Exercise 2: Roots of Equations Equa tions 1)
Numerical Methods Method s
Determine the root of 2 0.5 x 2 3 x using bisection method. Given the initial guesses are 1, 2,3 and 4 .Decide the appropriate lower and upper bound that bracket the root. Hence, carry out the computation until
a
10% . [Answer : 2.75 ]
2)
Find a root of f ( x) x 3 2 x 3 in the range 0 x 7 using the false position method for 5
a
5% . [Answer : 0.98814]
3)
Perform three iterations iteration s of iterative method to find a real root of the equation cos x xe x
which lies between [0,1] using (a) (b)
bisection method met hod false position method [Answer : a) 0.6250 b) 0.4940 ]
Universiti Malaysia Pahang
Exercise 2: Roots of Equations 4)
Numerical Methods
The upward velocity of a rocket can be computed by the following formula:
v u ln
m0 m0 qt
gt
where v = upward velocity, u = the velocity at which fuel is expelled to the rocket, m0 = the initial mass of the rocket at time t = 0, q = the fuel consumption -2
rate and g = the downward acceleration of gravity (assumed constant = 9.81ms ). If u 2000ms-1 , m0 150,000kg and q 2700kgs-1 , compute the time at which v 750ms . (Hint: t is somewhere between 20 and 30 s). Determine your result -1
using bisection method so that
a
4% . [Answer : 20.625]
5)
You are designing a spherical tank to hold water for a small village in developing country. The volume of liquid it can hold can be computed as V h
2
[3 R h ] 3
where V = volume, h = depth of water in tank, a nd R = the tank radius. I f
R 3m , to what depth must the tank be filled so that it holds 30 m 3 ?Compute three iterations of the false position method to determine your Answer . Determine the approximate relative error for each iteration. Employ initial guesses of 0 and R . (hint : use 3.142 ) [Answer : h 2.0237 , a 1.8432% ]
6)
Determine the lowest real root of f ( x) 2x3 11.7 x 2 17.7 x 5
using Newton Raphson method with x0 0.3 and perform three iterations. Compute a for each approximation. [Answer : lowest root ≈ 0.3651 ,
Universiti Malaysia Pahang
a
0.0035% ]
Exercise 2: Roots of Equations
7)
Numerical Methods
TEST 1 BUM 2313 1112I
Use Secant Method to estimate the root of f ( x ) e x x 2 .
Start with initial estimates x1 0.5 and x0 1 . Perform the computation until a
8)
3%.
[ Answer : 0.70095]
x Determine the lowest positive root of f ( x ) 8e sin( x ) 1
(a) (b)
using the Newton Raphson method (three iterations, xi=0.3) using the secant method (three iteration, xi-1=0.5 and xi=0.4) [Answer : a) 0.14501 b) 0.15805 ]
9)
The lateral surface area, S ,of a cone is given by:
S r r 2 h2 where r is the radius of the base and h is the height. Determine the radius of a cone which has a surface area of 1200 m 2 and a height of 20 m , by calculating the first three iterations using the newton raphson method. Use r 0 17 m. [ Answer : 15.2041]
FINAL EXAM
10)
BETto 2553 1011I I The concentration of pollutant bacteria, c in a lake decreases according
1.5 t 0.075 t c 75e 20e
Determine the time required for the bacteria concentration to be reduced to 15 using secant method with an initial guess of t 1 4 and t 0 5 . Calculate until second iterations only ( i 0,1, 2 ). Find f t i and
a after
each iterations.
[Answer: Time required = 4.0016, f t 1 0.0023, f t 0 1.2127,
f t 1 0.0004, f t 2 0,
Universiti Malaysia Pahang
a , 0
20%, a,1 24.9407%,
Exercise 2: Roots of Equations
Numerical Methods TEST2 BUM 2313 1213II
Determine the root of f ( x) 2 e x sin x 1 using the Newton Raphson method
11)
with three iteration and x0 0.5 (Hint: Use radian mode) [Answer : -0.3573 ]
TEST2 BUM 2313 1213II
12)
A flat of mass m failing freely in air with a velocity V is subject to downward gravitational force and an upward frictional drag force due to air. The drag force F D is given by the expression
F D
0.3V 2 500 (ln V )3
0.02V
Terminal velocity is reached when the drag force equals the gravitational force, that is
F F D mg 0
(a)
Find the terminal velocity using the Bisection method if m = 1 kg and g = 9.8 m/s2. Use an initial interval of V l 1 and V u 200 m/s. Show your work for computing the first three iterations.
