Recent questions in Application of Derivatives

Find points on the curve $\large\frac{x^2}{9} + \frac{y^2}{16} = 1$ at which the tangents are $(i)\; parallel\; to\; x - axis$

asked Nov 24, 2012 by thanvigandhi_1

1 answer

Find the equation of all lines having slope $0$ that are tangents to the curve $y=\large \frac{1}{x^2-2x+3}$

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Find the equation of all lines having slope $2$ that are tangents to the curve $y= \large\frac{1}{x-3},$ $\: x \neq 3$

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Find the equation of all lines having slope $–1$ that are tangents to the curve $y= \large\frac{1}{x-1},$ $\: x \neq 1$

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Find the point on the curve $y = x^3 - 11x + 5$ at which the tangent is $y = x -11$ .

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Find a point on the curve $y = (x – 2)^2$ at which the tangent is parallel to the chord joining the points $(2, 0)$ and $(4, 4)$ .

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Find points at which the tangent to the curve $y = x^3 - 3x^2 - 9x + 7$ is parallel to the $x$ - axis.

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Find the slope of the normal to the curve $x = 1 - a \sin\theta, y = b \cos^2\theta$ at $\theta = \Large {\frac{\pi}{2}}$

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Find the slope of the normal to the curve $x = a \cos^3\theta, \: y = a \sin^3\theta$ at $\theta =\Large {\frac{\pi}{4}}$ .

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Find the slope of the tangent to the curve $y = x^3 - 3x + 2$ at the point whose $x$ - coordinate is $3$ .

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Find the slope of the tangent to curve $y = x^3 - x + 1$ at the point whose $x$ - coordinate is $2$ .

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Find the slope of the tangent to the curve $\normalsize y =\large { \frac{x-1}{x-2},} \normalsize x \neq 2\; at\;x = 10$

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Find the slope of the tangent to the curve $y = 3x^4 – 4x \; at\; x = 4.$

asked Nov 23, 2012 by thanvigandhi_1

1 answer

The interval in which $y = x^2 e ^{- x}$ is increasing is

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Prove that the function given by $f (x) = x^3 – 3x^2 + 3x - 100$ is increasing in $R.$

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Prove that the function $f$ given by $f (x) = \log \cos\: x$ is strictly decreasing on $\left(0, \: \large {\frac{\pi}{2}}\right)$ and strictly increasing on, $\left( \large {\frac{\pi}{2},} \normalsize \pi\right)$

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Prove that the function $f$ given by $f (x) = \log \sin x$ is strictly increasing on $\left(0, \: \large {\frac{\pi}{2}}\right)$ and strictly decreasing on $\left( \large {\frac{\pi}{2}}, \pi\right)$

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Let $I$ be any interval disjoint from $[–1, 1]$ . Prove that the function $f$ given by $f(x) = x + \frac{1}{\large x}$ is strictly increasing on $I$ .

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Find the least value of a such that the function $f$ given by $f (x) = x^2 + ax + 1$ is strictly increasing on $(1, 2).$

asked Nov 23, 2012 by thanvigandhi_1

1 answer

On which of the following intervals is the function $f$ given by $f (x) = x^{100} + sin x –1$ strictly decreasing ?

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Which of the following functions are strictly decreasing on $\left(0, \: \large {\frac{\pi}{2}}\right)$

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Prove that the function $f$ given by $f (x) = x^2 – x + 1$ is neither strictly increasing nor strictly decreasing on $(– 1, 1).$

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Prove that the logarithmic function is strictly increasing on $(0, \infty).$

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Prove that $y =\large { \frac{4\sin\theta}{(2+\cos\theta)}}-\normalsize \theta$ is an increasing function of $\theta$ in $\left[0, \: \large {\frac{\pi}{2}}\right]$

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Find the values of $x$ for which $y = [x(x – 2)]^2$ is an increasing function.

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Show that $y = log(1+x)-\large {\frac{2x}{2+x}}, \normalsize x > – \normalsize 1$ , is an increasing function of $x$ throughout its domain.

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Find the intervals in which the following functions are strictly increasing or decreasing: $(a) \;x^2 + 2x - 5$

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Find the intervals in which the function f given by $f (x) = 2x^3 – 3x^2 – 36 x + 7$ is (a) strictly increasing and (b) strictly decreasing ?

