♣♦r ❧✐tr♦ ② ❡❧ s❛❧❛r✐♦
❞❡❧ ❝♦♥❞✉❝t♦r ❡s ❞❡ ✩ ✻ ♣♦r ❤♦r❛✱ ❞❡t❡r♠✐♥❡ ❧❛ ✈❡❧♦❝✐❞❛❞ q✉❡ ♠✐♥✐♠✐③❛ ❡❧ ❝♦st♦ t♦t❛❧ ❞❡
✉♥ ✈✐❛❥❡ ❞❡ ✼✵✵ ❦♠✱ ❛s✉♠✐❡♥❞♦ q✉❡ ❡❧ ❝❛♠✐ó♥ ✈✐❛❥❛ ❛ ✈❡❧♦❝✐❞❛❞ ❝♦♥st❛♥t❡✳
❊❥❡♠♣❧♦ ✷✶✳
✽
✹✳ ❈✉❛❞r♦ ❞❡ ✈❛r✐❛❝✐ó♥ ② tr❛③❛❞♦ ❞❡ ❣rá✜❝❛s
▲♦s ❝r✐t❡r✐♦s ❞❡ ♣r✐♠❡r❛ ② s❡❣✉♥❞❛ ❞❡r✐✈❛❞❛ ♣❡r♠✐t❡♥✱ ❥✉♥t♦ ❝♦♥ ❧♦s ❝♦♥♦❝✐♠✐❡♥t♦s ❜ás✐❝♦s
❞❡ ❧❛ ❣❡♦♠❡trí❛ ❛♥❛❧ít✐❝❛✱ tr❛③❛r ❧❛ ❣rá✜❝❛ ❞❡ ❝✉❛❧q✉✐❡r ❢✉♥❝✐ó♥✳
❊❧ ♣r♦❝❡❞✐♠✐❡♥t♦ ❝♦♥s✐st❡ ❡♥ ❛♥❛❧✐③❛r t♦❞❛ ❧❛ ✐♥❢♦r♠❛❝✐ó♥ q✉❡ ♣r♦♣♦r❝✐♦♥❛ ❡❧ ❝r✐t❡r✐♦ ❞❡
❧❛ ❢✉♥❝✐ó♥✿ ❞♦♠✐♥✐♦✱ ❝r❡❝✐♠✐❡♥t♦✱ ✈❛❧♦r❡s ❡①tr❡♠♦s✱ ❝♦♥❝❛✈✐❞❛❞ ② ❛sí♥t♦t❛s❀ ♣❛r❛ r❡s✉♠✐r❧❛
❡♥ ✉♥ ❝✉❛❞r♦ ②✱ ❝♦♥ ❡st♦✱ ❣r❛✜❝❛r❧❛ ❡♥ ✉♥ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✳
❊❧ ❡sq✉❡♠❛ ❡s ❡❧ s✐❣✉✐❡♥t❡✿
✶✳ ❍❛❧❧❛r ❡❧ ❞♦♠✐♥✐♦ ② ❧♦s ❝❡r♦s ❞❡ ❧❛ ❢✉♥❝✐ó♥✳
✷✳ ❈á❧❝✉❧♦ ❞❡ ❧❛ ♣r✐♠❡r❛ ❞❡r✐✈❛❞❛✱ ❡①tr❡♠♦s ❡ ✐♥t❡r✈❛❧♦s ❞❡ ♠♦♥♦t♦♥í❛✳
✸✳ ❈á❧❝✉❧♦ ❞❡ ❧❛ s❡❣✉♥❞❛ ❞❡r✐✈❛❞❛✱ ♣✉♥t♦s ❞❡ ✐♥✢❡①✐ó♥ ② ❝♦♥❝❛✈✐❞❛❞✳
✹✳ ❊①tr❡♠♦s ❞❡ ❧❛ ❢✉♥❝✐ó♥ ❡ ✐♠á❣❡♥❡s ❞❡ ❧♦s ♣✉♥t♦s ❝rít✐❝♦s ② ❞❡ ✐♥✢❡①✐ó♥ ❝❛❧❝✉❧❛❞♦s
❛♥t❡r✐♦r♠❡♥t❡✳
✺✳ ❆sí♥t♦t❛s✳ ❙♦♥ r❡❝t❛s ❛ ❧❛s ❝✉❛❧❡s ❧❛ ❣rá✜❝❛ ❞❡ ❧❛ ❢✉♥❝✐ó♥ s❡ ❛❝❡r❝❛ ✐♥❞❡✜♥✐❞❛♠❡♥t❡✳
❍❛② tr❡s t✐♣♦s✿
✮ ❱❡rt✐❝❛❧❡s✳ ❯♥❛ r❡❝t❛ x = a ❡s ❛sí♥t♦t❛ ✈❡rt✐❝❛❧ ❞❡ ✉♥❛ ❢✉♥❝✐ó♥ f s✐ lı́m f (x) =
x→a
±∞✳
❜ ✮ ❍♦r✐③♦♥t❛❧❡s✳ ❯♥❛ r❡❝t❛ y = b ❡s ❛sí♥t♦t❛ ❤♦r✐③♦♥t❛❧ ❞❡ ✉♥❛ ❢✉♥❝✐ó♥ f s✐
lı́m f (x) = b✳
❛
x→±∞
❝
✮ ❖❜❧í❝✉❛s✳ ❯♥❛ r❡❝t❛ y = mx + b ❡s ❛sí♥t♦t❛ ♦❜❧í❝✉❛ ❞❡ ✉♥❛ ❢✉♥❝✐ó♥ f s✐
lı́m
x→±∞
f (x)
= m ✭❝♦♥ m 6= 0✮ ② lı́m [f (x) − mx] = b
x→±∞
x
✻✳ ❈✉❛❞r♦ ❞❡ ✈❛r✐❛❝✐ó♥✳
✼✳ ❈♦♥str✉❝❝✐ó♥ ❞❡ ❧❛ ❣rá✜❝❛✳
❆ ❝♦♥t✐♥✉❛❝✐ó♥ s❡ ♠✉❡str❛♥ ✉♥❛ s❡r✐❡ ❞❡ ❡❥❡♠♣❧♦s q✉❡ ♠✉❡str❛♥ ❡❧ ♣r♦❝❡❞✐♠✐❡♥t♦ ❞❡❧
❛♥á❧✐s✐s ❡①❛❤✉st✐✈♦ ❛ r❡❛❧✐③❛r ♣❛r❛ ❧❛ ❣r❛✜❝❛❝✐ó♥ ❞❡ ✉♥❛ ❢✉♥❝✐ó♥✳
❊❥❡♠♣❧♦ ✷✷✳
❙♦❧✉❝✐ó♥✳
f (x) = 13 x3 − x2 − 3x + 3
✳
