Recent questions tagged medium

Differentiate w.r.t. $x$ the function in $ x^{\large x^2-3} + ( x - 3 )^{\large x^2} $, for $x > 3$

asked Nov 21, 2012 by thanvigandhi_1

1 answer

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

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Differentiate w.r.t. $x$ the function in $( \log x )^{\large\log x}, x > 1 $

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Differentiate w.r.t. $x$ the function in $\large\frac{\cos^{ -1}\Large\frac{ x}{ 2}}{\sqrt {2x + 7}}, \normalsize-2 < x < 2 $

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Differentiate w.r.t. $x$ the function in $ sin^{-1}(x \: \sqrt x ), 0 \leq x \leq 1 $

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Differentiate w.r.t. $x$ the function in $ sin^3x + cos^6x $

asked Nov 21, 2012 by thanvigandhi_1

1 answer

Verify Mean Value Theorem, if $ f (x) = x^3 - 5x^2 - 3x $ in the interval $ [a,b]$, where $ a = 1$ and $ b = 3$. Find all $ c \in (1,3) $ for which $ f'(c) = 0$.

asked Nov 21, 2012 by thanvigandhi_1

1 answer

If $ f : [ -5,5] \rightarrow R $ is a differentiable function and if $ f' (x) $ does not vanish anywhere, then prove that $ f (-5) \: \neq \: f (5)$.

asked Nov 21, 2012 by thanvigandhi_1

1 answer

If $ y = ( tan^{-1}x )^2$, show that $ (x^2 + 1 )^2 \: y_2 + 2x (x^2 +1 ) \: y_1 = 2 $

asked Nov 20, 2012 by thanvigandhi_1

1 answer

If $ y =\cos^{-1}x $ , find $ \large\frac{d^2y}{dx^2}$ in terms of $ y $ alone.

asked Nov 20, 2012 by thanvigandhi_1

1 answer

Find the second order derivatives of the function $(e^{\large x} \sin 5x)$

asked Nov 20, 2012 by thanvigandhi_1

1 answer

If $x = \sqrt {a^{\large sin^{-1}t}}, \: y = \sqrt {a^{\large cos^{-1}t}} $, show that $\frac {\large dy}{\large dx} = -\frac{\large y}{\large x} $

asked Nov 20, 2012 by thanvigandhi_1

1 answer

If $x$ and $y$ are connected parametrically by the equations given in $x = a \big(\cos t +\ log \tan \large\frac{t}{2}\big) $ and $ y = a \sin t $ without eliminating the parameter, Find $ \frac{\large dy}{\large dx}$

asked Nov 20, 2012 by thanvigandhi_1

1 answer

Let $f : R - \{ - \frac {4} {3} \} \to R $ be a function defined as $f(x)= \frac {4x} {3x+4} $ .The inverse of $f$ is the map $g$: Range $f \to R - \{ - \frac {4} {3} \} $ given by

asked Nov 19, 2012 by vaishali.a

1 answer

If $u,\: v$ and $w$ are functions of $x$, then show that $\large\frac{d}{dx} $$( u.v.w) =\large \frac{du}{dx} $$v.w + u . \large\frac{dv}{dx} $$. w + u.v. \large\frac{dw}{dx} $ in two ways - first by repeated application of product rule, second by logarithmic differentiation.

asked Nov 17, 2012 by thanvigandhi_1

1 answer

Find $\large\frac{dy}{dx}$ of the function $xy = e^{\large(x-y)}$

asked Nov 17, 2012 by thanvigandhi_1

1 answer

Find $\large\frac{dy}{dx}$ of the function $(\cos x)^{\large y} = (\cos y)^{\large x}$

asked Nov 17, 2012 by thanvigandhi_1

1 answer

Differentiate the functions given in w.r.t. $x : $ $ x^{\large x \cos x} + \large\frac{x^2 + 1}{x^2 - 1} $

asked Nov 17, 2012 by thanvigandhi_1

1 answer

Differentiate the functions given in w.r.t. $x$ : $ (\sin x )^{\large x} + \sin^{-1} \sqrt x $

asked Nov 17, 2012 by thanvigandhi_1

1 answer

Differentiate the functions given in w.r.t. $x : $ $y=(\log x)^{\large x}+x^{\large\log x}$

asked Nov 17, 2012 by thanvigandhi_1

1 answer

Differentiate the functions given in w.r.t. $x : $ $ (x + \large\frac{1}{x})^{\large x}$$ + x ^{\large (1 + \Large\frac{1}{x})} $

asked Nov 17, 2012 by thanvigandhi_1

1 answer

There are two types of fertilisers $F_{1} $ and $F_{2} $. $F_{1} $ consists of 10% nitrogen and 6% phosphoric acid and $F_{2} $ consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, a farmer finds that she needs atleast 14 kg of nitrogen and 14 kg of phosphoric acid for her crop. If $F_{1} $ costs Rs 6/kg and $F_{2} $ costs Rs 5/kg, determine how much of each type of fertiliser should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost?

