2nd Level
Numerical Analysis
Sheet No.: 5
Title: Newton-Raphson
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Textbook: Numerical Methods for Engineers by: Steven C. Chapra, Raymond P.
Canal. McGraw-Hill.
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By using Newton-Raphson method, find one root to the following
equations with an error less than 0.005%
(1) 3x − e x + sin x = 0 using x0 = 2
(2) 2 x tan x = 1 using x0 = ½
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(3) By using Newton-Raphson method, find an iterative formula to find the
value of m where m is a positive real number and hence use this formula
to approximate the following square roots with an error less than 0.001 %.
(a) 5 (b) 12
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gm
(4) The velocity v of a failing parachutist is given by v =
c
(1 − e−ct m )
where g = 9.8 m s . For a parachutist with a drag coefficient c = 15 kg s ,
2
compute the mass m so that the velocity is v = 35 m s at t = 9 s . Use the
Newton-Raphson method with m0 = 50 to find the value of the mass with
an error less than 0.1% . Compare the results between results produced by
graphically, bisection, and Newton Raphson.
Extra Problems:
Exercise of Chapter 6 p.171 numbers; 2(c), 3(b), 4(b), 5, 11.
1
Notes
examine sinx cost
costs six
Radin tax seen
fix 3X et sinx Xr Eat atx six
fix 3 extcosx 2 sect seaxtanx
5.257010
Xml Xr
Is 1.900158499 cscxs cscxc.tt
189012709 0.529010
1.890029738 0.0052901
1 890029729 0.000004
fix 2x tax l 2xtax 1 0
fixiscaxieseexactaxies
Arm xr seYih
3 Xr Eat
42
0.6897481217 27.5010
0.6549793363 5.31010
0.6532750374 0.2606
0.6532711871 0.0005
af jam X 5 fix X s o Xupper 59 3 Xlower542 inituilvale 2.5
satin
Xr Xr
Ys exactsolution2.23606
Xr Ead Et lot
2.5 11.8
2 25 11.11010 0.62010
223611 0.621010 0.002301
2.236067 0.0019 0.00031010
b
72 12 fix X 12 0 Tupper 16 4 Mower a 3 initialuate 3.5
fix s 2x
exactsolution 3464101615
Yr Xr 412 Xr Ead Et lot
3.5 1.036010
3,46428 1.0310010 0.005101
3.46410162 0.0051010 0.00000010k
3.464101615 0.0000001 0
I
initialpointmo s 50 Xu x nos Bi sections wides
v
ga l É a axis We
35 98,5m l e fonts m.me 53.57
no Egyptian
mCe mett
qysJ som
53.57sm me fins l m É 1m34 e
fu s m met53.57 0 l l e
fins I I é É Jew 06 1
wi m in
Xr Eat
50
59.223 15.5010
59.836 1.02010
59.838 0.0034010 0.1 to
simple fixed point
fixis e X 618,41666 6 1 11 di
X e