✶✳ P♦r s❡r ✉♥❛ ❢✉♥❝✐ó♥ ♣♦❧✐♥♦♠✐❛❧✱ Df = R✳
▲♦s ❝❡r♦s ❞❡ ❧❛ ❢✉♥❝✐ó♥ ✭♣✉♥t♦s ♣♦r ❞♦♥❞❡ ♣❛s❛ ❧❛ ❣rá✜❝❛ ♣♦r ❡❧ ❡❥❡ x✮ s❡ ❝❛❧❝✉❧❛♥
r❡s♦❧✈✐❡♥❞♦ ❧❛ ❡❝✉❛❝✐ó♥ 13 x3 − x2 − 3x + 3 = 0✳ ❊♥ ❡st❡ ❝❛s♦✱ ❧❛s s♦❧✉❝✐♦♥❡s s♦♥
♥ú♠❡r♦s ✐rr❛❝✐♦♥❛❧❡s✳
✾
✷✳ Pr✐♠❡r❛ ❞❡r✐✈❛❞❛✳
f ′ (x) = x2 − 2x − 3 = 0
(x − 3)(x + 1) = 0
⇒ x = 3 ② x = −1
−∞
x+1
x−3
f′
f
−1 3
−
−
+
ր
+
−
−
ց
∞
+
+
+
ր
✭♣✉♥t♦s ❝rít✐❝♦s✮
f ր ❡♥ ] − ∞, −1[ ② ❡♥ ]3, ∞[
f ց ❡♥ ] − 1, 3[✳
▲✉❡❣♦ x = −1 ❡s ✉♥ ♠á①✐♠♦ ② x = 3 ❡s ✉♥
♠í♥✐♠♦
✸✳ ❙❡❣✉♥❞❛ ❞❡r✐✈❛❞❛✳
f ′′ (x) = 2x − 2 = 0 ⇒ x = 1 ✭♣♦s✐❜❧❡ ♣✉♥t♦ ❞❡ ✐♥✢❡①✐ó♥✮
❙❡ t✐❡♥❡ q✉❡ f ′′ (x) > 0 s✐ x > 1 ② f ′′ (x) < 0 s✐ x < 1✳ ❊♥t♦♥❝❡s✱ f ❡s ❝ó♥❝❛✈❛ ❤❛❝✐❛
❛rr✐❜❛ ❡♥ ]1, ∞[ ② ❝ó♥✈❛✈❛ ❤❛❝✐❛ ❛❜❛❥♦ ❡♥ ] − ∞, 1[✳ ❆❞❡♠ás✱ x = 1 ❡s ✉♥ ♣✉♥t♦ ❞❡
✐♥✢❡①✐ó♥✳
✹✳ ❊①tr❡♠♦s ❡ ✐♠á❣❡♥❡s✳
f (−1) =
14
3
f (3) = −6
f (1) = − 23
lı́m f (x) = ∞
x→∞
lı́m f (x) = −∞
x→−∞
✺✳ ❆sí♥t♦t❛s✳ P♦r s❡r ✉♥❛ ❢✉♥❝✐ó♥ ♣♦❧✐♥♦♠✐❛❧✱ ♥♦ ❤❛② ❛sí♥t♦t❛s✳
✻✳ ❈✉❛❞r♦ ❞❡ ✈❛r✐❛❝✐ó♥✳
f ′ (x) ր
T ց
T ց
S ր
S
f ′′ (x)
✼✳ ●rá✜❝❛✳
✶✵
y
f (x) = 31 x3 − x2 − 3x + 3
3
x
❊❥❡♠♣❧♦ ✷✸
✭❈♦♠♦ ❡❥❡r❝✐❝✐♦✮✳ f (x) = x 3 + 5x 3
❊❥❡♠♣❧♦ ✷✹
x2
✭❈♦♠♦ ❡❥❡r❝✐❝✐♦✮✳ f (x) =
x+1
2
5
❊❥❡♠♣❧♦ ✷✺
❊❥❡♠♣❧♦ ✷✻
❊❥❡♠♣❧♦ ✷✼
✺✳
x2
x2 − 1
√
✭❈♦♠♦ ❡❥❡r❝✐❝✐♦✮✳ f (x) = x2 − 1
√
x+1
✭❈♦♠♦ ❡❥❡r❝✐❝✐♦✮✳ f (x) =
x
✭❈♦♠♦ ❡❥❡r❝✐❝✐♦✮✳ f (x) =
❘❡❣❧❛ ❞❡ ▲✬ ❍♦♣✐t❛❧✲❇❡r♥♦✉❧❧✐
0 ∞
♦ ✱ ❡♥ ♦❝❛s✐♦♥❡s ❡s út✐❧ ❧❛ s✐❣✉✐❡♥t❡ r❡❣❧❛✿
0 ∞
f ′ (x)
f (x)
= lı́m ′
lı́m
x→a g (x)
x→a g(x)
P❛r❛ ❡❧ ❝á❧❝✉❧♦ ❞❡ ❧í♠✐t❡s ❞❡ ❧❛ ❢♦r♠❛
▲❛ ❢ór♠✉❧❛ ❛♥t❡r✐♦r✱ ❧❧❛♠❛❞❛ ❘❡❣❧❛ ❞❡ ▲✬❍♦♣✐t❛❧✲❇❡r♥♦✉❧❧✐✶ ✱ s❡ ❛♣❧✐❝❛ ♣❛r❛ ❧í♠✐t❡s ❞❡ ❢✉♥✲
❝✐♦♥❡s r❛❝✐♦♥❛❧❡s q✉❡ ♥♦ ♣✉❡❞❡♥ s✐♠♣❧✐✜❝❛rs❡ ♣♦r ❧♦s ♠ét♦❞♦s ❛❧❣❡❜r❛✐❝♦s ❡st✉❞✐❛❞♦s✳
✶ ◗✉✐❡♥
❞❡s❛rr♦❧❧ó ❡st❛ r❡❣❧❛ ❢✉❡ ❡❧ ♠❛t❡♠át✐❝♦ s✉✐③♦ ❏♦❤❛♥♥ ❇❡r♥♦✉❧❧✐ ✭✶✻✻✼✲✶✼✹✽✮✱ ♣❡r♦ ❢✉❡ ❡❧ ▼❛rq✉és
