Numerical Methods

EXERCISES  2

1)

By using the trapezoidal rule with 6 subintervals, estimate the value of

4x

 1 sin x dx .

0

Give your answer correct to four decimal places.

5 12

2)

Approximate the value of

x

2

 tan x dx by using the Trapezoidal rule with 6

0

subintervals. Give your answer correct to four decimal places.  4

3)

Find an approximate value of

 1 tan x dx by using trapezoidal rule with 6 subintervals.

3

0

Give your answer correct to four decimal places.

2 3

4)

Use the trapezoidal rule to approximate

 1 x

sin x 2

dx with 6 subintervals. Give your

 6

answer correct to four decimal places.

5)

A region R is bounded by the curve y  3x 

5 , the x-axis and the lines x  3 and x  6. x

By using trapezoidal rule with 6 subintervals, estimate the area of R. Give your answer correct to four decimal places.

6)

Find a root of 4xe2x  3  0 correct to three decimal places by applying the NewtonRaphson method with an initial estimate x  0.5 .

7)

By using Newton-Raphson method, determine the root of the equation 2 x  2x 3 correct to three decimal places with the initial estimate of x  1.5 .

8)

 

Find the root of 2x 2  ln 2x 2  3  0 correct to four decimal places by applying the Newton-Raphson method with the initial approximation of x  2 .

Find the root of 2e x  xe x  1  0 correct to three decimal places using Newton-

9)

Raphson method starting at x  1 .

Find the root of 3 sin2x   2e 3 x  x  0 correct to four decimal places by applying the

10)

Newton-Raphson method with the initial estimate of x  1.5 .

ANSWERS :

1) 2) 3) 4) 5)

3.3324 1.9521 1.7001 0.5218 43.9743

6) 7) 8) 9) 10)

0.363 1 1.0507 0.443 -1.91