(b)
Compute the approximate percent relative error,
a for
each iteration. 2
[Answer : (a) 175.125 m/s (b) 14.20% ]
Universiti Malaysia Pahang
Exercise 2: Roots of Equations
Numerical Methods
SUPPLE M ENTA RY EXERCI SES
BISECTION M METHOD
1.
Determine the first root of f ( x) 14 20 x 19 x 2 3x 3 using bisection method with initial guesses of xl 1 and xu 2.5 and a stopping criterion of 15%. [Answer : root 1.9375,
2.
a
Determine the positive real root of ln( x 4 ) 0.7 using three iterations of the bisection method with initial guesses of xl 0.5 and x u
2.
[Answer : root 1.0625,
3.
= 9.6774%]
a
=17.65%]
Water is flowing in a trapezoidal channel at a rate of Q = 20ms 3 . The critical depth y for such a channel must satisfy the equation
0
1
Q2 B gAc3
FINAL EXAM BMM2112
where g = 9.81ms , A c = the cross-sectional area (m2), and B = the width of the channel at the surface (m). For this case, the width and the cross -sectional area can be related to depth y by B = 3+y and A c = 3y + y 2/2. Solve for the critical depth using bisection method with initial guesses of yl = 0.5 and yu = 2.5, and iterate until the approximate error falls below 1 % or t he number of iteration exceeds 10. [Answer : root 1.5078125, a =0.52%] 2
4.
In environmental engineering (a specialty area in civil engineering), the following equation can be used to compute the oxygen level c (mg/L) in a river downstream from a sewage discharge: c 10 20e 0.15 x e 0.5 x where x is the distance downstream in kilometers. Determine the distance downstream where the oxygen level first falls to a reading of 5 mg/L. Use an appropriate numerical method studied in this course to determine your Answer with error must be less than 50%. (Hint: Use x 0 km and x 2 km as initial distance downstream.) [Answer: root 0.75 ]
Universiti Malaysia Pahang
Exercise 2: Roots of Equations 5.
Numerical Methods
You have a spherical storage tank containing oil. The tank has a diameter of 6 ft. You are asked to calculate the height, h , to which a dipstick 8 ft long would be wet with oil when immersed in the tank when it contains 4 ft 3 of oil. TEST 1 BUM 2313 1011I I
Spherical storage tank problem The equation that gives the height, h , of liquid in the spherical tank for the given volume and radius is given by:
f (h) h3 9h2 3.8197 Use bisection method of finding roots of equations to find the height, h ,to which the dipstick is wet with oil. Conduct three iterations to estimate the root of the above problem and calculate
a
at the end of each iteration.
(Hint: The dipstick would be wet between h 0 and h 2r ,where r is the radius of the tank. Also note that diameter = 2r ) [Answer : root 0.75 ] 6.
Determine the lowest real root of TEST 1 BMM2112 1011I
f ( x) 2x 11.7 x 17.7 x 5 3
2
Use two iterations of the bisection method. Compute the estimated error a and the true error
t
for each approximation. Determine the initial guesses by using
graphical method with scale 1:1 in the interval [0, 4]. [Answer : root 0.5,
Universiti Malaysia Pahang
a
=100%,
t =
31.53%]
Exercise 2: Roots of Equations
Numerical Methods
FALSE-POSI TION M METHOD
7.
Use false-position to find the zero of y 3 x 3 19 x 2 20 x 13
TEST 1 BUM 2313 1112I
on the interval [3,5] with stopping criterion . Use four decimal places in your calculation. [Answer : root 4.6256, a =10.6418% ] 8.
Verify that the function f ( x ) x 2 sin x 2 x 3 has exactly one root in [0,2]. Find this root using the method of false position with three iterations. [Answer : root 1.0330]
9.