❞❡ ▲✬❍♦♣✐t❛❧ ✭✶✻✻✶✲✶✼✵✹✮ q✉✐❡♥ ❧❛ ♣✉❜❧✐❝ó ❡♥ ✉♥♦ ❞❡ s✉s tr❛❜❛ ❥♦s✳
✶✶
❊❥❡♠♣❧♦ ✷✽✳
❊❥❡♠♣❧♦ ✷✾✳
ex − 1
lı́m
x→0 sen 2x
sen x
lı́m
x→0 x
❊❥❡♠♣❧♦ ✸✵✳
1 − x + ln x
x→1 1 + cos πx
❊❥❡♠♣❧♦ ✸✶✳
ex
x→∞ x2 + x
✺✳✶✳
lı́m
lı́m
❋♦r♠❛s ✐♥❞❡t❡r♠✐♥❛❞❛s
∞−∞
②
0·∞
❊♥ ♦❝❛s✐♦♥❡s✱ ❛❧ ❡✈❛❧✉❛r ❞✐r❡❝t❛♠❡♥t❡ ❡❧ ❧í♠✐t❡✱ s❡ ♦❜t✐❡♥❡♥ ❢♦r♠❛s ❧❛s ✐♥❞❡t❡r♠✐♥❛❞❛s
∞ − ∞ ♦ 0 · ∞✳ ❊♥ ❡st♦s ❝❛s♦s✱ s❡ ❞❡❜❡♥ r❡❛❧✐③❛r tr❛♥s❢♦r♠❛❝✐♦♥❡s ❛❧❣❡❜r❛✐❝❛s ♣❛r❛ ❧❧❡❣❛r
0 ∞
♦
✱ ♣❛r❛ ❧✉❡❣♦ ❛♣❧✐❝❛r ❧❛ r❡❣❧❛ ❞❡ ▲✬ ❍♦♣✐t❛❧✲❇❡r♥♦✉❧❧✐✳
❛
0 ∞
❊❥❡♠♣❧♦ ✸✷✳
❊❥❡♠♣❧♦ ✸✸✳
✺✳✷✳
lı́m x3 e−x
x→∞
lı́m (1 − cos x) cot x
x→0
❋♦r♠❛s ✐♥❞❡t❡r♠✐♥❛❞❛s
1∞ ✱ ∞0
②
00
❙✐ ❛❧ ❡✈❛❧✉❛r ❡❧ ❧í♠✐t❡ s❡ ♦❜t✐❡♥❡♥ ❡st❛s ❢♦r♠❛s ✐♥❞❡t❡r♠✐♥❛❞❛s✱ ❡s ♣♦rq✉❡ ❧❛ ❢✉♥❝✐ó♥ ❡s ❞❡
❧❛ ❢♦r♠❛
y = [f (x)]g(x) ✳
y
ln y
ln y
lı́m ln y
x→a
[f (x)]g(x)
ln[f (x)]g(x)
g(x) · ln f (x)
lı́m [g(x) · ln f (x)]
=
=
=
=
▲✉❡❣♦✱ s❡ ❝❛❧❝✉❧❛
❊❧ ❡sq✉❡♠❛ ♣❛r❛ ❡❧ ❝á❧❝✉❧♦ ❞❡❧ ❧í♠✐t❡ ❡s ❡❧ s✐❣✉✐❡♥t❡✿
✭❛♣❧✐❝❛♥❞♦ ❧♥ ❛ ❛♠❜♦s ❧❛❞♦s ❞❡ ❧❛ ✐❣✉❛❧❞❛❞✮
✭♣r♦♣✐❡❞❛❞ ❞❡ ❧♦❣❛r✐t♠♦✮
✭❛♣❧✐❝❛♥❞♦ ❧í♠✐t❡ ❛ ❛♠❜♦s ❧❛❞♦s✮
x→a
lı́m [g(x) · ln f (x)]✳
x→a
❙✐
lı́m [g(x) · ln f (x)] = L✱
x→a
lı́m ln y = L
x→a
⇒ ln lı́m y = L
x→a
⇒
lı́m y = eL
x→a
1
❊❥❡♠♣❧♦ ✸✹✳
❊❥❡♠♣❧♦ ✸✺✳
lı́m (cos x) x2
x→0
lı́m xtan x
x→0
1
❊❥❡♠♣❧♦ ✸✻✳
❊❥❡♠♣❧♦ ✸✼✳
lı́m x x
x→∞
1
lı́m 1 + 2
x→0
x
x
✶✷
❡♥t♦♥❝❡s
✻✳
❊❥❡r❝✐❝✐♦s
✶✳ ❘❡s♦❧✈❡r ❧♦s s✐❣✉✐❡♥t❡s ♣r♦❜❧❡♠❛s✿
❛ ✮ ❯♥❛ ❜♦♠❜❛ ❞❡ ❛✐r❡ ❡s❢ér✐❝❛ s❡ ❞❡s✐♥✢❛ ❞❡ ♠❛♥❡r❛ q✉❡ s✉ ✈♦❧✉♠❡♥ ❞✐s♠✐♥✉②❡ ❛
r❛③ó♥ ❞❡ ✶ cm3 /min✳ ➽❆ q✉é ✈❡❧♦❝✐❞❛❞ ❞✐s♠✐♥✉②❡ s✉ ❞✐á♠❡tr♦ ❝✉❛♥❞♦ ♠✐❞❡ ✶✵
cm❄
❜ ✮ ❙❡ ❜♦♠❜❡❛ ❛✐r❡ ❛ ✉♥ ❣❧♦❜♦ ❡s❢ér✐❝♦ ❛ r❛③ó♥ ❞❡ ✹✳✺ cm3 /min✳ ❈❛❧❝✉❧❡ ❧❛ r❛♣✐❞❡③
❝♦♥ q✉❡ ❝❛♠❜✐❛ ❡❧ r❛❞✐♦ ❞❡❧ ❣❧♦❜♦ ❝✉❛♥❞♦ ♠✐❞❡ ✷ cm✳
❝ ✮ ❯♥❛ ❧á♠♣❛r❛ q✉❡ ❡stá s♦❜r❡ ❡❧ s✉❡❧♦ ✐❧✉♠✐♥❛ ✉♥ ❛♥✉♥❝✐♦ ❡♥ ✉♥❛ ♣❛r❡❞ q✉❡ ❡stá
