Recent questions tagged ch5

What is the value of $\;$ $\lim\limits_{x\to 0} (1+\frac{1}{x})^{x}$?

asked Apr 4, 2013 by balaji.thirumalai

1 answer

What is the value of $\;$ $\lim_{x\to 0} log(1+x)^{\frac{1}{x}}$?

asked Apr 4, 2013 by balaji.thirumalai

1 answer

What is the value of $\;$ $\lim_{x\to 0} \large \frac{log(1+x)}{x} $?

asked Apr 4, 2013 by balaji.thirumalai

1 answer

What is the value of $\;$ $\lim_{x\to 0} \large \frac{a^x-1}{x}$?

asked Apr 4, 2013 by balaji.thirumalai

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What is the value of $\;$ $\lim_{x\to 0} \large \frac{e^x-1}{x}$?

asked Apr 4, 2013 by balaji.thirumalai

1 answer

What is the value of $\lim_{x\to 0} \large \frac{sin^{-1}x}{x}$?

asked Apr 4, 2013 by balaji.thirumalai

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What is the value of $\lim_{x\to 0} \large \frac{tan^{-1}x}{x}$?

asked Apr 4, 2013 by balaji.thirumalai

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What is the value of $\;\lim_{x\to 0} \large \frac{tanx}{x}$?

asked Apr 4, 2013 by balaji.thirumalai

1 answer

What is the value of:$ \;\lim_{x\to a} \large \frac{x^n-a^n}{x-a}$?

asked Apr 4, 2013 by balaji.thirumalai

1 answer

Which of the following is not a property of coninuity? $\qquad$(A) Every Constant Function is Continous, $\\$ (B) Every Polinomial Function is Continous, $\\$(C) Every Differential Function is Continous, $\\$(D) Every Continous Function is Differntiable.

asked Apr 4, 2013 by balaji.thirumalai

1 answer

True or False: Every Constant Function is Continous

asked Apr 4, 2013 by balaji.thirumalai

1 answer

True or False; Every Continous Function is Differntiable.

asked Apr 4, 2013 by balaji.thirumalai

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Let f be a real function on a subset of the real numbers and let c be a point in the domain of f. For f to be continuous at c $\; \lim_{x\to c} f(x)$ should equal what?

asked Apr 4, 2013 by balaji.thirumalai

1 answer

True or False: A Function is Continous if it is continous on the whole of its domain.

asked Apr 4, 2013 by balaji.thirumalai

1 answer

Differentiate each of the following w.r.t $x$: $tan^{-1}\large\frac{3a^2x-x^3}{a^3-3ax^2}$, $\frac{1}{\sqrt 3}< \frac{x}{a} <\frac{-1}{\sqrt 3}$

asked Feb 19, 2013 by sreemathi.v

1 answer

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]$

asked Jan 28, 2013 by meena.p

1 answer

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).

asked Jan 28, 2013 by meena.p

1 answer

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).$

asked Jan 28, 2013 by meena.p

1 answer

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.

asked Jan 28, 2013 by meena.p

1 answer

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

asked Jan 28, 2013 by meena.p

1 answer

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

asked Jan 28, 2013 by meena.p

1 answer

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}$

asked Jan 28, 2013 by meena.p

1 answer

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]$

asked Jan 28, 2013 by meena.p

1 answer

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}$

asked Jan 28, 2013 by meena.p

1 answer

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

asked Jan 28, 2013 by meena.p

1 answer

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

asked Jan 28, 2013 by meena.p

1 answer

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

asked Jan 28, 2013 by meena.p

1 answer

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

asked Jan 28, 2013 by meena.p

1 answer

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}}$

asked Jan 28, 2013 by meena.p

1 answer

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}}$

asked Jan 25, 2013 by meena.p

1 answer

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}$

asked Jan 25, 2013 by meena.p

1 answer

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}$

asked Jan 25, 2013 by meena.p

1 answer

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

asked Jan 25, 2013 by meena.p

1 answer

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

asked Jan 25, 2013 by meena.p

1 answer

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

asked Jan 25, 2013 by meena.p

1 answer

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 $

asked Jan 25, 2013 by meena.p

1 answer

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}}$

asked Jan 25, 2013 by meena.p

1 answer

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

asked Jan 25, 2013 by meena.p

1 answer

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 $(i)\phi(c)=h(c)\quad(ii)\phi'(c)=h'(c)$

asked Jan 25, 2013 by meena.p

1 answer

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.

asked Jan 25, 2013 by meena.p

1 answer

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. $

asked Jan 25, 2013 by meena.p

1 answer

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

asked Jan 25, 2013 by meena.p

1 answer

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

asked Jan 25, 2013 by meena.p

1 answer

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.

asked Jan 25, 2013 by meena.p

1 answer

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

asked Jan 25, 2013 by meena.p

1 answer

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

asked Jan 25, 2013 by meena.p

1 answer

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 $

asked Jan 25, 2013 by meena.p

1 answer

Show that Sin | x | is continuous.

asked Jan 25, 2013 by meena.p

1 answer

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].

asked Jan 25, 2013 by meena.p

1 answer

Find a and b if the function :$ f(x)= \left\{ \begin{array}{1 1} [1+| \sin x | ]^{\Large\frac{a}{\sin |x|}}, & \quad \Large\frac{-\pi}{6}< \normalsize x<0 \\ 6 & \quad ,x=0 \\ e^{\Large\frac{\tan 2x}{\tan 3x}}, & \quad 0<x<\Large\frac{\pi}{6} \end{array} \right. $ is continuous on $\bigg(\Large\frac{-\pi}{6},\frac{\pi}{6}\bigg).$

asked Jan 25, 2013 by meena.p

0 answers

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