Answers posted by thanvigandhi_1

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If $ f(x)=\begin{vmatrix} x & x^2 & x^3 \\ 1 & 2x & 3x^2 \\ 0 & 2 & 6x \end{vmatrix}$ than find $ f' (x)$

answered Mar 12, 2013

Toolbox: Open the determinent and then differentiate. $ f(x) = 2\: x^3$ $ f ' ...

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Verify Rolles theorem for the function $f(x)=\sin 2x \;in\;\bigg[0,\Large\frac{\pi}{2}\bigg]$

answered Mar 12, 2013

Toolbox: Check $ f $ is continuous and differentiable in $ \bigg[ 0, \large\frac{\pi}{2} \...

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Find $\Large \frac{dy}{dx}$ when $ y=a^x.x^a$

answered Mar 12, 2013

Toolbox: If there is a variable in index then use log before differentiating. $ log\...

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Find $\Large \frac{dy}{dx}$ when $y=\tan^{-1}\Large\frac{1+x^2}{1-x^2}$

answered Mar 12, 2013

Toolbox: Use chain rule. $ tan^{-1}\large \frac{a+b}{1-ab} = tan^{-1}a+tan^{-1}b$ ...

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Given that $ f(x)= \left\{ \begin{array}{1 1} \Large\frac{1- \cos 4x}{x^2}, & \quad if\;x<0 \\ a & \quad ,if\;x=0 \\ \Large\frac{\sqrt x}{\sqrt{16+\sqrt{x}}-4}, & \quad if\;x>0 \end{array}. \right. $ If f(x) is continuous at x = 0, find the value of a.

answered Mar 11, 2013

Toolbox: $ LHL = RHL = f(0)$ for continuity $ 1-cos4x=2sin^22x$ LHL $ \l...

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Show that the function :$f(x)= \left\{ \begin{array}{1 1} \Large\frac{e^{\frac{1}{x}}-1}{e^{\frac{1}{x}}+1}, & \quad when\;x\neq0 \\ 0, & \quad when\; x=0 \end{array} \right. $ is discontinuous at x = 0.

answered Mar 11, 2013

Toolbox: Check for $ LHL = RHL $ or not Use LHL definition and RHL definition. ...

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Is the function $ f(x)=\Large\frac{3x+4\tan x}{x}$ continuous at x = 0? If not, how may the function be defined to make it continuous at this point.

answered Mar 11, 2013

Toolbox: For continuity $ LHL = RHL = f(0)$ $ \lim\limits_{x \to 0}\large\frac{tan\:...

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Show that the function : $f(x)= \left\{ \begin{array}{1 1} |x|, & \quad x\leq2 \\ [x], & \quad x>2 \end{array} \right. $ is continuous [0,2].

answered Mar 11, 2013

Toolbox: Check $ LHL = RHL = f(2)$ $ [x] = n \: where\: n \leq x\: n+1\: and \: n \i...

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Show that Sin | x | is continuous.

answered Mar 11, 2013

Toolbox:Check whether $ LHL = RHL = f(a)$ at any point $ x = a$LHL \( \lim\limits_{x \to \overli...

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Show that the function $ f(x)= \left\{ \begin{array}{1 1} x+\lambda, & \quad x< 1 \\ \lambda x^2+1, & \quad x\geq1 \end{array} \right. $ is continuous function, regardless of the choice of $ \lambda \in R $

answered Mar 11, 2013

Toolbox: For continuous function at $ x = 1. LHL = RHL = f(1)$ LHL $ \lim\limits_{...

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Determine the values of a, b and c for which the function :$f(x)= \left\{ \begin{array}{1 1} \Large\frac{\sin (a+1)x+\sin x}{x} & \quad ,\;x<0 \\ c & \quad ,\;x=0 \\ \Large\frac{\sqrt{x+bx^2}-\sqrt x}{b\sqrt{x^3}} & \quad ,\;x>0 \end{array} \right. $ may continuous at x = 0

answered Mar 11, 2013

Toolbox: For continuous at $ x = \pi, LHL = RHL = f(\pi)$ $ LHL \: \lim\limits_{x ...

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

answered Mar 11, 2013

Toolbox: For continuous at $ x = \pi, LHL = RHL = f(\pi)$ $ LHL\: \lim\limits_{h \...

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The function $ \Large\frac{log(1+ax)-log(1-bx)}{x}$ is not defined at x = 0. Find the value of f(x) so that f(x) is continuous at x = 0.

answered Mar 11, 2013

Toolbox: For $f(x)$ to be continuous at $ '0' LHL = RHL = f(0)$ $ \lim\limits_{x \to...

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Find all the points of discontinuity of f defined by f(x) = | x | - | x+1|.

answered Mar 11, 2013

Ans : $ f(x)$ is continuous at all the points in the domain. Take any point $ a \in R$ chec...