❛ ✶✷ m ❞❡ ❞✐st❛♥❝✐❛✳ ❯♥ ❤♦♠❜r❡ q✉❡ ♠✐❞❡ ✷ m ❞❡ ❛❧t♦✱ ❝❛♠✐♥❛ ❞❡ ❧❛ ❧á♠♣❛r❛
❤❛❝✐❛ ❧❛ ♣❛r❡❞ ❛ ✉♥❛ ✈❡❧♦❝✐❞❛❞ ❞❡ ✶✳✻ m/s✳ ➽❈♦♥ q✉é r❛♣✐❞❡③ ❞✐s♠✐♥✉②❡ s✉
s♦♠❜r❛ s♦❜r❡ ❡❧ ❛♥✉♥❝✐♦ ❝✉❛♥❞♦ s❡ ❡♥❝✉❡♥tr❛ ❛ ✹ m ❞❡ é❧❄
❞ ✮ ❯♥ ❛✈✐ó♥ q✉❡ ✈✉❡❧❛ ❤♦r✐③♦♥t❛❧♠❡♥t❡ ❛ ✉♥❛ ❛❧t✉r❛ ❞❡ ✶ km ② ❛ ✉♥❛ ✈❡❧♦❝✐❞❛❞
❞❡ ✺✵✵ km/h✱ ♣❛s❛ s♦❜r❡ ✉♥❛ ❡st❛❝✐ó♥ ❞❡ r❛❞❛r✳ ❊♥❝✉❡♥tr❡ ❧❛ ✈❡❧♦❝✐❞❛❞ ❛ ❧❛
❡✮
❢✮
❣✮
❤✮
✐✮
q✉❡ ❧❛ ❞✐st❛♥❝✐❛ ❞❡❧ ❛✈✐ó♥ ❛ ❧❛ ❡st❛❝✐ó♥ ❛✉♠❡♥t❛ ❝✉❛♥❞♦ ❡❧ ❛✈✐ó♥ s❡ ❡♥❝✉❡♥tr❛
❛ ✷ km ❞❡ ❧❛ ❡st❛❝✐ó♥✳
❯♥ ❝❛♠♣♦ ❞❡ ❜❡✐s❜♦❧ t✐❡♥❡ ❧❛ ❢♦r♠❛ ❞❡ ✉♥ ❝✉❛❞r❛❞♦ ❞❡ ✷✼✱✹✸ m ❞❡ ❧❛❞♦✳ ❯♥
❜❛t❡❛❞♦r ❣♦❧♣❡❛ ❧❛ ♣❡❧♦t❛ ② ❝♦rr❡ ❤❛❝✐❛ ♣r✐♠❡r❛ ❜❛s❡ ❛ ✉♥❛ ✈❡❧♦❝✐❞❛❞ ❞❡ ✼✳✸✶
m/s✳
✶✮ ➽❆ q✉é ✈❡❧♦❝✐❞❛❞ ❞✐s♠✐♥✉②❡ s✉ ❞✐st❛♥❝✐❛ ❛ ❧❛ s❡❣✉♥❞❛ ❜❛s❡ ❝✉❛♥❞♦ s❡ ❡♥✲
❝✉❡♥tr❛ ❛ ❧❛ ♠✐t❛❞ ❞❡❧ ❝❛♠✐♥♦ ❛ ❧❛ ♣r✐♠❡r❛ ❜❛s❡❄
✷✮ ➽❆ q✉é ✈❡❧♦❝✐❞❛❞ ❛✉♠❡♥t❛ s✉ ❞✐st❛♥❝✐❛ ❛ ❧❛ t❡r❝❡r❛ ❜❛s❡ ❡♥ ❡❧ ♠✐s♠♦ ♠♦✲
♠❡♥t♦❄
❉♦s ❛✉t♦♠ó✈✐❧❡s ♣❛rt❡♥ ❞❡ ✉♥ ♠✐s♠♦ ♣✉♥t♦✳ ❯♥♦ ✈✐❛❥❛ ❤❛❝✐❛ ❡❧ s✉r ❛ ✻✵ km/h
② ❡❧ ♦tr♦ ❤❛❝✐❛ ❡❧ ♦❡st❡ ❛ ✷✺ km/h✳ ➽❆ q✉é ✈❡❧♦❝✐❞❛❞ ❛✉♠❡♥t❛ ❧❛ ❞✐st❛♥❝✐❛ ❡♥tr❡
❡❧❧♦s ❞♦s ❤♦r❛s ❞❡s♣✉és ❞❡ ❤❛❜❡r s❛❧✐❞♦❄
❉❡ ✉♥ t❛♥q✉❡ ❝♦♥ ❢♦r♠❛ ❞❡ ❝♦♥♦ ✐♥✈❡rt✐❞♦ s❡ ❞❡❥❛ s❛❧✐r ❛❣✉❛ ❛ r❛③ó♥ ❞❡ ✶✵✵✵✵
cm3 /min✱ ❛❧ ♠✐s♠♦ t✐❡♠♣♦ q✉❡ s❡ ❜♦♠❜❡❛ ❛❣✉❛ ❛❧ ✐♥t❡r✐♦r ❛ ✉♥❛ ✈❡❧♦❝✐❞❛❞
❝♦♥st❛♥t❡✳ ❊❧ t❛♥q✉❡ t✐❡♥❡ ✻m ❞❡ ❛❧t✉r❛ ② ❡❧ ❞✐á♠❡tr♦ ❞❡ ❧❛ ♣❛rt❡ s✉♣❡r✐♦r ❡s
❞❡ ✹ m✳ ❙✐ ❡❧ ♥✐✈❡❧ ❞❡❧ ❛❣✉❛ ❡stá ❛✉♠❡♥t❛♥❞♦ ❛ r❛③ó♥ ❞❡ ✷✵ cm/min ❝✉❛♥❞♦ ❧❛
❛❧t✉r❛ ❞❡❧ ❛❣✉❛ ❡s ❞❡ ✷ m✱ ❡♥❝✉❡♥tr❡ ❧❛ ✈❡❧♦❝✐❞❛❞ ❛ ❧❛ q✉❡ s❡ ❜♦♠❜❡❛ ❛❣✉❛ ❛❧
✐♥t❡r✐♦r ❞❡❧ t❛♥q✉❡✳
❉❡ ✉♥❛ ❝✐♥t❛ tr❛♥s♣♦rt❛❞♦r❛ s❡ ❞❡s❝❛r❣❛ ❛r❡♥❛ ❛ r❛③ó♥ ❞❡ ✸✵ pies3 /min✱ ❢♦r✲
♠❛♥❞♦ ✉♥❛ ♣✐❧❛ ❝ó♥✐❝❛ ❝✉②♦ ❞✐á♠❡tr♦ ② ❛❧t✉r❛ s♦♥ s✐❡♠♣r❡ ✐❣✉❛❧❡s✳ ➽❈♦♥ q✉é
r❛♣✐❞❡③ ❛✉♠❡♥t❛ ❧❛ ❛❧t✉r❛ ❞❡ ❧❛ ♣✐❧❛ ❝✉❛♥❞♦ ♠✐❞❡ ✶✵ ♣✐❡s❄
