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Unformatted text preview: Version 208 – Exam 2 – Radin – (58305)
Explanation:
After implicit diﬀerentiation with respect
to x we see that
2xy cos(x2 ) + y ′ sin(x2 ) = 0 .
Consequently,
dy
2xy cos(x2 )
=−
= −2xy cot(x2 ) .
dx
sin(x2 )
004 10.0 points Find an equation for the tangent line to the
curve
6x2 + xy + 6y 2 = 13
at the point (1, 1).
1. y = x + 5
2. y = 3x − 4
3. y = 5x + 2
4. y = −3x + 2
5. y = −5x + 2
6. y = −x + 2 correct
Explanation:
Diﬀerentiating implicitly, we see that
6x2 + xy + 6y 2 = 13
12x + xy ′ + y · 1 + 12yy ′ = 0
xy ′ + 12yy ′ = −12x − y
y ′ (x + 12y ) = −12x − y
−12x − y
y′ =
x + 12y
When x = 1 and y = 1, we have
y′ = −13
−12 − 1
=
= −1
1 + 12
13 so an equation of the tangent line is
y − 1 = −1 (x − 1)
y = −x + 2 2 keywords:
10.0 points 005 If a tank holds 2000 gallons of water, and
the water can drain from the tank in 40 minutes, then Torricelli’s Law gives the volume V
of water remaining in the tank after t minutes
as
2
t
V = 2000 1 −
.
40
Find the rate at which water is draining from
the tank after 30 minutes.
1. rate = 27 gal/min
2. rate = 20 gal/min
3. rate = 22 gal/min
4. rate = 21 gal/min
5. rate = 25 gal/min correct
Explanation:
By the Chain Rule,
1
2000
1− t
20
40
1
= −100 1 − t ,
40 V ′ ( t) = − the negative sign indicating that the volume
is decreasing. Consequently, after 30 minutes
the water is draining from the tank at a
rate = 100 1 −
006 30
40 = 25 gal/min . 10.0 points A 15 foot ladder is leaning against a wall. If
the foot of the ladder is sliding away from the
wall at a rate of 5 ft/sec, at what speed is
the top of the ladder falling when the foot of ...
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 Spring '08
 CLARK,C.W./HOY,R.R

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