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Let $ f(x)= \left\{ \begin{array}{1 1} |x|\cos \frac{1}{x} & \quad if\;x\neq0 \\ 0 & \quad if\;x=0 \end{array} \right. $ then discuss the continuity of f(x) at x = 0

answered Mar 11, 2013

Toolbox: For continuity at '0' LHL = RHL = f(0) $ \lim\limits_{x \to 0} \: \large\fr...

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Discuss the continuity of the following function at x = 0 $f(x)= \left\{ \begin{array}{1 1} \Large\frac{x^4+x^3+2x^2}{\tan ^{-1}x} & \quad ,\;x\neq0 \\ 10 & \quad ,\;x=0 \end{array} \right. $

answered Mar 11, 2013

Toolbox: For continuity at '0' LHL = RHL = f(0) $ \lim\limits_{x \to 0} \: \large\fr...

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Let $f(x) = \left\{ \begin{array}{l l} \Large\frac{1-sin^3x}{3cos^2x}, & \quad if\; { x < \Large\frac{\pi}{2}}\\ a, & \quad if\; { x = \Large\frac{\pi}{2}} \\ \Large\frac{b(1-sin x)}{(\pi - 2x)^2}, & \quad if \;{ x > \Large\frac{\pi}{2}} \end{array}. \right.$ If f(x) is continuous function at x $ x = \Large\frac{\pi}{2} $, find a and b.

answered Mar 9, 2013

Toolbox: For a function to be continuous at any point c $L.H.L = R.H.L = f(c)$ $ 1-c...

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Let $\phi(x)\;andh(x)$ be derivable at x = c. Show that necessary and sufficient condition for the function defined as :\[f(x)= \left\{ \begin{array}{1 1} \phi(x), & \quad x \leq c \\ h(x), & \quad x>c \end{array} \right. \]to be derivable at x= c are $(i)\phi(c)=h(c)\quad(ii)\phi'(c)=h'(c)$

answered Mar 9, 2013

Toolbox: The necessary condition for any function to be differentiable at any point is it ...

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Find$ \Large\frac{dy}{dx}\;\normalsize when\;y=\sin ^{-1} \bigg[\Large\frac{5x+12 \sqrt{1-x^2}}{13}\bigg]$

answered Mar 9, 2013

Toolbox: $ sin^{-1}(x\sqrt{1-y^2}+y\sqrt{1-x^2} )=sin^{-1}x+sin^{-1}y$ Take x as...

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If $y=\sin ^{-1}\big [x^2\sqrt{1-x^2}+x\sqrt {1-x^4}\big],$ prove that $ \Large\frac{dy}{dx}=\frac{2x}{\sqrt{1-x^4}}+\frac{1}{\sqrt{1-x^2}}$

answered Mar 9, 2013

Toolbox: $ sin^{-1} [ x\sqrt{1-y^2}+y\sqrt{1-x^2} ] = sin^{-1}x+sin^{-1}y$ Consider ...

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If $ y=\frac{2}{\Large\sqrt {a^2-b^2}}\tan^{-1}\bigg[\sqrt{\frac{a-b}{a+b}}\tan \frac{x}{2} \bigg],$ prove that $ \Large\frac{dy}{dx}=\frac{1}{a+b \cos x}, \normalsize a>b>0 $

answered Mar 9, 2013

Toolbox: $ cosx=\large\frac{1-tan^2\large\frac{x}{2}}{1+tan^2\large\frac{x}{2}}$ Use...

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Let $y= \tan^{-1}\bigg[\Large\frac{4x}{1+5x^2}\bigg] \normalsize +\tan^{-1}\bigg[\Large\frac{2+3x}{3-2x}\bigg],$ show that $ \Large\frac{dy}{dy}=\frac{5}{1+25x^2}$

answered Mar 9, 2013

Toolbox: Write $\large \frac{4x}{1+5x^2}=\large\frac{5x-x}{1+5x^2}, \large\frac{2+3x}{3-2x...

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Differentiate w.r.t. x, $ y=\tan^{-1}\bigg[\Large\frac{a\cos x-b\sin x}{b\cos x+a \sin x}\bigg]$

answered Mar 9, 2013

Toolbox:Divide number and then by b cos x $ tan^{-1}\large\frac{x-y}{1+xy}=tan^{-1}x-tan^{-1}y$ ...

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Find $\Large\frac{dy}{dx},$ when $y=\sin^{-1}\bigg[x\sqrt{1-x}-\sqrt x \sqrt{1-x^2}\bigg]$

answered Mar 9, 2013

Toolbox: $ sin^{-1} ( x\sqrt{1-y^2}-y\sqrt{1-x^2})=sin^{-1}x-sin^{-1}y$ Take $ x=x\:...