❯♥❛ ♣✐❡❞r❛ q✉❡ s❡ ❞❡❥❛ ❝❛❡r ❡♥ ✉♥ ❡st❛♥q✉❡ ♣r♦❞✉❝❡ ✉♥❛ s❡r✐❡ ❞❡ ♦♥❞❛s ❝♦♥✲
❝é♥tr✐❝❛s✳ ❙✐ ❡❧ r❛❞✐♦ r ❞❡ ❧❛ ♦♥❞❛ ❡①t❡r✐♦r ❛✉♠❡♥t❛ ❝♦♥st❛♥t❡♠❡♥t❡ ❛ ✉♥❛ t❛s❛
❞❡ ✷ m/s✱ ❤❛❧❧❛r ❧❛ t❛s❛ ❛ ❧❛ q✉❡ ❛✉♠❡♥t❛ ❡❧ ár❡❛ ❞❡❧ ❛❣✉❛ ❛❢❡❝t❛❞❛ ❝✉❛♥❞♦
r = 3 ② ❝✉❛♥❞♦ r = 6✳
✶✸
✷✳ ❊❢❡❝t✉❛r ✉♥ ❛♥á❧✐s✐s ❡①❤❛✉st✐✈♦ ❞❡ ❧❛s s✐❣✉✐❡♥t❡s ❢✉♥❝✐♦♥❡s ② tr❛③❛r s✉s ❣rá✜❝❛s ❝♦♥
❛②✉❞❛ ❞❡❧ ❝✉❛❞r♦ ❞❡ ✈❛r✐❛❝✐ó♥✿
x4
❛ ✮ f (x) = 1 + x −
2
x2 − 3x − 4
❦ ✮ f (x) =
x−2
2
❜ ✮ g(x) =
x(x − 3)
(x + 3)2
❧ ✮ f (x) =
x2
x+1
2x − 1
❞ ✮ f (x) = 2
x −x
|x|
x2 + 4
♠ ✮ g(x) = (x − 1)−2 − (x + 3)−2
❝ ✮ h(x) =
♥ ✮ f (x) = (x3 − x) 3
1
x4
(x + 1)3
❡ ✮ g(x) = x3 − 3x2 − 24x
ñ ✮ g(x) =
x2 + x + 1
x−1
x−2
❣ ✮ f (x) = √
x2 + 1
4
x+1
❤ ✮ g(x) =
1−x
♦ ✮ h(x) = 8x5 − 5x4 − 20x3
❢ ✮ h(x) =
✐ ✮ f (x) =
❥ ✮ g(x) =
♣ ✮ f (x) = x3 e−x
2
x (x − 1)
(x + 1)2
q ✮ f (x) =
x3 + 2x2 + 7x − 3
2x2
r ✮ g(x) =
2 + x − x2
(x − 1)2
s ✮ f (x) = x 3 (x + 2)− 3
1
3
(x + 1)
(x − 1)2
✶✹
2
✸✳ ❆♣❧✐❝❛♥❞♦ ❧❛ ❘❡❣❧❛ ❞❡ ▲✬❍ô♣✐t❛❧✱ ❤❛❧❧❛r ❧♦s ❧í♠✐t❡s s✐❣✉✐❡♥t❡s ✭❧❛ r❡s♣✉❡st❛ ❛♣❛r❡❝❡
❡♥tr❡ ♣❛ré♥t❡s✐s✮✿
√
√
3
x− 3a
√
❛ ✮ lı́m √
x→a
x− a
ex − 1
❜ ✮ lı́m
x→0 sen x
3
a
x
1
−
❦ ✮ lı́m
x→1
ln x ln x
1
2
x
❧ ✮ lı́m x e − 1
(1)
(−1)
(0)
x→+∞
1
ln (cos x)
❝ ✮ lı́m
x→0
x
αx
e − cos αx
❞ ✮ lı́m βx
x→0 e
− cos βx
a
❡ ✮ lı́m x sen
x→+∞
x
πϕ
❢ ✮ lı́m (a2 − ϕ2 ) tan
ϕ→a
2a
1
❣ ✮ lı́m cot x −
x→0
x
x − sen x
❤ ✮ lı́m
x→0 x − tan x
2
√
6
(0)
α
β
π
(1)
1
ñ ✮ lı́m (ex + x) x
e2
x→0
x
1
♦ ✮ lı́m 1 + 2
x→+∞
x
x tan πx
2a
♣ ✮ lı́m 2 −
x→a
a
(0)
(1)
q ✮ lı́m xn e−x
(− 21 )
e
1
1
2
r ✮ lı́m x ln x−1
(e)
tan x
1
s ✮ lı́m
x→0
x
(1)
x→0
(−2)
2
π
(0)
x→+∞
2
ex − 1
✐ ✮ lı́m
x→0 cos x − 1
x
1
❥ ✮ lı́m
−
x→1
x − 1 ln x
♥ ✮ lı́m xsen x
x→0
(a)
4a2
(+∞)
x→0
♠ ✮ lı́m x2 e x2
✹✳ ❘❡s♦❧✈❡r ❧♦s s✐❣✉✐❡♥t❡s ♣r♦❜❧❡♠❛s✿
❛✮