The velocity v of a falling parachutist is given by v
gm c
(1 e
( c m ) t
)
FINAL EXAM BUM 2313 1011II
where g 9.8ms 2 . For a parachutist with a drag coefficient c 15kgs 1 , compute 1 the mass, m, so that the velocity is v 35ms at t 9s. Use the false-position
10.
method with ml 50 kg and mu 70 kg to determine m to a level of
s
[Answer : root 59.88461,
a
2%. =1.051%]
2 3 4 5 Given f ( x) 26 85 x 91x 44 x 8 x x . Use false position method to
determine the root to s 0.2% . Employ initial guesses of xl 0.5 and xu 1.0. [Answer : root 0.55705,
Universiti Malaysia Pahang
a
=0.051%]
Exercise 2: Roots of Equations
Numerical Methods
NEWTON R R APHSON M METHOD
11.
In ocean engineering, the equation for a reflected standing wave in a harbor is given by
FINAL EXAM BAM 2012
2 x 2 tv x cos e
0.5 sin
where
16,
t 12 and v 48 . Solve for the lowest positive value of x by
using Newton Raphson method with an initial guess, x 0 1 and stopping criterion of 5%. [Answer : x2 = 49.2062,
12.
13.
=1.892%]
Compute three iterations of Newton Raphson method to find the root of the given equation: (a)
f ( x) x 3 x 1 with x0 1.5.
[Answer : x3 = 1.32471 ]
(b)
f ( x) sin x 1 x 3 with x0 1.5.
[ Answer : x3 = -1.2491 ]
(c)
xe x 2 0 with x0 0.7.
[Answer : x3 = 0.85261 ]
2 Use Newton Raphson method to determine a root of f ( x ) x 1.8 x 2.5
using xo=5. Perform the computation until
a
is less than
s
0.05%.
[Answer : 2.719341 ,
14.
a
a
0.00% ]
x Determine the lowest positive root of f ( x) 8 sin( x)e 1 using the Newton
0.3. Raphson method with three iterations and x0 [Answer : 0.145012,
Universiti Malaysia Pahang
a
= 1.052%]
Exercise 2: Roots of Equations
Numerical Methods
SEC ANT M METHOD
15.
Find the solution of the equation
TEST 1 BAM 2012 1011I
1 0.08 tan 0 0.08 tan
0.8 tan
using secant method with 0 and 0.5 (in radians) as the initial approximations for .
Perform the computation until three iterations with
a for
each iteration.
[Answer : f = 0.3805,
16.
a
=20.8475% ]
Do three steps of the secant method for f ( x) x 3 2 , using x1 0 and x0 1. [Answer : 1.2097]
17.
If the secant method is used to find the zeros of f ( x) x3 3 x 2 2 x 6 with x1 1 and x0 2 , what is x2 ? [Answer : 2.3]
18.
The concentration of pollutant bacteria c in a lake decreases according to c 75e 1.5t 20e 0.075 t Determine the time required for the bacteria concentration to be reduced to 15 using secant method with t 1 3 and t 0 3.5 . Perform the computation until a
0.5% . [Answer : t 4.0013 and
19.
a
0.3% ]
The fourth degree polynomial f ( x) x 4 3x 2 6 x 2 has zero in x1 1 and x0 0 . Attempt to approximate this zero by performing three iterations of secant method. 0.4112 ] [Answer : x3
Universiti Malaysia Pahang
Exercise 2: Roots of Equations 20.
Numerical Methods
You are working for a start-up computer assembly company and have been asked to determine the minimum number of computers that the shop will have to sell to make a profit. The equation that gives the minimum number of computers to be sold after considering the total costs and the total sales is f (n) 40n1.5 875n 35000 Use the secant method of finding roots of equations to find the minimum number of computers that need to be sold to make a profit. Conduct three iterations to estimate the root of the above equation. Find the absolute relative approximate error at the end of each iteration. Let us take the initial guesses of the root of f (n) 0 as
n1 25 and n0 50. [Answer : n3 = 62.690 ,
Universiti Malaysia Pahang
a
0.19425% ]