asked Nov 16, 2012 by fingeazy

1 answer

A diet is to contain at least 80 units of vitamin A and 100 units of minerals. Two foods $F_{1} $ and $F_{2} $ are available. Food $F_1$ costs Rs 4 per unit food and $F_2$ costs Rs 6 per unit. One unit of food $F_{1} $ contains 3 units of vitamin A and 4 units of minerals. One unit of food $F_{2} $ contains 6 units of vitamin A and 3 units of minerals. Formulate this as a linear programming problem. Find the minimum cost for diet that consists of mixture of these two foods and also meets the minimal nutritional requirements.

asked Nov 16, 2012 by fingeazy

1 answer

A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A require 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours 20 minutes available for cutting and 4 hours for assembling. The profit is Rs 5 each for type A and Rs 6 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximise the profit?

asked Nov 16, 2012 by fingeazy

1 answer

A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of a grinding/cutting machine and a sprayer. A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of a grinding/cutting machine and a sprayer. It takes 2 hours on grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp. It takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer to manufacture a shade. On any day, the sprayer is available for at the most 20 hours and the grinding/cutting machine for at the most 12 hours. The profit from the sale of a lamp is Rs 5 and that from a shade is Rs 3. Assuming that the manufacturer can sell all the lamps and shades that he produces, how should he schedule his daily production in order to maximise his profit?

asked Nov 16, 2012 by fingeazy

1 answer

A factory manufactures two types of screws, A and B. Each type of screw requires the use of two machines, an automatic and a hand operated. It takes 4 minutes on the automatic and 6 minutes on hand operated machines to manufacture a package of screws A, while it takes 6 minutes on automatic and 3 minutes on the hand operated machines to manufacture a package of screws B. Each machine is available for at the most 4 hours on any day. The manufacturer can sell a package of screws A at a profit of Rs 7 and screws B at a profit of Rs 10. Assuming that he can sell all the screws he manufactures, how many packages of each type should the factory owner produce in a day in order to maximise his profit? Determine the maximum profit.

asked Nov 16, 2012 by fingeazy

1 answer

A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftmans time in its making while a cricket bat takes 3 hour of machine time and 1 hour of craftmans time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsmans time. (i) What number of rackets and bats must be made if the factory is to work at full capacity? (ii) If the profit on a racket and on a bat is Rs 20 and Rs 10 respectively, find the maximum profit of the factory when it works at full capacity.

asked Nov 16, 2012 by fingeazy

1 answer

Reshma wishes to mix two types of Food P and Q in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin A and 11 units of vitamin B. Food P costs Rs 60/kg and Food Q costs Rs 80/kg. Food P contains 3 units/kg of Vitamin A and 5 units / kg of Vitamin B while Food Q contains 4 units/kg of Vitamin A and 2 units/kg of vitamin B. Determine the minimum cost of the mixture

asked Nov 16, 2012 by fingeazy

1 answer

Find $ \frac {dy}{dx} $ in the following: $y = cos^{-1} \left(\frac{1 - x^2}{1 + x^2}\right), 0 < x < 1 $

asked Nov 15, 2012 by thanvigandhi_1

1 answer

Consider $f:R_+ \to [\;\text{–5}, \infty )$given by $f(x)=9x^2 +6x$-$5$.Show that $f$ is invertible with $ f^{-1} (y) = \bigg(\frac{(\sqrt{y+6}) -1} { 3}\bigg) $

asked Nov 15, 2012 by vaishali.a

1 answer

Consider $ f:R_+ \to [4,\infty)$given by $f(x)=x^2 +4$. Show that $f$ is invertible with the inverse $f^{-1}$of $f$ given by $f^{-1}(y) = \sqrt {y-4}$,where $R_+$ is the set of all non-negative real numbers.

asked Nov 14, 2012 by vaishali.a

1 answer

Consider $f:R \to R$ given by $f(x)=4x+3$. Show that $f$ is invertible. Find the inverse of $ f$

asked Nov 14, 2012 by vaishali.a

1 answer

Show that $f : [-1,1]$ $ \rightarrow R$, given by $f(x) =\frac {x } { (x+2)}$ is one-one. Find the inverse of of the $f : [-1,1] \rightarrow $ Range $ f$.

asked Nov 14, 2012 by vaishali.a

1 answer

Find $ \frac {dy}{dx} $ in the following: $ sin^2y + cos \: xy = k $

asked Nov 11, 2012 by thanvigandhi_1

1 answer

Find the values of $a$ and $b$ such that the function defined by $ f(x) = \left\{ \begin{array} {1 1} 5 ,& \quad\text{ if $ x $ $\leq 2$}\\ ax+b,& \quad \text{if $2$ < x<10}\\ 21,& \quad \text{if $ x $ $\geq 10$}\\ \end{array} \right.$ is a continuous function.

