Recent questions and answers in Continuity and Differentiability

If $ f^1(x)=g(x)\;and\;g^1(x)=-f(x)$ for all n and $f(2)=f^1(2)=4,$ find the value of $f^2(24)+g^2(24)$

answered Jun 24, 2019 by premkht

1 answer

A card from a pack of 52 playing cards is lost. Two cards are drawn from the remaining cards and found to be both spade. (i) Find the probability of lost card being a spade card.(ii)While playing in ground you found a purse containing rupees and some other documents,what will you do?

answered Feb 16, 2017 by meena.p

1 answer

If $x=a(\theta - \sin \theta) \quad ; y= a(1+\cos \theta) $ find $\large\frac{d^2y}{dx^2}$ at $\theta =\large\frac{\pi}{2}$

answered Feb 16, 2017 by meena.p

1 answer

Differentiate w.r.t. $x$ the function in $x^x + x^a +a^x+a^a$, for some fixed $a> 0$ and $x>0$

answered Dec 8, 2016 by priyanka.c

1 answer

Find the slope of the tangent to the curve $x^2+y^2=4x-2$at the point (3,1)

answered Apr 3, 2016 by sarveshchauhan185

1 answer

Find points on the curve $y=x^2+1$ at which the slope of the tangent is equal to its $x$ coordinate.

asked May 1, 2014 by meena.p

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Find the equation of normal to the curve $x^2+2y=8x-2 $ at the point $(6,5)$

asked May 1, 2014 by meena.p

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Find the equation of the tangent to the curve $\sqrt x +\sqrt y =a $ at the point $\bigg( \large\frac{a^2}{4},\frac{a^2}{4} \bigg)$

asked May 1, 2014 by meena.p

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Verify Mean value theorem for the following functions: $(a)\;f(x)=x(x+4)^2\;on\;[0,4]\\ (b)\;f(x)= 2 \sin x +\sin 2x \;on\;[0,\pi] \\ (c)\;f(x)=x^3-2x^2-x+3\;in\;[0,1] \\ (d)\;f(x)=x +\large\frac{1}{x} \normalsize \;on \;[1,3]$

asked May 1, 2014 by meena.p

0 answers

Verify Rolles theorem for the following functions: $(a) \; f(x)=x^2-5x+4\;on \;[0,4]\\ (b) \; f(x)=x(x-1)^2\;on \;[0,1]\\ (c) \; f(x)=x^3-7x^2+16x-12 \;on \;[2,3]\\ (d) \; f(x)=e^x(\sin x- \cos x)\;on \;[\large\frac{\pi}{4},\frac{5 \pi}{4} ] $

asked May 1, 2014 by meena.p

0 answers

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

asked May 1, 2014 by meena.p

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

asked May 1, 2014 by meena.p

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If $\sin x =y \sin (x+b)$ show that $\large\frac{dy}{dx} =\frac{\sin b}{\sin ^2 (x+b)}$

asked May 1, 2014 by meena.p

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If $y=[x+\sqrt {x^2+1} ]^m$ show that $(x^2+1)y_2 +x y_1-m^2y=0$

asked May 1, 2014 by meena.p

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If $x^my^n=(x+y)^{m+n}$ show that $\large\frac{dy}{dx}=\frac{y}{x}$

asked May 1, 2014 by meena.p

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Differentiate the following function: $(x \cos x)^x +(x \sin x)^{\Large\frac{1}{x}}$

asked May 1, 2014 by meena.p

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Differentiate the following function: $x^{\large \sin x +\cos x}$

asked May 1, 2014 by meena.p

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Differentiate the following function: $x ^{\large \cos ^{-1}x}$

asked May 1, 2014 by meena.p

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Differentiate the following function: $\tan^{-1} \bigg( \large \frac{\cos x}{1+\sin x} \bigg)$

asked May 1, 2014 by meena.p

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Differentiate the following function: $ \tan ^{-1}( \sqrt{1+x^2}+x)$

asked May 1, 2014 by meena.p

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Show that $f(x) =|x -2| $ is continuous but not derivable at $x=2$

asked May 1, 2014 by meena.p

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The function $f(x)$ is defined as follows : $f(x) = \left\{ \begin{array}{l l} x^2+ax+b & \quad 0 \leq x < 2 \\ 3x+2, & \quad 2 \leq x \leq 4 \\ 2ax+5b & \quad 4 < x \leq 8 \end{array} \right. $ If $f $ is continuous on $[0,8]$ find the values of a and b

