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Recent questions tagged medium

Verify mean value theorem for each of the function $f(x)=x^3-2x^2-x+3\;in\;[0,1].$

asked Jan 2, 2013 by sreemathi.v

1 answer

Using Rolle's theorem,find the point on the curve $y=x(x-4),x\in[0,4]$,where the tangent is parallel to $x$-axis.

asked Jan 2, 2013 by sreemathi.v

1 answer

Find the points on the curve $y=(\cos x-1)$ in [0,2$\pi$],where the tangent is parallel to $x$-axis.

asked Jan 2, 2013 by sreemathi.v

1 answer

Discuss the applicability of Rolle's theorem on the function given by $f(x)=\left \{\begin{array}{1 1}x^2+1, & if\;0\leq x\leq1\\3\;x, & if\;1<x<2\end{array}\right.$

asked Jan 2, 2013 by sreemathi.v

1 answer

Verify the Rolle's theorem for the function $f(x)=\sqrt {4-x^2}\;in\;[-2,2]$

asked Jan 2, 2013 by sreemathi.v

1 answer

Verify the Rolle's theorem for the function $f(x)=x(x+3)e^{\large-x/2}\;in\;[-3,0]$

asked Jan 2, 2013 by sreemathi.v

1 answer

Verify the Rolle's theorem for the function $log(x^2+2)-log 3\;in\;[-1,1]$

asked Jan 2, 2013 by sreemathi.v

1 answer

Verify the Rolle's theorem for the function $f(x)=\sin^4x+\cos^4x\;in\;\begin{bmatrix}0,\large\frac{\pi}{2}\end{bmatrix}$

asked Jan 2, 2013 by sreemathi.v

1 answer

If $\sqrt {1-x^2}+\sqrt {1-y^2}=a(x-y),$prove that $\large \frac{dy}{dx}=\frac{\sqrt{1-y^2}}{\sqrt{1-x^2}}$.

asked Jan 2, 2013 by sreemathi.v

1 answer

If $x=e^{\Large \frac{x}{y}},prove\; that \; \large {\frac{dy}{dx}=\frac{x-y}{xlog x}}$

asked Jan 2, 2013 by sreemathi.v

1 answer

If $ax^2+2hxy+by^2+2gx+2fy+c=0$,then show that $\large \frac{dy}{dx}.\frac{dx}{dy}\normalsize =1$

asked Jan 2, 2013 by sreemathi.v

1 answer

Differentiate $ tan^{-1}\bigg (\Large \frac{\sqrt {1+x^2}-1}{x}\bigg)\;\normalsize w.r.t\;\tan^{-1}x\;when\;x\neq 0$

asked Jan 1, 2013 by sreemathi.v

1 answer

If $x=3\sin t-\sin 3t,y=3\cos t-\cos 3t,find \;\large \frac{dy}{dx}\;\normalsize at \;t=\large \frac{\pi}{3}$

asked Dec 31, 2012 by sreemathi.v

1 answer

If $x=e^{\large\cos 2t}\; and\;y=e^{\large\sin 2t},prove\;that\;\large \frac{dy}{dx}=\frac{-ylog x}{xlog y}$

asked Dec 31, 2012 by sreemathi.v

1 answer

Find $\large \frac{dy}{dx}$ of the function expressed in parametric form $x=\large\frac{1+log t}{t^2},$$y=\large\frac{3+2log t}{t}$

asked Dec 31, 2012 by sreemathi.v

1 answer

Differentiate the following w.r.t x: $\tan^{-1}\Large {\bigg(\frac{\sqrt {1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}\bigg)},\normalsize -1 < x < 1,x\neq 0$

asked Dec 31, 2012 by sreemathi.v

1 answer

Differentiate the following w.r.t $x$ : $\sec^{-1}\bigg(\Large \frac{1}{4x^3-3x}\bigg),\normalsize 0 < x <\Large \frac{1}{\sqrt 2}$

asked Dec 31, 2012 by sreemathi.v

1 answer

Differentiate the following w.r.t x: $\tan^{-1}\bigg(\large {\frac {a \cos x - b \sin x}{b \cos x + a \sin x}}\bigg)$, where $\frac{-\pi}{2} < x < \frac{\pi}{2}$ and $\frac{a}{b} \tan x > -1$

asked Dec 31, 2012 by sreemathi.v

1 answer

Differentiate the following w.r.t $x$: $(x+1)^2(x+2)^3(x+3)^4$.

