Recent questions tagged numerical-methods

A fifth-degree polynomial in $x$ is defined for $x>0$. All coefficients of the polynomial are positive. The first derivative of the polynomial is obtained numerically at ...

Starting with the first approximation as $x=0.5$, the second approximation for the root of the following function by the Newton-Raphson method is $\_\_\_\_$ (rounded off ...

Values of $y$ for different values of $x$ are tabulated below.\[\begin{array}{|c|c|c|c|}\hlinex & -2 & 1 & 2 \\\hliney & 28 & 4 & 16 \\\hline\end{array}\]If a second-degr...

Consider the differential equation given below. Using the Euler method with the step size $(h)$ of $0.5$, the value of $y$ at $x=1.0$ is equal to $\_\_\_\_\_$ (rounded of...

​​​​​The second derivative of a function $f$ is computed using the fourth-order Central Divided Difference method with a step length $h$.The CORRECT expression for the se...

Activated carbon is used to remove a pollutant from wastewater in a mixed batch reactor, which follows first-order reaction kinetics.At a reaction rate of $0.38 /$ day, t...

A function $f(x)$, that is smooth and convex-shaped between interval $\left(x_l, x_u\right)$ is shown in the figure. This function is observed at odd number of regularly ...

The differential equation,$$\frac{\mathrm{d} u}{\mathrm{~d} t}+2 t u^2=1,$$is solved by employing a backward difference scheme within the finite difference framework. The...

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...

Consider the differential equation $$\frac{\mathrm{dy} }{\mathrm{d} x} = 4 \left ( x + 2 \right ) - y$$ For the initial condition $y = 3$ at $x = 1$, the value of $y$ at ...

The value of $\int_{0}^{1}\:e^{x}\:dx$ using the trapezoidal rule with four equal subintervals is$1.718$$1.727$$2.192$$2.718$

The value of abscissa $(x)$ and ordinate $(y)$ of a curve are as follows:$$\begin{array}{|cl|cI|}\hline&x & y\\ \hline & \text{$2.0$} & \text{$5.00$} \\ \hline & \text{$2...

The true value of $\ln(2)$ is $0.69$. If the value of $\ln(2)$ is obtained by linear interpolation between $\ln(1)$ and $\ln(6)$, the percentage of absolute error (round ...

The integral $\int\limits_{0}^{1} (5x^3 + 4x^2 + 3x + 2) dx$ is estimated numerically using three alternative methods namely the rectangular,trapezoidal and Simpson’s rul...

The diameter and height of a right circular cylinder are $3\: cm$ and $4\: cm$, respectively. The absolute error in each of these two measurements is $0.2\: cm$. The abso...

The value of the function $f(x)$ is given at $n$ distinct values of $x$ and its value is to be interpolated at the point $x^*$, using all the $n$ points. The estimate is ...

The quadratic approximation of $f(x)=x^3 – 3x^2 -5$ at the point $x=0$ is$3x^2 -6x-5$$-3x^2-5$$-3x^2+6x-5$$3x^2-5$

Newton-Raphson method is to be used to find root of equation $3x-e^x+ \sin x=0$.If the initial trial value for the root is taken as $0.333$, the next approximation for th...

In newton-Raphson iterative method, the initial guess value $(x_{\text{ini}})$ is considered as zero while finding the roots of the equation: $f(x) = -2+6x-4x^2+0.5x^3$. ...

For step-size, $\Delta x =0.4$, the value of following integral using Simpson’s $1/3$ rule is _______.$$\int_0^{0.8} (0.2 + 25 x – 200 x^2 + 675 x^3 – 900 x^4 +400 x^5) d...

The integral $\int_{x_{1}}^{x_{2}}x^{2}\:dx$ with $x_{2}>x_{1}>0$ is evaluated analytically as well as numerically using a single application of the trapezoidal rule. If ...

The quadratic equation $x^2 – 4x +4 =0$ is to be solved numerically, starting with the initial guess $x_0=3$. The Newton-Raphson method is applied once to get a new estim...

Find the magnitude of the error (correct to two decimal places) in the estimation of following integral using Simpson’s $\dfrac{1}{3}$ Rule. Take the step length as $1$ _...

There is no value of $x$ that can simultaneously satisfy both the given equations. Therefore, find the ‘least squares error’ solution to the two equations, i.e., find the...

The error in $\dfrac{d}{dx} f(x) \mid_{x=x_0}$ for a continuous function estimated with $h=0.03$ using the central difference formula $\dfrac{d}{dx} f(x) \mid_{x=x_0} \ap...

The estimate of $$\int_{0.5}^{1.5} \dfrac{dx}{x}$$ obtained using Simpson’s rule with three-point function evaluation exceeds the exact value by$0.235$$0.068$$0.024$$0.01...