Quiz Chapter 07.04: Romberg Method – Holistic Numerical Methods

1. If $I_{n}$ is the value of integral $\, \displaystyle\int_{a}^{b} f(x)dx$ using $n$ -segment Trapezoidal rule, a better of the integral can be found using Richardson’s extrapolation as

I_{2n} + \dfrac{I_{2n} - I_{n}}{15}

I_{2n} + \dfrac{I_{2n} - I_{n}}{3}

I_{2n}

I_{2n} + \dfrac{I_{2n} - I_{n}}{I_{2n}}

2. The estimate of an integral of $\, \displaystyle\int_{3}^{19} f(x)dx$ is given as $1860.9$ using $1$ -segment Trapezoidal rule. Given $f(7)=20.27, \, f(11)=45.125,$ and $f(14)=82.23$ , the value of the integral using $2$ -segment Trapezoidal rule would most nearly be

787.32

1072.0

1144.9

1291.5

3. The value of an integral $\, \displaystyle\int_{a}^{b} f(x)dx$ given using $1-, \, 2-,$ and $4$ -segment Trapezoidal Rule is given as $5.3460, \, 2.7708,$ and $1.7536$ , respectively. The best estimate of the integral you can find using Romberg Integration is most nearly

1.3355

1.3813

1.4145

1.9124

4. Without using the formula for one-segment Trapezoidal Rule for estimating $\, \displaystyle\int_{a}^{b} f(x)dx$ the true error, $E_{t}$ , can be found directly as well as exactly by using the formula

- - $E_{t} = -\dfrac{ \left( b - a \right)^{3}}{12} {f}''(\xi), \, a \leq \xi \leq b$

f(x)=e^{x}

f(x) = x^{3} + 3x

f(x) = 5x^{2} + 3

f(x) = 5x^{2} + e^{x}

5. For $\, \displaystyle\int_{a}^{b} f(x)dx$ , the true error, $E_{t}$ , in $1$ -segment Trapezoidal Rule is given by

- - $E_{t} = - \dfrac{ \left( b - a \right)^{3}}{12} {f}'' ( \xi ), \, a \leq \xi \leq b$

The value of $\xi$ for the integral $\, \displaystyle\int_{2.5}^{7.2} 3e^{0.2x}dx$ is most nearly

2.7998

4.8500

4.9601

5.0327

6. Given the velocity vs time data for a body

$t(s)$	$2$	$4$	$6$	$8$	$10$	$25$
$v$ (m/s)	$0.166$	$0.55115$	$1.8299$	$6.0755$	$20.172$	$8137.5$

The best estimate for distance covered between $2$ s and $10$ s by using the Romberg Rule based on the Trapezoidal Rule’s results would be

33.456

36.877

37.251

81.350