asked Dec 31, 2012 by sreemathi.v

1 answer

Differentiate the following w.r.t x: $\sin^mx.\cos ^nx$.

asked Dec 31, 2012 by sreemathi.v

1 answer

Differentiate the following w.r.t $x$ : $\cos\bigg(\tan\sqrt {x+1}\bigg)$.

asked Dec 31, 2012 by sreemathi.v

1 answer

Differentiate the following w.r.t x: $\sin^n(ax^2+bx+c)$.

asked Dec 31, 2012 by sreemathi.v

1 answer

Differentiate the following w.r.t $x$ : $log\begin{bmatrix}log\bigg(log x^5\bigg)\end{bmatrix}$.

asked Dec 31, 2012 by sreemathi.v

1 answer

Differentiate the following w.r.t $x$ :$\large \frac{8^x}{x^8}$

asked Dec 31, 2012 by sreemathi.v

1 answer

Differentiate the following w.r.t $x$: $2^{\large \cos^2x}$

asked Dec 31, 2012 by sreemathi.v

1 answer

A function $f\;:\;R\rightarrow R$ satisfies the equation $f(x+y)=f(x)f(y)$ for all $x,y\in R,f(x)\neq 0.$ Suppose that the function is differentiable at $x=0$ and $f'(0)=2$.Prove that $f'(x)=2f(x).$

asked Dec 31, 2012 by sreemathi.v

1 answer

Examine the differentiability of f,where f is defined by $f(x)=\left \{\begin{array}{1 1}1+x, & if\;x\leq 2\\5-x, & if\;x>2\end{array}\right.$at $x=2$.

asked Dec 31, 2012 by sreemathi.v

1 answer

Examine the differentiability of $f$,where $f$ is defined by $f(x)=\left \{\begin{array}{1 1}x^2\sin\large\frac{1}{x}, & if\;x\neq 0\\0, & if\;x=0\end{array}\right.$at $x=0$

asked Dec 31, 2012 by sreemathi.v

1 answer

Show that the function $f(x)=\mid \sin x+\cos x\mid$ is continuous at $x=\pi$.

asked Dec 31, 2012 by sreemathi.v

1 answer

Find all points of discontinuity of the function $f(t)=\large\frac{1}{t^2+t-2},$where $\;t=\large\frac{1}{x-1}.$

asked Dec 31, 2012 by sreemathi.v

1 answer

Given the function $ f(x)=\large\frac{1}{x+2}.$Find the points of discontinuity of the composite function $y=f(f(x)).$

asked Dec 31, 2012 by sreemathi.v

1 answer

Find the values of $a$ and $b$ such that the function $f$ defined by $f(x)=\left \{\begin{array}{1 1}\large\frac{x-4}{|x-4|}\normalsize+a, & if\;x<4\\a+b, & if\;x=4\\\large\frac{x-4}{|x-4|}\normalsize +b, & if\;x>4\end{array}\right.$ is a continuous function at $x=4.$

asked Dec 31, 2012 by sreemathi.v

1 answer

Prove that the function f defined by $ f(x)=\left \{\begin{array}{1 1}\large\frac{x}{\mid x\mid+2x^2}, & x\neq 0\\k, &x=0\end{array}\right.$remains discontinuous at x=0,regardless the choice of k.

asked Dec 31, 2012 by sreemathi.v

1 answer

Find the value of $k$ in the following so that the function $f$ is continuous at the indicated point $f(x)=\left \{\begin{array}{1 1}\large\frac{1-\cos kx}{x\sin x}, & if\;x\neq 0\\\large\frac{1}{2}, & if\;x=0\end{array}\right.\;at\;x=0$

asked Dec 31, 2012 by sreemathi.v

1 answer

Find the value of $k$ in the following so that the function $f$ is continuous at the indicated point $f(x)=\left \{\begin{array}{1 1}\large\frac{\sqrt {1+kx}-\sqrt{1-kx}}{x}, & if\;-1\leq x< 0\\\large\frac{2x+1}{x-2}, & if\;0\leq x\leq 1\end{array}\right.\; at\; x=0$

asked Dec 31, 2012 by sreemathi.v

1 answer

Find the value of k in the following so that the function $f$ is continuous at the indicated point: $f(x)=\left \{\begin{array}{1 1}\frac{\large 2^{x+2}-16}{4^{\large x}-16}, & if\;x\neq 2\\k, & if\;x=2\end{array}\right.$ at $x=2$

asked Dec 31, 2012 by sreemathi.v

1 answer

Find whether $f(x)$ is continuous or discontinuous at the indicated point $f(x)=\left \{\begin{array}{1 1}\large\frac{x^2}{2}, & if\;0\leq x\leq 1\\2x^2-3x+\large\frac{3}{2}\normalsize,& if\;0< x\leq 2\end{array}\right.$ at $x=1$.