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Given that $ \cos \frac{x}{2}.\cos \frac{x}{4}.\cos \frac{x}{8}.......=\Large\frac {\sin x}{x},$ \[Prove\;that\;\frac{1}{2^2}\sec^2\frac{x}{2}+\frac{1}{2^4}\sec^2\frac{x}{4}+.......=cosec^2x-\frac{1}{x^2}\]

answered Mar 9, 2013

Toolbox: Take log on both sides. $ log (a.b.c.d.....) = loga+logb+logc+........$ ...

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Prove that $ \Large\frac{d}{dx}\bigg[\Large\frac{1}{4 \sqrt 2} \normalsize log\bigg|\Large\frac{x^2+\sqrt 2 x+1}{x^2- \sqrt 2 x+1} \bigg|+\Large\frac {1}{2 \sqrt 2} \normalsize \tan^{-1} \Large\frac{\sqrt 2 x}{1-x^2}\bigg]=\Large\frac{1}{1+x^4}$

answered Mar 9, 2013

Toolbox: Write $ log \: \large\frac{a}{b}=log\: a - log\: b$ Simplify to get the res...

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

answered Mar 9, 2013

Toolbox: Put $ x = sin \theta\: and \: y = sin\phi$ Simplify and differentiate after...

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If $ y=\Large\frac{1}{\sqrt{b^2-a^2}} \normalsize log \bigg[\Large\frac{\sqrt{b+a}+\sqrt{b-a}\tan \frac{x}{2}} {\sqrt{b+a}-\sqrt{b-a}\tan \frac{x}{2}}\bigg]$ prove that $ \Large\frac {dy}{dx}=\frac{\sec^2\frac{x}{2}}{(b+a)-(b-a) \tan^2\frac{x}{2}}$

answered Mar 9, 2013

Toolbox: Differentiate using chain rule and quotient rule. $ \large\frac{dy}{dx}=\...

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If $ y=|\cos x|+|\sin x|,\;find\; \Large\frac{dy}{dx}\; \normalsize at\;x=\Large\frac{2\pi}{3}$

answered Mar 9, 2013

Toolbox: before differentiating eliminate modulus by its definition. $ | cos\: x|=...

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Find $\Large\frac{dy}{dx}$when$y=\tan^{-1}\bigg[\Large\frac{x^{\frac{1}{3}}+a^{\frac{1}{3}}}{1-x^{\frac{1}{3}}a^{\frac{1}{3}}}\bigg]$

answered Mar 9, 2013

Toolbox: $ tan^{-1}\large\frac{x+y}{1-xy} = tan^{-1}x+tan^{-1}y$ $ y=tan^{-1}\: x^...

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If $ y=f \bigg(\Large\frac{2x-1}{x^2+1}\bigg)\;\normalsize and\;f' (x)=\sin x^2,\;find\; \Large\frac{dy}{dx}$

answered Mar 9, 2013

Toolbox: Use chain rule for differentiating y $\large\frac{dy}{dx}=f ' \bigg(\larg...

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If $ y^{ \cos x}+(\tan^{-1}x)^y=1,find \Large\frac{dy}{dx}$

answered Mar 9, 2013

Toolbox: When you have any function on the index then use log before differentiating. ...

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If $ x=\cos \theta + log \tan \Large\frac{\theta }{2},\normalsize y=\sin \theta \;find\; \Large\frac {d^2y}{dx^2}\normalsize \;at\;\theta =\Large\frac{\pi}{4}$

answered Mar 9, 2013

Toolbox: $\large \frac{dy}{dx}=\large\frac{\large\frac{dy}{d\theta}}{\large\frac{dx}{d\the...

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Differentiate $ \cos^{-1} \bigg[\large\frac{3 \cos x-2 \sin x}{\sqrt {13}}\bigg] w.r.t \; \sin^{-1} \bigg[\large\frac{5\sin x +4 \cos x}{\sqrt{41}}\bigg]$

answered Mar 9, 2013

Toolbox: Assume $ cos^{-1} \bigg( \large\frac{3cosx-2sinx}{\sqrt{13}} \bigg) = u \: and $ ...

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If $ y=\sqrt{x^2-1} - log \bigg(\Large\frac{1}{x}+\sqrt{1+\frac{1}{x^2}}\bigg),\;\normalsize find\;\Large\frac{dy}{dx}$

answered Mar 9, 2013

Toolbox: Use chain rule and differentiate $\large \frac{dy}{dx}=\large\frac{2x}{2\...

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If $ x\sqrt{1+y}+y\sqrt{1+x}=0,\normalsize\; Prove\; that\; \Large\frac{dy}{dx}=\frac{-1}{(1+x)^2}$

answered Mar 9, 2013

Toolbox: Shift one term to R.H.S square and then differentiate Step 1 $ x \sqr...