❉✐✈✐❞✐r ❡❧ ♥ú♠❡r♦ ✽ ❡♥ ❞♦s s✉♠❛♥❞♦s t❛❧❡s q✉❡ ❧❛ s✉♠❛ ❞❡ s✉s ❝✉❜♦s s❡❛ ❧❛
♠❡♥♦r ♣♦s✐❜❧❡✳
❜✮
❝✮
❘✴ ✹ ② ✹
➽◗✉é ♥ú♠❡r♦ ♣♦s✐t✐✈♦ s✉♠❛❞♦ ❛ s✉ ✐♥✈❡rs♦ ❞❛ ❧✉❣❛r ❛ ❧❛ s✉♠❛ ♠í♥✐♠❛❄
❉❡ ✉♥❛ ❤♦❥❛ ❞❡ ❝❛rtó♥ ❞❡
18 × 18
❘✴ ✶
❝♠✱ ❞❡❜❡♥ s❡r r❡❝♦rt❛❞♦s ❝✉❛❞r❛❞♦s ✐❣✉❛❧❡s
❡♥ ❧❛s ❡sq✉✐♥❛s ❞❡ ♠♦❞♦ q✉❡✱ ❞♦❜❧❛♥❞♦ ❧❛ ❤♦ ❥❛✱ r❡s✉❧t❡ ✉♥❛ ❝❛❥❛ q✉❡ t❡♥❣❛
❧❛ ♠❛②♦r ❝❛♣❛❝✐❞❛❞ ♣♦s✐❜❧❡✳ ❍❛❧❧❛r ❧❛ ♠❡❞✐❞❛ ❞❡❧ ❧❛❞♦ ❞❡ ❧♦s ❝✉❛❞r❛❞♦s ♣❛r❛
♦❜t❡♥❡r ❡s❡ ✈♦❧✉♠❡♥ ♠á①✐♠♦✳
❞✮
❘✴ ✸ ❝♠
❯♥ ❛❣r✐❝✉❧t♦r ❞✐s♣♦♥❡ ❞❡ ✶✵✵ ♠❡tr♦s ❞❡ ❝❡r❝❛ ♣❛r❛ ❧✐♠✐t❛r ✉♥ t❡rr❡♥♦ r❡❝t❛♥❣✉✲
❧❛r ❝♦♥t✐❣✉♦ ❛ ❧❛ ♦r✐❧❧❛ ❞❡ ✉♥ rí♦✳ ❙✐ ♥♦ r❡q✉✐❡r❡ ❝❡r❝❛r ❛ ❧❛ ♦r✐❧❧❛ ❞❡❧ rí♦✱ ➽❝✉á❧❡s
❞❡❜❡♥ s❡r ❧❛s ❞✐♠❡♥s✐♦♥❡s ❞❡❧ t❡rr❡♥♦ ❞❡ ♠❛②♦r ár❡❛ ♣♦s✐❜❧❡❄
❡✮
❙❡ ❞❡s❡❛ ❢❛❜r✐❝❛r ✉♥❛ ❧❛t❛ ❝✐❧í♥❞r✐❝❛ ❝♦♥ ❝❛♣❛❝✐❞❛❞ ♣❛r❛ ❛❧♠❛❝❡♥❛r
100 cm3
❞❡
r❡❢r❡s❝♦✳ ❍❛❧❧❛r ❧❛s ❞✐♠❡♥s✐♦♥❡s ❞❡ ❧❛ ❧❛t❛ q✉❡ ♠✐♥✐♠✐③❛♥ ❡❧ ❝♦st♦ ❞❡❧ ♠❛t❡r✐❛❧
r❡q✉❡r✐❞♦ ♣❛r❛ ❤❛❝❡r ❡❧ ❡♥✈❛s❡✳
✶✺
❢
✮ ❍❛❧❧❛r ❧❛s ❞✐♠❡♥s✐♦♥❡s ❞❡❧ ❝♦♥♦ ❝✐r❝✉❧❛r r❡❝t♦✱ ❞❡ ✈♦❧✉♠❡♥ ♠á①✐♠♦✱ q✉❡ ♣✉❡❞❡
√
✐♥s❝r✐❜✐rs❡ ❡♥ ✉♥❛ ❡s❢❡r❛ ❞❡ r❛❞✐♦ R✳
❘✴ h =
2R 2
4R
②r=
3
3
✮ ❍❛❧❧❛r ❧❛s ❞✐♠❡♥s✐♦♥❡s ❞❡❧ ❝♦♥♦ ❝✐r❝✉❧❛r r❡❝t♦✱ ❞❡ ✈♦❧✉♠❡♥ ♠í♥✐♠♦✱ q✉❡ ♣✉❡❞❡
√
❝✐r❝✉♥s❝r✐❜✐rs❡ ❡♥ ✉♥❛ ❡s❢❡r❛ ❞❡ 8 ❝♠ ❞❡ ❞✐á♠❡tr♦✳ ❘✴ h = 12 ❝♠ ② r = 4 2
❝♠✳
❤ ✮ ❯♥ ❝❛♠♣♦ ❞❡ ❞❡♣♦rt❡s ❝♦♥st❛ ❞❡ ✉♥❛ r❡❣✐ó♥ r❡❝t❛♥❣✉❧❛r ❝♦♥ ✉♥❛ r❡❣✐ó♥ s❡♠✐✲
❝✐r❝✉❧❛r ❛❞❥✉♥t❛ ❡♥ ❝❛❞❛ ❡①tr❡♠♦✳ ❙✐ ❡❧ ♣❡rí♠❡tr♦ ❞❡❧ ❝❛♠♣♦ ❡s ❞❡ 400 ♠✱ ❤❛❧❧❛r
❧❛s ❞✐♠❡♥s✐♦♥❡s ❞❡❧ ❝❛♠♣♦ ❞❡ ♠❛②♦r ár❡❛ ♣♦s✐❜❧❡✳ ❘✴ ❈✐r❝✉♥❢❡r❡♥❝✐❛ ❞❡ r❛❞✐♦
200
♠✳
π
3
✐ ✮ ❯♥❛ t✐♥❛ t✐❡♥❡ ❢♦r♠❛ ❞❡ ❝✐❧✐♥❞r♦ r❡❝t♦✳ ❙✐ s✉ ✈♦❧✉♠❡♥ ❡s ✸✷ cm ✱ ➽❝✉á♥t♦ ❞❡❜❡♥
♠❡❞✐r ❡❧ r❛❞✐♦ ❞❡ ❧❛ ❜❛s❡ ② ❧❛ ❛❧t✉r❛ ♣❛r❛ q✉❡ s✉ s✉♣❡r✜❝✐❡ t♦t❛❧qs❡❛ ❧❛ ♠❡♥♦r
q
♣♦s✐❜❧❡❄
❘✴ r = 2 3 π2 ✱ h = 4 3 π2
❣
✮ ❯♥ r❡❝✐♣✐❡♥t❡ r❡❝t❛♥❣✉❧❛r ❞❡❜❡ t❡♥❡r ✉♥ ✈♦❧✉♠❡♥ ❞❡ ✶✵ m3 ✳ ▲❛ ❧♦♥❣✐t✉❞ ❞❡ ❧❛