asked Nov 11, 2012 by thanvigandhi_1

1 answer

If $f(x) = \frac { (4x+3) } { (6x-4) }, x \neq \frac {2} {3}$, show that $ f(x) =x $, for all $ x \neq \frac {2} {3}$. What is the inverse of $f$

asked Nov 9, 2012 by vaishali.a

1 answer

Discuss the continuity of the function $f$, where $f$ is defined by $f(x) = \left\{ \begin{array} {1 1} -2 ,& \quad\text{ if $ x $ $\leq -1$}\\ 2x,& \quad\text{ if $ -1 $ < x $\leq1$}\\ 2 ,& \quad\text{ if $ x $ > 1}\\ \end{array} \right. $

asked Nov 9, 2012 by thanvigandhi_1

1 answer

Discuss the continuity of the function $f$, where $f$ is defined by $ f(x) = \left\{ \begin{array} {1 1} 2x ,& \quad\text{ if $ x $ < 0}\\ 0,& \quad\text{if $ 0 $ $\leq x$ $\leq1$}\\ 4x ,& \quad\text{ if $ x $ > 1}\\ \end{array} \right. $

asked Nov 9, 2012 by thanvigandhi_1

1 answer

Find all points of discontinuity of $f$, where $f$fined by $ f(x) = \left\{ \begin{array} {1 1} x^{10} -1 ,& \quad\text{ if $ x $ $\leq 1$}\\ x^2,& \quad \text{if $x$ > 1}\\ \end{array} \right. $

asked Nov 9, 2012 by thanvigandhi_1

1 answer

Find all points of discontinuity of $f$, where $f$ is defined by $f(x) = \left\{ \begin{array} {1 1} \large \frac x { |\;x\;| },& \quad\text{ if $x$ < 0 }\\ -1 ,& \quad \text{if $x$ $ \geq 0$}\\ \end{array} \right. $

asked Nov 9, 2012 by thanvigandhi_1

1 answer

Let $A=R-\{3\}$ and $B=R-\{1\}$. Consider the function $ f:A \to B$ defined by $f(x)=\large\frac {x-2}{ x-3}$. Is $f$ one-one and onto?

asked Nov 9, 2012 by vaishali.a

1 answer

Find all points of discontinuity of $f$, where $f$ is defined by $ f(x) = \left\{ \begin{array} {1 1} |\;x\;| + 3,& \quad\text{ if $ x $ $\leq -3$}\\ -2x ,& \quad \text{if $-3$ <x< 3}\\ 6x+2, & \quad\text{if $x$ $\geq 3$}\\ \end{array} \right. $

asked Nov 8, 2012 by thanvigandhi_1

1 answer

Prove that the function $ f $ given by $f(x) = | x - 1 |, x \in R\;$ is not differentiable at $\; x = 1$.

asked Nov 8, 2012 by thanvigandhi_1

1 answer

Let $A$ and $B$be sets. Show that$f : A × B \to\; B×A$such that $ f(a,b)\,=\,(b,a)$ is bijective function

asked Nov 8, 2012 by vaishali.a

1 answer

Differentiate the functions with respect to x: $ cos \; x^3. sin^2 (x)^5 $

asked Nov 8, 2012 by thanvigandhi_1

1 answer

Show that the Modulus Function $f : R \to R$, given by $ f(x) = |\;x\;|$, is neither one-one nor onto, where$ |\;x\;|\; is\; x$, if $x$ is positive or $0$ and $|\;x\;|\; is\; $ $-$$x$, if $ x$ is negative.

asked Nov 8, 2012 by vaishali.a

1 answer

Show that the relation $R$ defined in the set $A$ of all triangles as $R = \{(T_1, T_2) : T_1\,is\, similar\, to\, T_2\}$, is equivalence relation. Consider three right angle triangles $(T_1 \;$ with sides $3, 4, 5, \; T_2$ with sides $5, 12, 13$, and $T_3$, with sides $6, 8, 10$) Which triangles among $T_1\, T_2\, and\, T_3$ are related?

asked Nov 8, 2012 by vaishali.a

1 answer

Find all points of discontinuity of $f$, where $f$ is defined by $f(x) = \left\{ \begin{array} {1 1} 2x + 3,& \quad\text{ if $ x $ $\leq 2$}\\ 2x - 3,& \quad \text{if $x$ > 1}\\ \end{array} \right. $

asked Nov 7, 2012 by thanvigandhi_1

1 answer

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

asked Nov 6, 2012 by thanvigandhi_1

1 answer

Examine that sin | x | is a continuous function.

asked Nov 6, 2012 by thanvigandhi_1

1 answer

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