Recent questions tagged functions-of-single-variable

Match each entry of $\text{List-1}$ with a suitable entry in $\text{List-2}$ and choose the correct option.$\textbf{List-1}$$\textbf{List-2}$$\mathrm{P}$$\text{The sum of...

Let $y(x)$ be the solution of the initial value problem,$$\begin{array}{c}x^{2} y^{\prime \prime}+x y^{\prime}-y=0, \quad x>0, \\y(1)=0, \quad y^{\prime}(1)=2 .\end{array...

Let\[f(x)=\left\{\begin{array}{cr}\pi+x, & -\pi \leq x<0, \\0, & 0 \leq x<\pi,\end{array}\right.\]with $f(x+2 \pi)=f(x)$. If $F(x)$ represents the Fourier series of $f(x)...

Let $y$ be a non-zero quadratic polynomial satisfying the differential equation\[\left(2+x^{2}\right) \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}-k y=0,\]where $k$ is a rea...

Assume that $f:[0,1] \rightarrow \mathbb{R}$ is continuous on $[0,1]$ and differentiable on $(0,1)$ such that $f(x+h)=f(x)+h f^{\prime}(x+\theta h)$ for some $0<\theta<1$...

The approximate value $\text{(rounded off to two decimal places)}$ of the integral\[\int_{0}^{1 / 2} e^{-x^{2}} d x\]using the Trapezoidal rule with step-size $h=\frac{1}...

If\[\int_{0}^{\alpha} \int_{\sqrt{x / \alpha}}^{1} e^{y^{3}} d y d x=e-1, \quad \alpha>0,\]then the value $\text{(in integer)}$ of $\alpha$ is __________.

If the quadrature formula $\int_{-1}^{1} f(x) d x \approx \frac{1}{9}\left(c_{1} f(-1)+c_{2} f\left(\frac{1}{2}\right)+c_{3} f(1)\right)$is exact for all polynomials of d...

If the polynomial $P(x)=a_{0}+a_{1} x+a_{2} x(x-1)+a_{3} x(x-1)(x-2)$interpolates the points $(0,2),(1,3),(2,2)$, and $(3,5)$, then the value of $P\left(\frac{5}{2}\right...

Which one of the given figures $P, Q, R$ and $S$ represents the graph of the following function?$$f(x)=|| x+2|-| x-1||$$$\text{P}$$\text{Q}$$\text{R}$$\text{S}$

The value of$\lim _{x \rightarrow 0} \frac{1}{x} \int_{2}^{2+x}\left(t+\sqrt{t^{2}+5}\right) d t$$0$$4$$5$$6$

Let $y_{1}(x)$ and $y_{2}(x)$ be two linearly independent solutions of\[x^{2} \frac{d^{2} y}{d x^{2}}-2 x \frac{d y}{d x}+2 y=0, \quad x>0\]Let $W\left(y_{1}, y_{2}\right...

Let $f:(0, \infty) \rightarrow \mathbb{R}$ be the continuous function such that $f(x)=2+\frac{g(x)}{x}$ for all $x>0$, where $g(x)=\int_{1}^{x} f(t) d t$ for all $x>0$. T...

If the quadrature rule$$\int_{-1}^1 f(x) d x \approx f(\alpha)+\gamma f(\beta),$$where $\alpha, \beta$ and $\gamma$ are real constants, is exact for all polynomials of de...