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## JDM Chap 3 Nemerical Method

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# JDM Chap 3 Nemerical Method

### JDM Chap 3 Nemerical Method

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

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 ex  1 for x < 2. Give your answer correct to 3 decimal places.
2x

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.

BKS 2019/20 Page 2

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

BKS 2019/20 Page 1

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

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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