Graphics Programs Reference
In-Depth Information
7. Apopularmethod of hand computationis the
secant formula
where the improved
estimate of the root (
x
i
+
1
) isobtainedbylinear interpolationbased two previous
estimates (
x
i
and
x
i
−
1
):
x
i
−
x
i
−
1
x
i
+
1
=
x
i
−
f
(
x
i
)
f
(
x
i
)
−
f
(
x
i
−
1
)
SolveProb. 6 using the secantformula.
8. Draw a plot of
f
(
x
)
=
cosh
x
cos
x
−
1inthe range4
≤
x
≤
8
.
(a)Verify from the
plot that the smallest positive, nonzero root of
f
(
x
)
.
(b) Showgraphically that the Newton-Raphson formulawouldnot convergeto
this root if it isstartedwith
x
=
0lies in the interval(4
,
5)
=
4
.
The equation
x
3
2
x
2
De-
terminethis root with the Newton-Raphsonmethodwithin four decimal places.
10.
Write aprogram thatcomputes all the roots of
f
(
x
)
9.
−
1
.
−
8
.
19
x
+
13
.
23
=
0has adouble root close to
x
=
2
.
0 in a giveninterval with
Brent's method. Utilize the functions
rootsearch
and
brent
. You may use the
program in Example 4.3 as amodel. Test the program by finding the roots of
x
sin
x
=
6).
11.
SolveProb. 10 with the Newton-Raphsonmethod.
12.
Determine all real roots of
x
4
+
3 cos
x
−
x
=
0 in (
−
6
,
+
0
.
9
x
3
−
2
.
3
x
2
+
3
.
6
x
−
25
.
2
=
0.
13.
Compute all positive real roots of
x
4
2
x
3
7
x
2
+
−
+
3
=
0
.
14.
Find all positive, nonzero roots of sin
x
−
0
.
1
x
=
0
.
15.
The naturalfrequencies of a uniform cantileverbeam are related to the roots
β
i
of the frequency equation
f
(
β
)
=
cosh
β
cos
β
+
1
=
0
,
where
f
i
)
2
mL
3
E I
4
i
β
=
(2
π
f
i
=
i
th naturalfrequency (cps)
m
=
mass of the beam
L
=
length of the beam
E
=
modulusofelasticity
I
=
momentofinertia of the cross section
Determine the lowest twofrequencies of a steel beam 0.9 m long, with arectan-
gular cross section25mmwide and 2.5 mm in. high. The mass density of steel is
7850kg/m
3
and
E
=
200 GPa.
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