Transcript POWERPOINT JEOPARDY

Partial
Derivatives
Limits &
Continuity
Directional
Derivatives
Chain Rule
Approximations
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QUESTION:
𝜕𝑓 3 2
2
𝑥𝑦
𝑥 𝑦 + sin(𝑥 ) − 𝑦𝑒
𝜕𝑥
ANSWER:
2 2
2
2 𝑥𝑦
3𝑥 𝑦 + 2𝑥 cos(𝑥 ) − 𝑦 𝑒
QUESTION:
𝜕𝑓
𝑥
𝑦2 3
ln 𝑥𝑦 − + 3𝑒 𝑦
𝜕𝑦
𝑦
ANSWER:
𝑥 𝑥
𝑦2 2
4 𝑦2
− 2 + 9𝑒 𝑦 + 6𝑦 𝑒
𝑦 𝑦
QUESTION:
3 2
𝐹𝑥𝑦 𝑥 𝑦 + sin 𝑥
2
− 𝑦𝑒
𝑥𝑦
ANSWER:
6𝑥 2 𝑦 − 𝑥𝑦 2 𝑒 𝑥𝑦 − 2𝑦𝑒 𝑥𝑦
QUESTION:
3 2
𝐹𝑥𝑥 𝑥 𝑦 + sin 𝑥
ANSWER:
3
2
− 𝑦𝑒
𝑥𝑦
at (0,-1)
QUESTION:
𝐹𝑦
csc(𝑦)
ANSWER:
3
1
− cos 𝑦 csc 2 (𝑦)
2
QUESTION:
lim
-18
3
(3𝑥 𝑦 − 𝑥𝑦 )
(𝑥,𝑦)→(1,3)
ANSWER:
2
QUESTION:
2
3𝑥
lim (
)
(𝑥,𝑦)→(1,0) 𝑥 + 𝑦 2
ANSWER:
3
QUESTION:
4
4
𝑥 −𝑦
lim ( 2
)
(𝑥,𝑦)→(0,0) 𝑥 + 𝑦 2
ANSWER:
0
QUESTION:
𝑥𝑦
lim ( 2
)
(𝑥,𝑦)→(0,0) 𝑥 + 𝑦 2
ANSWER:
DNE
QUESTION:
lim
(
(𝑥,𝑦)→(0,0)
ANSWER:
0
𝑥𝑦
𝑥2 + 𝑦2
)
QUESTION:
𝑓 𝑥, 𝑦 = sin 𝑥 − cos 𝑦 ;
𝜋 3𝜋
𝑝=
,
; 𝑎 = −𝒋
4 2
ANSWER:
1
QUESTION:
2
𝑓 𝑥, 𝑦 = 𝑥 𝑦; 𝑝 = 1,2 ; 𝑎
= 3𝒊 − 4𝒋
ANSWER:
8
5
QUESTION:
𝜋
𝑥
𝑓 𝑥, 𝑦 = 𝑒 sin 𝑦 ; 𝑝 = 0, ;
4
𝑎 = 𝒊 + 3𝒋
ANSWER:
( 2 + 6)
≈ 0.9659
4
QUESTION:
Find the unit vector in the direction in which f increases most
rapidly at p:
3
5
𝑓 𝑥, 𝑦 = 𝑥 − 𝑦 ; 𝑝 = 2, −1
ANSWER:
12, −5
13
QUESTION:
Find the plane tangent to
2
𝑓 𝑥, 𝑦, 𝑧 = 𝑥𝑧 − 𝑦𝑧 + 𝑒
at the point (-2,0,-2).
ANSWER:
𝑥
𝑧
− − 3𝑦 − = 2
2
2
𝑦
QUESTION:
𝑧 = 𝑥2𝑦3; 𝑥 = 𝑡3, 𝑦 = 𝑡2
ANSWER:
𝑑𝑧
11
= 12𝑡
𝑑𝑡
QUESTION:
2
𝑧 = ln(𝑥𝑦); 𝑥 = 3𝑡 , 𝑦 = t
ANSWER:
𝑑𝑧 3
=
𝑑𝑡 𝑡
QUESTION:
2
2
𝑧 = 𝑥 𝑦 − 𝑦 𝑥; 𝑥 = cos(𝑡), 𝑦
= sin(𝑡)
ANSWER:
𝑑𝑧
𝑑𝑡
3
2
= sin 𝑡 − 2 sin 𝑡 cos 𝑡
2
3
− 2 cos 𝑡 sin 𝑡 + cos (𝑡)
QUESTION:
𝜕𝑧
Find
:
𝜕𝑡
2
𝑧 = 𝑥 𝑦; 𝑥 = 𝑠𝑡, 𝑦 = 𝑠 − 𝑡
ANSWER:
2
𝑠 𝑡(2𝑠 − 3𝑡)
QUESTION:
Find
3
2
𝑑𝑦
:
𝑑𝑥
3
𝑥 + 2𝑥 𝑦 − 𝑦 = 0
ANSWER:
2
3𝑥 + 4𝑥𝑦
3𝑦 2 − 2𝑥 2
QUESTION:
Find the equation of the tangent plane at the given point.
2
2
2
𝑥 + 𝑦 + 𝑧 = 16; 𝑝 = (2,3, 3)
ANSWER:
2𝑥 + 3𝑦 + 3𝑧 = 16
QUESTION:
Find the equation of the tangent plane at the given point.
2
4
𝑥
𝑦
𝑧=
+
4
4
ANSWER:
𝑥+𝑦−𝑧 =2
QUESTION:
Approximate ∆𝑧 moving from P to Q.
2 3
𝑧 = 2𝑥 𝑦 ; 𝑃 1,1 , 𝑄(0.99,1.02)
ANSWER:
∆𝑧 ≈ 0.08017992
QUESTION:
Find all points where the tangent plane is horizontal.
2
2
𝑧 = 𝑥 − 2𝑥𝑦 − 𝑦 − 8𝑥 + 4𝑦
ANSWER:
(3, −1, −14)
QUESTION:
The radius and height of a right circular cone
are measured with errors of at most 2% and
3%, respectively. Use differentials to estimate
the maximum percentage error in the
calculated volume.
ANSWER:
7%