❜❛s❡ ❡s ❡❧ ❞♦❜❧❡ ❞❡ ❧❛ ❛♥❝❤✉r❛✳ ❊❧ ♠❛t❡r✐❛❧ ❞❡ ❧❛ ❜❛s❡ ❝✉❡st❛ ❝✴✻✵✵✵ ❡❧ ♠❡tr♦
❝✉❛❞r❛❞♦✳ ❊❧ ♠❛t❡r✐❛❧ ❞❡ ❧♦s ❧❛❞♦s ❝✉❡st❛ ❝✴✸✻✵✵ ❡❧ ♠❡tr♦ ❝✉❛❞r❛❞♦✳ ❉❡t❡r♠✐♥❡
❡❧ ❝♦st♦ ❞❡ ❧♦s ♠❛t❡r✐❛❧❡s ❞❡❧ r❡❝✐♣✐❡♥t❡ ♠ás ❜❛r❛t♦✱ s✉♣♦♥✐❡♥❞♦ q✉❡ ❧❛ t❛♣❛
❡stá ❤❡❝❤❛ ❞❡❧ ♠✐s♠♦ ♠❛t❡r✐❛❧ q✉❡ ❧♦s ❧❛❞♦s✳
❘✴ ❝✴✶✶✹ ✼✻✽
2
❦ ✮ ❊❧ ár❡❛ ❞❡ ✉♥❛ s✉♣❡r✜❝✐❡ r❡❝t❛♥❣✉❧❛r ❡s ❞❡ 18 m ✳ ❙❛❜✐❡♥❞♦ q✉❡ ❡♥ s✉ ✐♥t❡r✐♦r
❤❛② ♦tr❛ s✉♣❡r✜❝❡ r❡❝t❛♥❣✉❧❛r ❝✉②♦s ♠ár❣❡♥❡s s✉♣❡r✐♦r ❡ ✐♥❢❡r✐♦r s♦♥ ❞❡ 0,75
m ② ❧♦s ♠ár❣❡♥❡s ❧❛t❡r❛❧❡s s♦♥ ❞❡ 0,5 m✱ ❤❛❧❧❛r ❧❛s ❞✐♠❡♥s✐♦♥❡s ❞❡ ❧❛ s✉♣❡r✜❝✐❡
❡①t❡r✐♦r ♣❛r❛ q✉❡ ❡❧ ár❡❛ ❝♦♠♣r❡♥❞✐❞❛ ❡♥tr❡ ❧♦s ♠ár❣❡♥❡s s❡❛ ♠á①✐♠❛✳
❘✴
❥
3,46 × 5,19
❧
✮ ❍❛❧❧❛r ❧❛s ❞✐♠❡♥s✐♦♥❡s ❞❡❧ r❡❝tá♥❣✉❧♦ ❞❡ ár❡❛ ♠á①✐♠❛ q✉❡ s❡ ♣✉❡❞❡ ✐♥s❝r✐❜✐r
8
3
❡♥ ❧❛ ♣♦r❝✐ó♥ ❞❡ ❧❛ ♣❛rá❜♦❧❛ y 2 = 12x✱ ❧✐♠✐t❛❞❛ ♣♦r ❧❛ r❡❝t❛ x = 4✳ ❘✴ 8 × ✳
✮ ❯♥ tr♦③♦ ❞❡ ❛❧❛♠❜r❡ ❞❡ ✶✵ ♠ ❞❡ ❧♦♥❣✐t✉❞ s❡ ❝♦rt❛ ❡♥ ❞♦s ♣❛rt❡s✳ ❈♦♥ ✉♥❛ ❞❡
❡❧❧❛s s❡ ❢♦r♠❛ ✉♥ ❝✉❛❞r❛❞♦ ② ❝♦♥ ❧❛ ♦tr❛ ✉♥ tr✐á♥❣✉❧♦ ❡q✉✐❧át❡r♦✳ ➽❈ó♠♦ ❞❡❜❡
❝♦rt❛rs❡ ❡❧ ❛❧❛♠❜r❡ ♣❛r❛ q✉❡ ❡❧ ár❡❛ t♦t❛❧ ❧✐♠✐t❛❞❛ s❡❛ ❧❛ ♠á①✐♠❛ ♣♦s✐❜❧❡❄ ❘✴
❚♦❞♦ ❡❧ ❛❧❛♠❜r❡ ♣❛r❛ ❡❧ ❝✉❛❞r❛❞♦
♥ ✮ ❯♥ ❞❡♣ós✐t♦ ❛❜✐❡rt♦ ❞❡ ❛❧✉♠✐♥✐♦✱ ❝♦♥ ❢♦♥❞♦ ❝✉❛❞r❛❞♦✱ ❞❡❜❡ t❡♥❡r ❝❛♣❛❝✐❞❛❞
♣❛r❛ ❛❧♠❛❝❡♥❛r v ❧✐tr♦s✳ ➽◗✉é ❞✐♠❡♥s✐♦♥❡s ❞❡❜❡ t❡♥❡r ❡❧ ❞❡♣ós✐t♦ ♣❛r❛ q✉❡ ❡♥
s✉ ❢❛❜r✐❝❛❝✐ó♥ s❡ ♥❡❝❡s✐t❡ ❧❛ ♠❡♥♦r ❝❛♥t✐❞❛❞ ❞❡ ❛❧✉♠✐♥✐♦❄
ñ ✮ ❯♥ r❡❝✐♣✐❡♥t❡ ❛❜✐❡rt♦ ❡stá ❢♦r♠❛❞♦ ♣♦r ✉♥ ❝✐❧✐♥❞r♦ ② ✉♥❛ s❡♠✐❡s❢❡r❛ ❡♥ s✉ ♣❛rt❡
✐♥❢❡r✐♦r✳ ➽◗✉é ❞✐♠❡♥s✐♦♥❡s ❞❡❜❡ t❡♥❡r ❡❧ r❡❝✐♣✐❡♥t❡ ♣❛r❛ q✉❡ ♣✉❡❞❛ ❛❧♠❛❝❡♥❛r
✷✺✵ ml ② s❡ ❣❛st❡✱ ♣❛r❛ ❝♦♥str✉✐r❧♦✱ ❧❛ ♠❡♥♦r ❝❛♥t✐❞❛❞ ❞❡ ♠❛t❡r✐❛❧ ♣♦s✐❜❧❡❄
♠
♦
6−x
✳ ➽◗✉é ♠❡❞✐❞❛s
2
3
❞❡❜❡ t❡♥❡r ❡❧ r❡❝tá♥❣✉❧♦ ♣❛r❛ q✉❡ s✉ ár❡❛ s❡❛ ♠á①✐♠❛❄
❘✴ 3 ×
2
✮ ❯♥ r❡❝tá♥❣✉❧♦ ❡stá ❛❝♦t❛❞♦ ♣♦r ❧♦s ❡❥❡s ② ♣♦r ❧❛ r❡❝t❛ y =
✶✻
♣✮
❉❡t❡r♠✐♥❛r !