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Find the intervals in which the function $f$ given by $f (x) = 2x^2 - 3x$ is $(a)$ strictly increasing

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Show that the function given by $f (x) = \sin x$ is neither increasing nor decreasing in $(0, \pi)$

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Show that the function given by $f (x) = \sin x$ is strictly decreasing in $(\large {\frac{\pi}{2}},$ $\pi)$

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Show that the function given by $f (x) = \sin \;x$ is strictly increasing in $\left(0, \: \large {\frac{\pi}{2}}\right)$

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Show that the function given by $f(x) = e^{2x}$ is strictly increasing on $R$ .

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Show that the function given by $f (x) = 3x + 17$ is strictly increasing on $R$ .

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Choose the correct answer in the total revenue in Rupees received from the sale of $x$ units of a product is given by $Rx = 3x^2 +36x + 5$ . The marginal revenue, when $x = 15$ is

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Choose the correct answer in the rate of change of the area of a circle with respect to its radius $r$ at $r = 6$ cm is

asked Nov 23, 2012 by thanvigandhi_1

1 answer

The total revenue in Rupees received from the sale of $x$ units of a product is given by $R(x) = 13x^2 + 26x + 15.$ Find the marginal revenue when $x = 7$ .

asked Nov 23, 2012 by thanvigandhi_1

1 answer

The total cost $C(x)$ in Rupees associated with the production of $x$ units of an item is given by $C(x) = 0.007x^3 - 0.003x^2 + 15x + 4000.$ Find the marginal cost when 17 units are produced.

asked Nov 23, 2012 by thanvigandhi_1

1 answer

Sand is pouring from a pipe at the rate of $12 cm^3/s$ . The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?

asked Nov 23, 2012 by thanvigandhi_1

1 answer

A balloon, which always remains spherical, has a variable diameter $\large\frac{3}{2}$ $(2x+1)$ . Find the rate of change of its volume with respect to $x.$

asked Nov 23, 2012 by thanvigandhi_1

1 answer

The radius of an air bubble is increasing at the rate of $\large\frac{1}{2}$ cm/s. At what rate is the volume of the bubble increasing when the radius is $1\; cm$ ?

asked Nov 23, 2012 by thanvigandhi_1

1 answer

A particle moves along the curve $6y = x^3 +2$ . Find the points on the curve at which the $y$ -coordinate is changing $8$ times as fast as the $x$ -coordinate.

asked Nov 23, 2012 by thanvigandhi_1

1 answer

A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall ?

asked Nov 23, 2012 by thanvigandhi_1

1 answer

A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the later is $10\; cm$ .

asked Nov 22, 2012 by thanvigandhi_1

1 answer

A balloon, which always remains spherical on inflation, is being inflated by pumping in $900$ cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is $15\; cm$ .

asked Nov 22, 2012 by thanvigandhi_1

1 answer

The length $x$ of a rectangle is decreasing at the rate of $5\; cm/minute$ and the width y is increasing at the rate of $4\; cm/minute$ . When $x = 8$ cm and $y = 6$ cm, find the rates of change of (a) the perimeter

asked Nov 22, 2012 by thanvigandhi_1

1 answer

The radius of a circle is increasing at the rate of $0.7\; cm/s$ . What is the rate of increase of its circumference?

asked Nov 22, 2012 by thanvigandhi_1

1 answer

A stone is dropped into a quiet lake and waves move in circles at the speed of $5 \;cm/s$ . At the instant when the radius of the circular wave is $8 \;cm$ , how fast is the enclosed area increasing?

asked Nov 22, 2012 by thanvigandhi_1

1 answer

An edge of a variable cube is increasing at the rate of $3\; cm/s$ . How fast is the volume of the cube increasing when the edge is $10\; cm$ long?

asked Nov 22, 2012 by thanvigandhi_1

1 answer

The radius of a circle is increasing uniformly at the rate of $3\; cm/s.$ Find the rate at which the area of the circle is increasing when the radius is $10\; cm$ .

asked Nov 22, 2012 by thanvigandhi_1

1 answer

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