Unit Matematik
Kolej Matrikulasi Johor
Kementerian Pendidikan Malaysia
JOM-DO-MATHS (JDM)
Chapter 3 Numerical Method ( Set 1 )
1. Show that a root of the equation f x x2 3 3x 7 lies in the interval [3,5].
4x
Taking 4 as the first approximation, apply the Newton-Raphson process to f x to
obtain the root. Give your answer to 2 decimal places.
2. Show that the equation 2x4 5 x has a root between x 1 and x 2 . By taking
x 1.4 as the first approximation, evaluate this root to three significant figures using the
Newton-Raphson method.
3. By using the Newton-Raphson method, solve the equation ex x2 x, with initial value
x0 0.5, correct to three decimal places.
4. Given the equation ex 2 1
x
(a) Show that there is a real root between 1 and 2.
(b) By using Newton- Raphson method, find the root of the equation correct
to three decimals places, taking 1.5 as the first approximation.
5. Show that the equation 2x ln 5 x2 has a root that lies between 0.6 and 0.8. By using
the Newton- Raphson method, determine the root of the equation 2x ln 5 x2 correct
to three decimal places.
BKS 2019/20 Page 1
6. Show that the equation 2x ex 2 0 has a root between x 0 and x 1. Using the
Newton-Raphson method and taking x 0.7 , find the root correct to four decimal
places.
7. Use the Newton-Raphson method to solve the equation 3x cos x 3 0 correct to four
decimal places by use x1 0.5 .
8. Show that the equation x3 2x 3cos x 0 has a root between x 2 and x 1. With
the initial value xo 1, use the Newton-Raphson method to obtain the root, correct to 3
decimal places
9. Sketch the graphs of y = ex and y = 2 – x on the same axes. Hence, find an approximate
solution for ex = 2 – x with 0 < x0 < 1. By using the Newton Raphson method, find the
solution for ex 1 for x < 2. Give your answer correct to 3 decimal places.
2x
10. Given y ln(x 1) and y 4 2x .
a) Sketch the graphs of the two functions on the same coordinate axes.
b) Show that there is a solution of ln(x 1) 2x 4 0 between 1 and 2.
c) By using the Newton-Raphson method, solve the equation in (b) correct to three
decimal places.
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Unit Matematik
Kolej Matrikulasi Johor
Kementerian Pendidikan Malaysia
JOM-DO-MATHS (JDM)
Chapter 3 Numerical Method ( Set 2 )
1
1. By using the trapezoidal rule, find the approximate value for x x 1 dx when n 4 ,
0
correct to four decimal places.
2. Use the trapezoidal rule with five subintervals to find an approximate value for
3 5 dx correct to three decimal places.
2 3x2 2
decimal
4
3. Use the trapezoidal rule with 6 ordinates to estimate 1 sin xdx correct to 2
0
places
4
4. Use the trapezoidal rule with 5 ordinates to approximate cos4 x dx Give your answer
0
correct to three decimal places.
5. Use the trapezium rule with 5 ordinates to find an estimate of value of
2 3cos x dx , giving your answer correct to 4 decimal places.
(2 sin x)2
2
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6. Sketch the graph of y 1 for x 0 . Hence, use the TRAPEZOIDAL RULE with 5
x2 1
ordinates to estimate the area covered by the curve, the line x 1 , the x and y axis. Give
your answer correct to three decimal places .
7. (a) Using the trapezoidal rule with 5 ordinates, estimate the value of
1
x sin x2dx , giving your answer correct to 4 decimal places.
0
1
(b) By using a suitable substitution, find the value of x sin x2dx .
0
Comment on the difference between the answers to part (a) and part (b).
1
8. Evaluate x2e x3 dx correct to three decimal places by using
0
a) an appropriate substitution.
b) the trapezoidal rule with five subintervals.
What is the percentage of error when evaluating the integral using the trapezoidal rule.
Give a suggestion to reduce the error.
BKS 2019/20 Page 2
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