asked Dec 31, 2012 by sreemathi.v

1 answer

Find whether $f(x)$ is continuous or discontinuous at the indicated point $f(x)=\left \{\begin{array}{1 1}( x-a)\sin\large\frac{1}{x-a},\normalsize & if\;x\neq 0\\0, & if\;x=0\end{array}\right.$ at $x=a$.

asked Dec 31, 2012 by sreemathi.v

1 answer

If $ A = \begin{bmatrix} 1 & 2 & 2 \\[0.3em] 2 & 1 & 2 \\[0.3em] 2 & 2 & 1 \end{bmatrix}$ what is $ A^2-4A-5I=0$

asked Dec 29, 2012 by thanvigandhi_1

1 answer

Solve for \( x : \sin^{-1} (1-x)-2 \sin^{-1}\: x = \large\frac{\pi}{2} \)

asked Dec 29, 2012 by thanvigandhi_1

1 answer

True or False: The maximum value of $\begin{vmatrix}1 & 1 & 1\\1 &(1-\sin x) & 1\\1 & 1 & 1-cos x\end{vmatrix}\;is\;\Large \frac{1}{2}$.

asked Dec 28, 2012 by sreemathi.v

1 answer

True or False: Let $\begin{vmatrix}a & p & x\\b & q & y\\c & r & z\end{vmatrix}$ = 16, then $\Delta_1=\begin{vmatrix}p+x & a+x & a+p\\q+y & b+y & b+q\\r+z & c+z & c+r\end{vmatrix}$ = 32

asked Dec 28, 2012 by sreemathi.v

1 answer

True or False: The determinant $ \begin{vmatrix}\sin A & \cos A & \sin A+\cos B\\\sin B & \cos A & \sin B+\cos B\\\sin C & \cos A & \sin C+\cos B\end{vmatrix}$ is equal to zero

asked Dec 28, 2012 by sreemathi.v

1 answer

True or False: $(aA)^{-1}= \Large{ \frac{1}{a}}\normalsize A^{-1}$, where $a$ is a any real number and $A$ is a square matrix.

asked Dec 28, 2012 by sreemathi.v

1 answer

If $f(x)$ = $\small\begin{vmatrix}(1+x)^{17} & (1+x)^{19} & (1+x)^{23}\\(1+x)^{23} & (1+x)^{29} & (1+x)^{34}\\(1+x)^{41} & (1+x)^{43} & (1+x)^{47}\end{vmatrix}$ = $A+Bx+Cx^2+.....$, then $A$ =

asked Dec 28, 2012 by sreemathi.v

1 answer

$\begin{vmatrix}0 & xyz & x-z\\y-x & 0 & y-z\\z-x & z-y & 0\end{vmatrix}$ =

asked Dec 28, 2012 by sreemathi.v

1 answer

If $\cos 2\theta=0$, then ${\begin{vmatrix}0 & \cos\theta & \sin\theta\\\cos\theta & \sin\theta & 0\\\sin\theta & 0 & \cos\theta\end{vmatrix}}^2$ =

asked Dec 28, 2012 by sreemathi.v

1 answer

There are two values of $a$ which makes the determinant $\Delta=\begin{vmatrix}1 & -2 & 5\\2 & a & 1\\0 & 4 & 2a\end{vmatrix}= 86,$ then sum of these number is

asked Dec 27, 2012 by sreemathi.v

1 answer

The value of the determinant $\begin{vmatrix}x & x+y & x+2y\\x+2y & x & x+y\\x+y & x+2y & x\end{vmatrix}$ is

asked Dec 27, 2012 by sreemathi.v

1 answer

If $x,y,z$ are all different from zero and $\begin{vmatrix}1+x & 1 & 1\\1 & 1+y & 1\\1 & 1 & 1+z\end{vmatrix}$ = 0, then value of $x^{-1}+y^{-1}+z^{-1}$ is

asked Dec 27, 2012 by sreemathi.v

1 answer

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