Math 137 Winter 2010
Assignment 4
Due Friday, February 12
All solutions must be clearly stated and fully justified.
Use the format given on UW-Ace under
Content, in the folder Assignments; it is the file
Math 137 Assignment Templates
.
Text problems:
Section 2.5:
40, 52
Section 2.7: 14, 20, 22, 32
Section 2.8:
12, 26
Section 3.1:
22, 30, 36, 60
Section 3.2:
10, 18, 52
Section 3.3: 10, 14, 34
Section 2.5:
40.
The gravitational force exerted by the earth on a unit mass at a distance r from
the center of the planet is F(r) = GMr/R
3
if f < R, and GM/r
2
if r
≥
R, where M is Earth’s mass, R
its radius and G the gravitational constant.
Is F a continuous function of r?
52.
a) Prove that ln x = 3 – 2x has at least one real root, and b) use your calculator to find an
interval of length 0.01 that contains a root.
a)
Let f(x) = ln x – 3 + 2x; f(1) = – 1 < 0 and f(2) = ln 2 + 1
≈
1.7 > 0.
Since f(x) is continuous
on [1,2] because ln x and any polynomial are, by the IVT there is a real number c
∈
(1, 2) such
that f(c) = 0.
This is a root of ln x = 3 – 2x.
b) f(1.34)
≈
– 0.33 < 0 and f(1.35)
≈
0.0001 > 0, so (1.34, 1.35) is an interval of length 0.01 that
contains a root.
Section 2.7: 14.
If a rock is thrown upward on the planet Mars with a velocity of 10 m/sec, its
height in meters after t seconds is given by H(t) = 10t – 1.86t
2
.
a) Find the velocity of the rock after one second.
h
h
h
h
h
h
h
H
v
h
h
86
.
1
10
86
.
1
72
.
3
86
.
1
10
10
lim
86
.
1
10
)
1
(
86
.
1
)
1
(
10
lim
)
1
(
)
1
(
2
0
2
0
+
−
−
−
−
+
=
+
−
+
−
+
=
′
=
→
→
28
.
6
86
.
1
28
.
6
lim
)
86
.
1
28
.
6
(
lim
86
.
1
72
.
3
10
lim
0
0
2
0
=
−
=
−
=
−
−
=
→
→
→
h
h
h
h
h
h
h
h
h
h
h
m / s
b) Find the velocity of the rock when t = a.

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