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.

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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.

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