Recent questions in Continuity and Differentiability

Find $\large \frac{dy}{dx}$ of the function expressed in parametric form $x=e^\theta\bigg(\theta+\frac{1}{\theta}\bigg),y=e^{-\theta}\bigg(\theta-\frac{1}{\theta}\bigg)$

asked Dec 31, 2012 by sreemathi.v

1 answer

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

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: $\tan^{-1}(\sec x+\tan x),\large \frac{-\pi}{2}<$ $x<\large\frac{\pi}{2}$ .

asked Dec 31, 2012 by sreemathi.v

1 answer

Differentiate the following w.r.t x: $\tan^{-1}\bigg(\sqrt{\large\frac{1-\cos x}{1+\cos x}}\bigg),$ $\large\frac{-\pi}{4}<$ $x<\large\frac{\pi}{4}$ .

asked Dec 31, 2012 by sreemathi.v

1 answer

Differentiate the following w.r.t $x$ : $\sin^{-1} \big[\large\frac{1}{\sqrt 2}$ $\sin x+\large\frac{1}{\sqrt 2}$ $\cos x\big],\large\frac{-\pi}{4}<$ $x<\large\frac{\pi}{4}$ .

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: $(\sin x)^{\large\cos x}$ .

asked Dec 31, 2012 by sreemathi.v

1 answer

Differentiate the following w.r.t x: $\sin^{-1}\bigg|\large\frac{1}{\sqrt {x+1}}\bigg|$ .

asked Dec 31, 2012 by sreemathi.v

1 answer

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

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 $x$ : $\sin\sqrt x+\cos^2\sqrt x$ .

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 $x$ : $log\bigg(x+\sqrt{ x^2+a}\bigg)$ .

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

Show that $f(x)=|x-5|$ is continuous but not differentiable at $x=5$

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

Examine the differentiability of f,where f is defined by $f(x)=\left \{\begin{array}{1 1}x[x], & if\;0\leq x<2\\(x-1)x, & if\;2\leq x<3\end{array}\right.$ at $x=2$

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 the value of k in the following so that the function f is continuous at the indicated point $f(x)=\left \{\begin{array}{1 1}3x-8, & if\;x\neq 5\\2k, & if\;x=5\end{array}\right.\;at\;x=5$

asked Dec 31, 2012 by sreemathi.v

1 answer

Find whether f(x) is continuous or discontinuous at the indicated point $f(x)= x ^2+x-1\;at\;x=1.$

asked Dec 31, 2012 by sreemathi.v

1 answer

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

asked Dec 31, 2012 by sreemathi.v

1 answer

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

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

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

asked Dec 31, 2012 by sreemathi.v

1 answer

Find whether f(x) is continuous or discontinuous at the indicated point $f(x)=\left \{\begin{array}{1 1}x-4, & if\;x\neq 4\\2(x-4)\\0, & if\;x=4\end{array}\right.$ at x=4.

asked Dec 31, 2012 by sreemathi.v

1 answer

Examine the continuity of f(x) at the indicated point $f(x)=\left \{\begin{array}{1 1}\large\frac{2x^2-3x-2}{x-2} &\normalsize if\;x\neq 2\\5 & if\;x=2\end{array}\right.$ at x=2.

asked Dec 31, 2012 by sreemathi.v

1 answer

Examine the continuity of f(x) at the indicated point $f(x)=\left \{\begin{array}{1 1}\large\frac{1-\cos 2x}{x^2} & \normalsize if\;x\neq 0\\5 & if\;x=0\end{array}\right.$ at x=0.

asked Dec 31, 2012 by sreemathi.v

1 answer

Examine the continuity of f(x) at the indicated point $f(x)=\left \{\begin{array}{1 1}3x+5 & if\;x\geq 2\\x^2 & if\;x<2\end{array}\right.$ at $x=2.$

asked Dec 31, 2012 by sreemathi.v

1 answer

Examine the continuity of the function $f(x)=x^3+2x^2-1$ at $x=1$

asked Dec 31, 2012 by sreemathi.v

1 answer

What-will-be-derivative-if-x-is-in-the-interval-(\pi/2,\pi)-Ans: -1/2

asked Dec 17, 2012 by rvidyagovindarajan_1

0 answers

If $y = e^{\large a \: cos^{-1}x}, -1 \leq x \leq 1$ , show that $(1- x^2)\large\frac{d^2y}{dx^2}-\normalsize x\large\frac{dy}{dx} -\normalsize a^2y \; = \; 0.$

asked Dec 5, 2012 by thanvigandhi_1

1 answer

If $y = \begin{vmatrix} f(x) & g(x) & h(x) \\l & m &n \\ a & b & c \end{vmatrix}$ , prove that $\large\frac{dy}{dx}\normalsize = \begin{vmatrix} f'(x) & g'(x) & h'(x) \\l & m &n \\ a & b & c \end{vmatrix}$

asked Dec 5, 2012 by thanvigandhi_1

1 answer

Does there exist a function which is continuous everywhere but not differentiable at exactly two points? Justify your answer.

asked Dec 4, 2012 by thanvigandhi_1

1 answer

Using the fact that $\sin (A + B) = \sin A \cos B + \cos A \sin B$ and the differentiation, obtain the sum formula for cosines.

asked Dec 4, 2012 by thanvigandhi_1

1 answer

Using mathematical induction prove that $\large\frac{d}{dx}\normalsize (x^{\large n}) = nx^{\large{n-1}}$ for all positive integers $n$ .

asked Dec 4, 2012 by thanvigandhi_1

1 answer

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