❧♦s ♣✉♥t♦s ❞❡ ❧❛
!♣❛rá❜♦❧❛
❘✴
q✮
r
3 5
,
2 2
②
−
r
3 5
,
2 2
y = 4 − x2
(0, 2)✳
❉❡t❡r♠✐♥❛r ❧❛ ♠❡❞✐❞❛ ❞❡❧ r❛❞✐♦ ❞❡❧ ❝✐❧✐♥❞r♦ ❞❡ ♠❛②♦r ✈♦❧✉♠❡♥ q✉❡ ♣✉❡❞❡ ✐♥s✲
❝r✐❜✐rs❡ ❡♥ ✉♥❛ ❡s❢❡r❛ ❞❡ r❛❞✐♦ ✶ ♠✳
r✮
♠ás ❝❡r❝❛♥♦s ❛❧ ♣✉♥t♦
❘✴
r ≈ 0,817
❯♥ ❜♦t❡ s❛❧❡ ❞❡ ✉♥ ♠✉❡❧❧❡ ❛ ❧❛s ✷ ♣✳♠✳ ❝♦♥ ❞✐r❡❝❝✐ó♥ ❤❛❝✐❛ ❡❧ s✉r✱ ❛ ✉♥❛ ✈❡❧♦❝✐❞❛❞
❞❡ ✷✵ ❦♠✴❤✳ ❖tr♦ ❜♦t❡ ❤❛ ❡st❛❞♦ ♥❛✈❡❣❛♥❞♦ ❤❛❝✐❛ ❡❧ ❡st❡ ❛ ✶✺ ❦♠✴❤ ② ❧❧❡❣❛
❛❧ ♠✐s♠♦ ♠✉❡❧❧❡ ❛ ❧❛s ✸ ♣✳♠✳ ➽❊♥ q✉é ♠♦♠❡♥t♦ ❡st✉✈✐❡r♦♥ ❧♦s ❜♦t❡s ♠ás ❝❡r❝❛
✉♥♦ ❞❡❧ ♦tr♦❄
s✮
❉♦s ♣♦st❡s✱ ✉♥♦ ❞❡ ✶✷
m ② ♦tr♦ ❞❡ ✷✽ m✱ ❡stá♥ ❛ ✸✵ m ❞❡ ❞✐st❛♥❝✐❛✳ ❙❡ s♦st✐❡♥❡♥
♣♦r ♠❡❞✐♦ ❞❡ ❞♦s ❝❛❜❧❡s✱ ❝♦♥❡❝t❛❞♦s ❛ ✉♥❛ s♦❧❛ ❡st❛❝❛ ❝❧❛✈❛❞❛ ❡♥ ❡❧ s✉❡❧♦ ②
❤❛st❛ ❧❛ ♣❛rt❡ s✉♣❡r✐♦r ❞❡ ❝❛❞❛ ♣♦st❡✳ ➽❆ q✉é ❞✐st❛♥❝✐❛ ❞❡ ❝❛❞❛ ♣♦st❡ ❞❡❜❡
❝♦❧♦❝❛rs❡ ❧❛ ❡st❛❝❛ ♣❛r❛ ✉s❛r ❧❛ ♠❡♥♦r ❝❛♥t✐❞❛❞ ❞❡ ❝❛❜❧❡❄ ❘✴ ❆ ✾
m
❞❡❧ ♣♦st❡
♣❡q✉❡ñ♦✳
t✮
❯♥❛ ❧á♠♣❛r❛ ❡stá ❝♦❧❣❛❞❛ s♦❜r❡ ❡❧ ❝❡♥tr♦ ❞❡ ✉♥❛ ♠❡s❛ r❡❞♦♥❞❛ ❞❡ r❛❞✐♦
r✳
➽❆
q✉é ❛❧t✉r❛ ❞❡❜❡rá ❡st❛r ❧❛ ❧á♠♣❛r❛ ♣❛r❛ q✉❡ ❧❛ ✐❧✉♠✐♥❛❝✐ó♥ ❞❡ ✉♥ ♦❜❥❡t♦ q✉❡
s❡ ❡♥❝✉❡♥tr❛ ❡♥ ❡❧ ❜♦r❞❡✱ s❡❛ ❧❛ ♠❡❥♦r ♣♦s✐❜❧❡❄
◆♦t❛✿ ▲❛ ✐❧✉♠✐♥❛❝✐ó♥ ❡s ❞✐r❡❝t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛❧ ❛❧ ❝♦s❡♥♦ ❞❡❧ á♥❣✉❧♦ ❞❡ ✐♥❝✐❞❡♥❝✐❛ ❞❡ ❧♦s
r❛②♦s ❧✉♠✐♥♦s♦s ❡ ✐♥✈❡rs❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛❧ ❛❧ ❝✉❛❞r❛❞♦ ❞❡ ❧❛ ❞✐st❛♥❝✐❛ ❛❧ ❢♦❝♦ ❞❡ ❧✉③✳
✶✼
✼✳
❇✐❜❧✐♦❣r❛❢í❛
❇❛r❛♥❡♥❦♦✈✱ ●✳ ② ♦tr♦s ✭✶✾✼✼✮✳
Pr♦❜❧❡♠❛s ② ❡❥❡r❝✐❝✐♦s ❞❡ ❛♥á❧✐s✐s ♠❛t❡♠át✐❝♦✳
◗✉✐♥t❛
❡❞✐❝✐ó♥✳ ❊❞✐t♦r✐❛❧ ▼■❘✱ ▼♦s❝ú✳
●♦♥③á❧❡③✱ ❋✳✱
■♥tr♦❞✉❝❝✐ó♥ ❛❧ ❈á❧❝✉❧♦
▲❛rs♦♥✱ ❘✳ ② ♦tr♦s ✭✷✵✵✻✮✳
P✐s❦✉♥♦✈✱ ◆ ✭✶✾✻✾✮✳
❙t❡✇❛rt✱ ❏✳ ✭✶✾✾✹✮✳
❈á❧❝✉❧♦✳
✭✈❡rs✐ó♥ ♣r❡❧✐♠✐♥❛r✮✱ ❊❯◆❊❉✱ ❙❛♥ ❏♦sé✳
▼❝●r❛✇ ❍✐❧❧✱ ▼é①✐❝♦✳
❈á❧❝✉❧♦ ❞✐❢❡r❡♥❝✐❛❧ ❡ ✐♥t❡❣r❛❧✱ ❚♦♠♦ ■✳
❈á❧❝✉❧♦✳
❊❞✐t♦r✐❛❧ ▼■❘✱ ▼♦s❝ú✳
●r✉♣♦ ❊❞✐t♦r✐❛❧ ■❜❡r♦❛♠ér✐❝❛✱ ▼é①✐❝♦✳
✶✽