Recent questions and answers in Calculus

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$\dfrac{d}{dx} \left [ \ln (2x) \right ]$ is equal to$1/2x \\$$1/x \\$$1 /2 \\$$x$
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The value of $\underset{x \to 0} \lim \left [ \dfrac{x- \sin 2x}{x-\sin 5x} \right]$ (rounded off to two decimal places) is __________
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If the area of a triangle with the vertices $(k,0), (2,0)$ and $(0, -2)$ is $2$ square units, the value of $k$ is _________
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6
Calculate the following integral $\int \limits_0^{\pi^2/4} \sin \sqrt{x} dx =$____________.
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The solution of $ \underset{x\rightarrow 8 }\lim\left ( \dfrac{x^{2}-64}{x-8} \right )$ is _____________.
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8
A function $f$ is given as :$$f(X)=4X-X^{2}$$The function $f$ is maximized when $X$ is equal to _________________.
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9
An infinite series $S$ is given as:$S=1+2/3+3/9+4/27+5/81+\:.\dots$ (to infinity)The value of $S$ is ______________________ (round off to $2$ decimal places).
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A function $f$ is as follows:$$f(x) = \begin{cases} 15 & \text{if }x<1 \\ cx& \text{if } x\geq 1 \end{cases}$$The function $f$ is a continuous function when $c$ is equal ...
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12
The value of the integral $\displaystyle \int_0^{0.9} \dfrac{dx}{(1-x)(2-x)}$ is ____________
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13
The surface area (in $m^2$) of the largest sphere that can fit into a hollow cube with edges of length $1$ meter is ______Given data: $\pi=3.14$
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14
The angle (in degrees) between the vectors $\overrightarrow{x}= \hat{i}-\hat{j}+2 \hat{k}$ and $\overrightarrow{y} = 2 \hat{i} – \hat{j}-1.5 \hat{k}$ is _________
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Consider the following infinite series:$$1+ r+r^2 +r^3+ \dots \dots \infty$$ If $r = 0.3$, then the sum of this infinite series is ____________
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16
The limit of the function $\bigg (1 + \dfrac{x}{n} \bigg )^n$ as $n \to \infty$ is$\ln x$$\ln \dfrac{1}{x}$$e^{-x}$$e^x$
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17
The limit of the function $e^{-2t}\sin (t)$ as t $\rightarrow\infty$ so, is
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18
If $y =x^x$, then $\frac{dy}{dx}$ is$x^x(x-1)$$x^{x-1}$$x^x(1 + \log x)$$e^x(1 + \log x)$
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20
The graph of the function $F(x) =\frac{x}{k_1x^2+k_2x+1}$ for $0<x<\infty$ is
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21
Evaluate $\underset{x\rightarrow\infty }{\lim}x\tan\frac{1}{x}$$\infty$$1$$0$$-1$
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22
If $u=\log (e^x+e^y),$ then $\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=$$e^x+e^y$$e^x-e^y$$\frac{1}{e^x+e^y}$$1$
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If $1+r + r^2+ r^3 +\dots \infty = 1.5$, then, $1 + 2r + 3r^2 + 4r^3 + \dots \infty = $ (up to two dcimal places) ________
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24
Calculate the following integral (up to two decimal places)$$\displaystyle \int_0^1 (x + 3)(x + 1)dx = \text{ ___________}$$
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25
Which one of the following is the solution for $\cos^2 x + 2 \cos x + 1 = 0$, for values of $x$ in the range of $0^\circ < x < 360^\circ$$45^\circ$$90^\circ$$180^\circ$$2...
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