Version 098 – EXAM 2 – spice – (52890)
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18
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00110.0pointsFind the absolute minimum value offonthe intervalbracketleftbigg12,2bracketrightbiggwhenf(x) =x2+2x1.
1.none of the other answers2.abs. min. value = 2correct
3.abs. min. value =52
on the interval
bracketleftbigg
1
2
,
2
bracketrightbigg
.
00210.0pointsLetfbe a twicedifferentiable function on(∞,∞) such that the equationf(x) = 0has exactly 3 real roots, all distinct. Considerthe following possibilities:A. the equationf′(x) = 0
has at least 1 root.Which of these properties willfhave?
Version 098 – EXAM 2 – spice – (52890)
2
and
2
4
6

2

4

6
2
4
6

2

4

6
From these and the MVT we thus see that
A. True: apply MVT to roots of
f
(
x
) = 0.
B. True: by MVT
f
has at least two critical
points. Apply MVT to these.
003
10.0points
The height of a triangle is increasing at a
rate of 5 cm/min while its area is increasing
at a rate of 3 sq. cms/min.
At what speed is the base of the triangle
changing when the height of the triangle is 3
cms and its area is 9 sq. cms?
Thus by the Product Rule,
dA
dt
=
1
2
parenleftBig
b
dh
dt
+
h
db
dt
parenrightBig
,
and so
db
dt
=
1
h
parenleftBig
2
dA
dt

b
dh
dt
parenrightBig
=
2
h
parenleftBig
dA
dt

A
h
dh
dt
parenrightBig
,
since
b
= 2
A/h
. Thus, when
dh
dt
= 5
,
and
dA
dt
= 3
,
we see that
db
dt
=
2
h
parenleftBig
3

5
A
h
parenrightBig
cms/min
.
Consequently, at the moment when
h
= 3
and
A
= 9
,
the base length is changing at a
speed = 8 cms/min
(recall: speed is nonnegative).
004
10.0points
Version 098 – EXAM 2 – spice – (52890)
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 Fall '09
 Calculus