Recent questions in Differential Calculus Applications - I - Clay6.com, a Free resource for your JEE, AIPMT and Board Exam preparation

Find the intervals on which $f$ is increasing or decreasing. $f(x)=x-2\sin x,[0 , 2\pi ]$

asked May 4, 2013 by poojasapani_1

1 answer

Find the intervals on which $f$ is increasing or decreasing. $f(x)=x^{3}+x+1$

asked May 4, 2013 by poojasapani_1

1 answer

Find the intervals on which $f$ is increasing or decreasing. $f(x)= x^{3}-3x+1$

asked May 4, 2013 by poojasapani_1

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Find the intervals on which $f$ is increasing or decreasing. $f(x)=20-x-x^{2}$

asked May 4, 2013 by poojasapani_1

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Prove that the following functions are not monotonic in the intervals given. $\tan x + \cot x$ on $[0 , \large\frac{\pi}{2}]$

asked May 4, 2013 by poojasapani_1

1 answer

Prove that the following functions are not monotonic in the intervals given. $x \sin x$ on $[0 , \pi ]$

asked May 4, 2013 by poojasapani_1

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Prove that the following functions are not monotonic in the intervals given. $x(x-1)(x+1)$ on $[0 , 2 ]$

asked May 4, 2013 by poojasapani_1

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Prove that the following functions are not monotonic in the intervals given. $2x^{2}+x-5$ on $[-1 ,0 ]$

asked May 4, 2013 by poojasapani_1

2 answers

Which of the following function are increasing or decreasing on the interval given? $x\sin x$ on $[0 , \large\frac{\pi}{4}]$

asked May 4, 2013 by poojasapani_1

1 answer

Which of the following function are increasing or decreasing on the interval given? $x(x-1)(x+1)$ on $[-2 , 1 ]$

asked May 4, 2013 by poojasapani_1

1 answer

Which of the following function are increasing or decreasing on the interval given? $e^{-x}$ on $[0 , 1 ]$

asked May 4, 2013 by poojasapani_1

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Which of the following function are increasing or decreasing on the interval given? $2x^{2}+3x$ on $[-\large\frac{1}{2} ,\frac{1}{2}]$

asked May 4, 2013 by poojasapani_1

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Which of the following function are increasing or decreasing on the interval given? $x^{2}-1$ on $[0 , 1]$

asked May 4, 2013 by poojasapani_1

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Prove that $\log _e x$ is strictly increasing function on $(0 ,\infty)$ .

asked May 4, 2013 by poojasapani_1

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Prove that $e^{x}$ is strictly increasing function on $R$ .

asked May 4, 2013 by poojasapani_1

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Evaluate the limit for the following if exists, $\;\lim\limits_{x \to 2} \large\frac{\sin\pi x}{2-x}$

asked May 4, 2013 by poojasapani_1

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Obtain the Maclaurin's series expansion for: $\;\tan x,- \large\frac{\pi}{2} \lt \normalsize x \lt \large\frac{\pi}{2}$

asked May 4, 2013 by poojasapani_1

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Obtain the Maclaurin's series expansion for: $\;\large\frac{1}{1+x}$

asked May 4, 2013 by poojasapani_1

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Obtain the Maclaurin's series expansion for: $\;\cos^{2}x$

asked May 4, 2013 by poojasapani_1

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Obtain the Maclaurin's series expansion for: $\;e^{2x}$

asked May 4, 2013 by poojasapani_1

1 answer

At $2.00$ pm a car's speedometer reads $30$ miles/hr, at $2.10$ pm it reads $50$ miles/hr. Show that sometime between $2.00$ and $2.10$ the acceleration is exactly $120$ miles/hr $^{2}$ .

asked May 4, 2013 by poojasapani_1

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If $f(1)=10$ and $f '(x)\geq 2$ for $1\leq x \leq 4$ how small can $f(4)$ possibly be?

asked May 4, 2013 by poojasapani_1

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Verify Lagrange's theorem for the following function; $f(x)=x^{3}-5x^{2}-3x\;[1,3]$

asked May 3, 2013 by poojasapani_1

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Verify Lagrange's theorem for the following function; $f(x)=x^{\large\frac{2}{3}}[-2,2]$

asked May 3, 2013 by poojasapani_1

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Verify Lagrange's theorem for the following function; $f(x)=2x^{3}+x^{2}-x-1\quad [0,2]$

asked May 3, 2013 by poojasapani_1

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Verify Lagranges theorem for the following function; $f(x)=\large\frac{1}{x}$ $\quad[1,2]$

asked May 3, 2013 by poojasapani_1

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Verify Lagrange's law of mean for the following function; $f(x)=1-x^{2} \quad [0,3]$ .

