Numerical Methods

 Question 1
A function $f(x)$, that is smooth and convex-shaped between interval $\left(x_{1}, x_{u}\right)$ is shown in the figure. This function is observed at odd number of regularly spaced points. If the area under the function is computed numerically, then ____

 A the numerical value of the area obtained using the trapezoidal rule will be less than the actual. B the numerical value of the area obtained using the trapezoidal rule will be more than the actual. C the numerical value of the area obtained using the trapezoidal rule will be exactly equal to the actual. D with the given details, the numerical value of area cannot be obtained using trapezoidal rule
GATE CE 2023 SET-1   Engineering Mathematics
Question 1 Explanation:
Approximated function has under estimation so numerical value of the area obtained using trapezoidal rule will be less than the actual.

 Question 2
Consider the definite integral
$\int_{1}^{2}(4x^2+2x+6)dx$
Let $I_e$ be the exact value of the integral. If the same integral is estimated using Simpson's rule with 10 equal subintervals, the value is $I_s$. The percentage error is defined as $e=100 \times (I_e-I_s)/I_e$. The value of $e$ is
 A 2.5 B 3.5 C 1.2 D 0
GATE ME 2022 SET-2   Engineering Mathematics
Question 2 Explanation:
Exact value
\begin{aligned} &=\int_{1}^{2} (4x^2+2x+6)dx\\ &=\frac{4x^3}{3}+\frac{2x^2}{2}+6x\\ &=\frac{4}{3}(\pi) \times (3)+6\\ &=\frac{28}{3}+9=\frac{55}{3} \end{aligned}
Approximate value
Here f(x) is a polynomial of degree 2, so Simpsons rule gives exact value with zero error
\begin{aligned} \therefore \;\; \text{Approx value}&=\frac{55}{3}\\ \frac{I_e-I_s}{I_e}&=0\\ \therefore \;\;e=\left (\frac{I_e-I_s}{I_e} \right )\times 100&=0 \end{aligned}

 Question 3
Let, $f(x,y,z)=4x^2+7xy+3xz^2$. The direction in which the function $f(x,y,z)$ increases most rapidly at point $P = (1,0,2)$ is
 A $20\hat{i}+7\hat{j}$ B $20\hat{i}+7\hat{j}+12\hat{k}$ C $20\hat{i}+12\hat{j}$ D $20\hat{i}$
GATE EE 2022   Engineering Mathematics
Question 3 Explanation:
Given: $f(x,y,z)=4x^2+7xy+3xz^2$
The directional derivative at point P is given by
$=\triangledown f|_{point \; P}$
$\therefore \; \triangledown f=(8x+7y+3z^2)\hat{i}+(0+7x+0)\hat{j}+(0+0+6xz)\hat{k}$
at point (1, 0, 2)
$\triangledown f|_{(1,0,2)}=20\hat{i}+7\hat{j}+12\hat{k}$
 Question 4
Consider the following recursive iteration scheme for different values of variable P with the initial guess $x_1=1$:
$x_{n+1}=\frac{1}{2}\left ( x_n+\frac{P}{x_n} \right ),\;\;\;n=1,2,3,4,5$
For $P=2,x_5$ is obtained to be 1.414, rounded-off to three decimal places. For $P=3,x_5$ is obtained to be 1.732, rounded-off to three decimal places.
If $P=10$, the numerical value of $x_5$ is _________ . (round off to three decimal places)
 A 2.155 B 3.162 C 1.125 D 4.568
GATE CE 2022 SET-1   Engineering Mathematics
Question 4 Explanation:
$x_{n+1}=\frac{1}{2}\left ( x_n+\frac{P}{x_n} \right )$
Converges when $x_{n+1}=x_{n}=\alpha$
\begin{aligned} \alpha &=\frac{1}{2}\left ( \alpha+\frac{P}{\alpha} \right )\\ \frac{\alpha}{2}&=\frac{P}{2\alpha}\\ \alpha&=\sqrt{P} \end{aligned}
When $P=2, x_5=\sqrt{2}=1.4124$
When $P=3, x_5=\sqrt{3}=1.732$
When $P=10, x_5=\sqrt{10}=3.162$
 Question 5
Numerically integrate, $f(x)=10 x-20 x^{2}$ from lower limit a=0 to upper limit b=0.5. Use Trapezoidal rule with five equal subdivisions. The value (in units,round off to two decimal places) obtained is ____________
 A 0.78 B 0.65 C 0.4 D 0.56
GATE CE 2021 SET-2   Engineering Mathematics
Question 5 Explanation:
\begin{aligned} y&=10 x-20 x^{2} \\ a&=0, b=0.5, n=5 \\ \text { So, } \qquad \qquad \qquad h&=\frac{b-a}{n}=0.1 \end{aligned}
And

\begin{aligned} I &=\int_{0}^{0.5} f(x) d x=\frac{h}{2}\left[y_{0}+y_{5}+2\left(y_{1}+y_{2}+y_{3}+y_{4}\right)\right] \\ &=\frac{0.1}{2}[0+0+2(0.8+1.2+1.2+0.8)] \\ &=0.40 \end{aligned}

There are 5 questions to complete.

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