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Verify Rolles theorem for the function $ f(x)=\Large e^{1-x^2}$in the interval [-1,1]

answered Mar 8, 2013

Toolbox: Check whether the function is continuous or not in the given closed interval [ a,...

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It is given that for the function $ f(x)=x^3-6x^2+px+q\;on\;[1,3].$Rolles theorem holds with $ c=2+\large\frac{1}{\sqrt 3}.$Find the values of p and q.

answered Mar 8, 2013

Toolbox: As per Rolle's theorem $ f(a)=f(b) \: and \: f'(c) = 0$ $ f(1) = -5...

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If f(x) and g(x) are functions derivable in [a,b] such that f(a) = 4, f(b) = 10, g(a) =1, g(b) =3.Show that for a < c < b, we have $ f'(c)=3g'(c).$

answered Mar 8, 2013

Toolbox: If a function is differentiable in an interval it is continuous in that interval....

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Using LMV Theorem, find a point on the curve $ y=(x-3)^2,$ where the tangent is parallel to the chord joining (3,0) and (5,4).

answered Mar 8, 2013

Toolbox: M.V.T is Check whether the function is continuous in [ a, b ] and different...

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Verify the Rolles Theorem for the function $ f(x)=\sin x- \cos x, $in the interval $ \bigg[\Large\frac{\pi}{4},\frac{5\pi}{4}\bigg]$

answered Mar 8, 2013

Toolbox: Check whether the function is continuous or not in the given closed interval [ a,...

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Distance of the point $(\alpha,\beta,\gamma)$ from y-axis is

answered Mar 7, 2013

Toolbox: Drop perpendicular from the point to Y axis and XY plane. Join the foot of ...

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If $l_1,m_1,n_1,l_2,m_2,n_2,l_3,m_3,n_3$ are direction cosines of three mutually perpendicular lines,prove that the line whose direction cosines are proportional to $l_1+l_2+l_3,m_1+m_2+m_3,n_1+n_2+n_3$ make equal angles with them.

answered Mar 7, 2013

Toolbox: d.c is proportional to (a, b,c) means d.r = (a, b, c) To prove angle betwee...

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Show that the straight lines whose direction cosines are given by $2l+2m-n=0\;and\;mn+nl+lm=0$ are at right angles.

answered Mar 7, 2013

Toolbox: Solve the two equations to get the d.r of both the lines $ d_1, d_2$ Prove ...

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$\overrightarrow{AB}=3 \hat i-\hat j+\hat k \;and\; \overrightarrow{CD}=-3\hat i+2 \hat j+4\hat k$ are two vectors. The position vectors of the points $A$ and $C$ are $6\hat i+7\hat j+4\hat k\;and\;-9\hat j+2\hat k,$ respectively.Find the position vector of a point $P$ on the line $AB$ and a point $Q$ on the line $CD$ such that $\overrightarrow{PQ}$ is perpendicular to $\overrightarrow{AB}\;and\;\overrightarrow{CD}$ both.

answered Mar 7, 2013

Toolbox: Write the cartesian equations of the lines $ \overline{AB}\: and \: \overline{CD}...

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Show that the points $ (\hat i-\hat j+3 \hat k)\; and\;3(\hat i+\hat j+\hat k)$ are equidistant from the plane $ \overrightarrow{r}.(5 \hat i+2 \hat j-7 \hat k)+9=0$ and lies on opposite side of it.

answered Mar 7, 2013

Toolbox: Distance of a point from plane is $ \bigg| \large\frac{ax_1+by_1+cz_1+d}{\sqrt{a^...

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Find the distance of a point(2,4,-1) from the line $\large \frac{x-5}{1}\;=\frac{y-3}{4}\;=\frac{z-6}{-9}$

answered Mar 6, 2013

Toolbox:To get distance between a point and line get foot of $ \perp$ of the point on the line and ...

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What is the value of $\tan({\frac{1}{2}}\sin^{-1}{\frac{3}{4}})$?

answered Mar 2, 2013

Toolbox:Take $\sin^{-1}\frac{3}{4}=x$ and proceed$\cos x=2cos^2\frac{x}{2}-1$$\cos x=\sqrt{1-\sin^2x...

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Find the simplified form of $cos^{-1}(\frac{3}{5}cos x+\frac{4}{5}sin x),$where $x\in (\frac{-3}{4},\frac{3}{4})$

answered Mar 2, 2013

Toolbox:$ cosA\: cosB+sinA\: sinB=cos(A-B)$$sinx=\sqrt{1-cos^2x}$Let $ \frac{3}{5}=cos\theta$\...

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Show that $2tan^{-1}(-3)=\frac{-\pi}{2}+tan^{-1}\bigg(\frac{-4}{3}\bigg)$

answered Mar 2, 2013

Toolbox: $ 2tan^{-1}x=-\pi +tan^{-1}\large\frac{2x}{1-x^2} \: if \: x < -1$ \(t...

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