Recent questions tagged calculus

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The function $\cos(x)$ is approximated using Taylor series around $x=0$ as $\cos(x) \approx 1 + ax + bx^2 + cx^3 + dx^4$. The values of $a,b,c$ and $d$ are$a=1, \: b=-0....
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A three-dimensional velocity field is given by $V=5x^2y \: i + Cy \: j-10xyz \: k$, where $i,j,k$ are the unit vectors in $x,y,z$ directions, respectively, describing a c...
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For the function $f(x) = \begin{cases} -x, & x<0 \\ x^2, & x \geq 0 \end{cases}$ the $\text{CORRECT}$ statement(s) is/are$f(x)$ is continuous at $x=1$$f(x) $ is different...
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To solve an algebraic equation $f(x)=0$, an iterative scehme of the type $x_{n+1} = g(x_n)$ is proposed, where $g(x)=x-\dfrac{f(x)}{f’(x)}$.At the solution $x=s,\: g’...
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The maximum value of the function $f(x)=-\dfrac{5}{3} x^3 +10x^2-15x+16$ in the interval $(0.5,3.5)$ is$0$$8$$16$$48$
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Consider the following continuously differentiable function $$\textbf{v}(x,y,z)=3x^2y \textbf{ i} + 8y^2z \textbf{ j} + 5xyz \textbf{ k}$$ where $\textbf{i, j,}$ and $\te...
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The value of the expression $\underset{x\rightarrow \frac{\pi }{2}}{\lim}\: \mid \frac{\tan\:x}{x} \mid $ is $\infty$$0$$1$$-1$
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If $x,y$ and $z$ are directions in a Cartesian coordinate system and $i$, $j$ and $k$ are the respective unit vectors, the directional derivative of the function $u\left ...
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The value of $\underset{x\rightarrow 0}{\lim}\:\frac{\tan\left ( x \right )}{x}$ is ________________.
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The number of positive roots of the function $f(x)$ shown below in the range $0<x<6$ is ___________________.
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Let $i$ and $j$ be the unit vectors in the $x$ and $y$ directions, respectively. For the function $F\left ( x,y \right )=x^{3}+y^{2}$ the gradient of the function. i.e.. ...
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The Lagrange mean-value theorem is satisfied for $f\left ( x \right )=x^{3}+5$, in the interval $\left ( 1,4 \right )$ at a value (rounded off to the second decimal place...
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The following set of the three vectors$$\begin{pmatrix} 1\\2\\1 \end{pmatrix}, \begin{pmatrix} x\\6\\x \end{pmatrix}\:and \:\begin{pmatrix} 3\\4 \\2 \end{pmatrix},$$is li...
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A scalar function in the $xy$-plane is given by $\phi \left ( x,y \right )=x^{2}+y^{2}$. If $\hat{i}$ and $\hat{j}$ are unit vectors in the $x$ and $y$ directions, the d...
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A vector $u=-2y\hat{i}+2x\hat{j}$, where $\hat{i}$ and $\hat{j}$ are unit vectors in $x$ and $y$ directions, respectively. Evaluate the line integral$$I=\oint _{C}u.dr$$w...
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Integral of the time-weighted absolute error $(ITAE)$ is expressed as$\int _{0}^{\infty }\frac{\left | \varepsilon \left ( t \right ) \right |}{t^{2}}dt$$\int _{0}^{\inft...
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If $f*(x)$ is the complex conjugate of $f(x)=\cos(x) + i\: \sin(x)$, then for real $a$ and $b$, $\int _{a}^{b}f*\left ( x \right )f\left ( x \right )$ is $ALWAYS$positive...
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Gradient of a scalar variable is alwaysa vectora scalara dot productzero
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An equation of state is explicit in pressure $p$ and cubic in the specific volume $v$. At the critical point $‘c’$ , the isotherm passing through $‘c’$ satisfies$...
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Evaluate ${\displaystyle \int \frac{dx}{e^x – 1}}$(Note: C is a constant of integration)$\frac{e^x}{e^x -1}$ + C$\frac {In(e^x -1)}{e^x}$ + CIn$(\frac {e^x}{e^x -1})$ ...
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If $a$ is a constant, then the value of the integral $a^{2}\int^\infty_0 xe^{-ax}dx$ is$1/a$$a$$1$$0$
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For the function $f(t) = e^{-t}/\tau$,the Taylor series approximation for $t\ll$$\tau$ is$1+\frac{t}{\tau}$$1-\frac{t}{\tau}$$1-\frac{t^2}{2\tau^2}$$1+t$
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If $y=e^{-x^{2}}$ then the value of $\underset{x\rightarrow \infty }{\lim}\frac{1}{x}\frac{dy}{dx}$ is __________
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The figure which represents for $y=\frac{sin \:x}{x}$ for $x>0$ ($x$ in radians) is
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