asked May 1, 2014 by meena.p

0 answers

Let $f(x) = \left\{ \begin{array}{l l} \frac{1-\cos 4x}{x^2}, & \quad x < 0 \\ a, & \quad x = 0 \\ \frac{\sqrt x)}{(\sqrt {16+\sqrt x})-4} & \quad x < 0 \end{array} \right. $ Find $a$ if $f $ is continuous at $x=0$

asked May 1, 2014 by meena.p

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If $ y = 500e^{\large 7x} + 600e^{\large -7x}$, what is $\large\frac{d^2y}{dx^2}$

answered Apr 10, 2014 by balaji.thirumalai

1 answer

If $ y = 3 \cos \: (\log x ) + 4 \sin ( \log x )$, show that $x^2y_2 + xy_1 $

answered Apr 10, 2014 by balaji.thirumalai

1 answer

If $ y = 5 \cos x - 3 \sin x $ , what is $\large\frac{d^2y}{dx^2} $

answered Apr 10, 2014 by balaji.thirumalai

1 answer

For the curve $\sqrt x + \sqrt y=1,\large \frac{dy}{dx}$ at $\bigg(\large\frac{1}{4},\frac{1}{4}\bigg)$ is ____________.

answered Mar 26, 2014 by balaji.thirumalai

1 answer

If $ \sqrt{1-y^2}-\sqrt{1-x^2}=a(x-y),\normalsize$ then find $\large\frac{dy}{dx}$

answered Feb 5, 2014 by rvidyagovindarajan_1

1 answer

If $e^{x+y}=xy,$ what is $\large\frac{dy}{dx}$?

answered Jan 29, 2014 by balaji

1 answer

If $( x = \sqrt {a^{\large sin^{-1}t}}, \: y = \sqrt {a^{\large cos^{-1}t}} ),$ what is the value of $\frac {\large dy}{\large dx}$

answered Dec 22, 2013 by balaji.thirumalai

1 answer

If $x^y\; y^x = 100$, evaluate $\large \frac{dy}{dx}$

answered Dec 22, 2013 by balaji.thirumalai

1 answer

If $\;x\sqrt{1+y}+y\sqrt{1+x}=0, \;$ Evaluate $ \large\frac{dy}{dx}$

answered Dec 22, 2013 by balaji.thirumalai

1 answer

Find $\large \frac{dy}{dx}$ if $( (x^2+y^2)^2=xy)$

answered Sep 23, 2013 by sreemathi.v

1 answer

Discuss the continuity of the following functions $(c)\;f(x)=\sin x.\cos x$

answered Sep 11, 2013 by sreemathi.v

1 answer

Discuss the continuity of the following functions $(b)\;f(x)=\sin x-\cos x$

answered Sep 11, 2013 by sreemathi.v

1 answer

Discuss the continuity of the following functions $(a)\;f(x)=\sin x+\cos x$

answered Sep 11, 2013 by sreemathi.v

1 answer

For what value of $ \lambda $ is the function defined by $f$ defined by $ f(x) = \left\{ \begin{array} {1 1} \lambda(x^2 - 2x) ,& \quad\text{ if $ x $ $\leq 0$}\\ 4x + 1,& \quad \text{if $x$ > 0}\\ \end{array} \right. $ Continuous at $x = 0$?

answered Aug 19, 2013 by sreemathi.v

1 answer

A car, traveling with a uniform acceleration, has a velocity of 18 km/hour at a certain time and 54 km/hour after covering a distance of 500m. How much further will it travel to attain a velocity of 72 km/hour?

answered Aug 18, 2013 by vijayalakshmi_ramakrishnans

1 answer

A body falls freely from the top of a tower; It covers of the whole distance in the last second. Find the height of the tower and the total by the

asked Aug 17, 2013 by anilglpl_1

0 answers

A car, traveling with a uniform acceleration, has a velocity of 18 km/hour at a certain time and 54 km/hour after covering a distance of 500m. How much further will it travel to attain a velocity of 72 km/hour?

asked Aug 17, 2013 by anilglpl_1

0 answers

Examine if Rolle's theorem is applicable to any of the following functions. Can you say some thing about the converse of Rolle's theorem from these example? $(i)\;f (x) = [x] \;for\; x \: \in [5,9] $

answered Aug 16, 2013 by sreemathi.v

1 answer

Examine if Rolle’s theorem is applicable to any of the following functions. Can you say some thing about the converse of Rolle’s theorem from these example? $(iii)\;f (x) = x^2 - 1 \; for\; x \: \in [1,2] $

answered Aug 16, 2013 by sreemathi.v

1 answer