asked May 3, 2013 by poojasapani_1

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Using Rolle's theorem find the points on the curve $y=x^{2}+1,-2\leq x \leq 2$ where the tangent is parallel to $x$ - axis.

asked May 3, 2013 by poojasapani_1

1 answer

Verify Rolle's theorem for the following function; $f(x)=4x^{3}-9x;-\large\frac{3}{2}\leq x\leq \frac{3}{2}$ .

asked May 3, 2013 by poojasapani_1

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Verify Rolle's theorem for the following functions; $f(x)=|x-1|,0\leq x\leq 2.$

asked May 3, 2013 by poojasapani_1

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Verify Rolle's theorem for the following function; $f(x)=x^{2}, 0\leq x\leq 1$

asked May 3, 2013 by poojasapani_1

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Verify Rolle's theorem for the following function; $f(x)=\sin x, 0\leq x\leq\pi$ .

asked May 3, 2013 by poojasapani_1

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If the curve $y^{2}=x$ and $xy=k$ are orthogonal than prove that $8k^{2}=1.$

asked May 3, 2013 by poojasapani_1

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Show that the equation of the normal to the curve $x=a\cos^{3}\theta ; y=a\sin^{3}\theta$ at $\;'\theta'$ is $\; x\cos\theta-y\cos\theta=a\cos 2\theta.$

asked May 3, 2013 by poojasapani_1

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At what angle $\theta$ do the curves $y=a^{x}$ and $y=b^{x}$ intersect $(a{\neq}b)$ ?

asked May 3, 2013 by poojasapani_1

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Prove that the curve $2x^{2}+4y^{2}=1$ and $6x^{2}-12y^{2}=1$ cut each other at right angles.

asked May 3, 2013 by poojasapani_1

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Let $P$ be a point on the curve $y=x^{3}$ and suppose that the tangent line at $P$ intersects the curve again at $Q$ . Prove that the slope at $Q$ is four times the slope at $P$ .

asked May 3, 2013 by poojasapani_1

1 answer

Find the equations of normal to $y=x^{3}-3x$ that is parallel to $2x+18y-9=0.$

asked May 3, 2013 by poojasapani_1

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Find the equations of those tangents to the circle $x^{2}+y^{2}=52,$ which are parallel to the straight line $2x+3y=6$ .

asked May 3, 2013 by poojasapani_1

1 answer

At what points on the curve $x^{2}+y^{2}-2x-4y+1=0$ the tangent is parallel to $y$ - axis.

asked May 3, 2013 by poojasapani_1

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At what points on the curve $x^{2}+y^{2}-2x-4y+1=0$ the tangent is parallel to $x$ - axis.

asked May 3, 2013 by poojasapani_1

1 answer

Find at what points on the circle $x^{2}+y^{2}=13$ , the tangent is parallel to the line $2x+3y=7$ .

asked May 3, 2013 by poojasapani_1

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Find the point on curve $x^{2}-y^{2}=2$ at which the slope of the tangent is $2$

asked May 1, 2013 by poojasapani_1

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Find the equation of the tangent and normal to the curves. $\;y=\large\frac{1+\sin x}{\cos x}$ at $\;x=\large\frac{\pi}{4}$

asked May 1, 2013 by poojasapani_1

1 answer

Find the equation of the tangent and normal to the curves. $\;y=2\sin^{2} 3x$ at $\;x=\large\frac{\pi}{6}$

asked May 1, 2013 by poojasapani_1

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Find the equation of the tangent and normal to the curves. $\;y=x-\sin x\cos x,$ at $\;x=\large\frac{\pi}{2}$

asked May 1, 2013 by poojasapani_1

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Find the equation of the tangent and normal to the curves. $y=x^{2}-4x-5$ at $\;x=-2$

asked May 1, 2013 by poojasapani_1

1 answer

Gravel is being dumped from a conveyor belt at a rate of $30\;ft^{3}/min$ and its coarsened such that it forms a pile in the shape of the cone whose base diameter and hight are always equal. How fast is the hight of the pile increasing when the pile is $10$ ft hight?

asked May 1, 2013 by poojasapani_1

1 answer

Two sides of a tringle have length $12$ m and $15$ m. The angle between them is increasing at a rate of $2^{\circ}/min.$ How fast is the length of third side increasing when the angle between the sides of fixe is $60^{\circ}$ ?

asked May 1, 2013 by poojasapani_1

1 answer

Two sides of a tringle are $4$ m and $5$ m in length and the angle between them is increasing at a rate of $0.06$ rad/sec. Find the rate at which the area of tringle is increasing when the angle between the sides of fixed length is $\pi/3$ .

asked May 1, 2013 by poojasapani_1

1 answer

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Recent questions in Differential Calculus Applications - I