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Recent questions tagged ch6

Show that the normal at any point \(\theta\) to the curve $ x = a \cos\theta + a \: \theta \sin\: \theta, y = a\sin\theta - a\theta \cos\theta$ is at a constant distance from the origin.

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Find the equation of the normal to curve \(y^2 = 4x\) which passes through the point \((1, 2).\)

asked Nov 28, 2012 by thanvigandhi_1

1 answer

The two equal sides of an isosceles triangle with fixed base $b$ are decreasing at the rate of 3 cm per second. How fast is the area decreasing when the two equal sides are equal to the base ?

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Show that the function given by \( f(x) =\large \frac{\log \: x}{x}\) has maximum at \(x = e.\)

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Using differentials, find the approximate value of each of the following: $(33)^{-\large\frac{1}{5}}$

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Using differentials, find the approximate value of each of the following: $ \left(\large\frac{17}{81}\right)^{\frac{1}{4}}$

asked Nov 28, 2012 by thanvigandhi_1

1 answer

The maximum value of $ [ x(x-1) +1]^{\frac{1}{3}}, 0 \leq \: x \leq$ is

asked Nov 27, 2012 by thanvigandhi_1

1 answer

For all real values of \(x\), the minimum value of \( \large\frac{1-x+x^2}{1+x+x^2}\) is

asked Nov 27, 2012 by thanvigandhi_1

1 answer

Choose the correct answer in the point on the curve \( x^2 = 2y\) which is nearest to the point \((0, 5)\) is

asked Nov 27, 2012 by thanvigandhi_1

1 answer

Show that semi-vertical angle of right circular cone of given surface area and maximum volume is \( \sin^{-1} \left(\frac{1}{3}\right)\)

asked Nov 27, 2012 by thanvigandhi_1

1 answer

Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is \(\tan^{-1} \sqrt {2}\).

asked Nov 27, 2012 by thanvigandhi_1

1 answer

Show that the right circular cone of least curved surface and given volume has an altitude equal to \(\sqrt 2\) time the radius of the base.

asked Nov 27, 2012 by thanvigandhi_1

1 answer

Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is\(\large\frac{8}{27}\) of the volume of the sphere.

asked Nov 27, 2012 by thanvigandhi_1

1 answer

A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?

asked Nov 27, 2012 by thanvigandhi_1

1 answer

Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area?

asked Nov 27, 2012 by thanvigandhi_1

1 answer

Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base.

asked Nov 27, 2012 by thanvigandhi_1

1 answer

Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

asked Nov 27, 2012 by thanvigandhi_1

1 answer

A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square in cms to be cut off so that the volume of the box is maximum ?

asked Nov 27, 2012 by thanvigandhi_1

1 answer

A square piece of tin of side 18 cm is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible.

asked Nov 27, 2012 by thanvigandhi_1

1 answer

Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.

asked Nov 27, 2012 by thanvigandhi_1

1 answer

Find two positive numbers \(x\) and \(y\) such that their sum is 35 and the product \(x^2 y^5\) is a maximum.

asked Nov 27, 2012 by thanvigandhi_1

1 answer

Find two positive numbers $x$ and $y$ such that $x + y = 60$ and $xy^3$ is maximum.

asked Nov 27, 2012 by thanvigandhi_1

1 answer

Find two numbers whose sum is 24 and whose product is as large as possible.

asked Nov 27, 2012 by thanvigandhi_1

1 answer

Find the maximum and minimum values of \( x + \sin\: 2x\) on \([0, 2\pi].\)

asked Nov 27, 2012 by thanvigandhi_1

1 answer

It is given that at $x = 1$, the function $x^4 – 62x^2 + ax + 9$ attains its maximum value, on the interval $[0, 2].$ Find the value of $a.$

asked Nov 27, 2012 by thanvigandhi_1

1 answer

Find the maximum value of \(2x^3 – 24x + 107\) in the interval \([1, 3]\).

asked Nov 27, 2012 by thanvigandhi_1

1 answer

What is the maximum value of the function $\sin \: x + \cos x$?

asked Nov 27, 2012 by thanvigandhi_1

1 answer

At what points in the interval \([0, 2\pi]\), does the function \(\sin\: 2x\) attain its maximum value?

asked Nov 27, 2012 by thanvigandhi_1

1 answer

Find both the maximum value and the minimum value of $3x^4 - 8x^3 + 12x^2 - 48x + 25$ on the interval $[0, 3].$

asked Nov 27, 2012 by thanvigandhi_1

1 answer

Find the maximum profit that a company can make, if the profit function is given by $ p(x) = 41 - 72x - 18x^2$

asked Nov 27, 2012 by thanvigandhi_1

1 answer

Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals: $\: f (x) = x^3, x \in [– 2, 2] $

asked Nov 26, 2012 by thanvigandhi_1

1 answer

Prove that the following functions do not have maxima or minima: $ (i)\: f (x) = e^x $

asked Nov 26, 2012 by thanvigandhi_1

1 answer

Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be: $ (i)\: f (x) = x^2 $

asked Nov 26, 2012 by thanvigandhi_1

1 answer

Find the maximum and minimum values, if any, of the following functions given by $ f (x) = |x + 2| - 1$

asked Nov 26, 2012 by thanvigandhi_1

1 answer

Find the maximum and minimum values, if any, of the following functions given by?$\;f(x)=(2x-1)^2+3$

asked Nov 26, 2012 by thanvigandhi_1

1 answer

The approximate change in the volume of a cube of side $x$ metres caused by increasing the side by $3\%$ is

asked Nov 26, 2012 by thanvigandhi_1

1 answer

If $f(x) = 3x^2 + 15x + 5$, then the approximate value of $f (3.02)$ is

asked Nov 26, 2012 by thanvigandhi_1

1 answer

If the radius of a sphere is measured as $9 \: m$ with an error of $0.03 \: m$, then find the approximate error in calculating its surface area.

asked Nov 26, 2012 by thanvigandhi_1

1 answer

If the radius of a sphere is measured as $7 \: m$ with an error of $0.02 \: m$, then find the approximate error in calculating its volume.

asked Nov 26, 2012 by thanvigandhi_1

1 answer

Find the approximate change in the surface area of a cube of side $x$ metres caused by decreasing the side by $1\%$.

asked Nov 26, 2012 by thanvigandhi_1

1 answer

Find the approximate change in the volume V of a cube of side $x$ metres caused by increasing the side by $1\%$.

asked Nov 26, 2012 by thanvigandhi_1

1 answer

Find the approximate value of \(f (5.001)\), where $f (x) = x^3 – 7x^2 + 15.$

asked Nov 26, 2012 by thanvigandhi_1

1 answer

Find the approximate value of $ f (2.01)$, where $f (x) = 4x^2 + 5x + 2$.

asked Nov 26, 2012 by thanvigandhi_1

1 answer

Using differentials, find the approximate value of each of the following up to $3$ places of decimal. $(i)\;\sqrt{25.3}$

asked Nov 26, 2012 by thanvigandhi_1

1 answer

Choose the correct answer in the line $y = x + 1$ is a tangent to the curve $y^2 = 4x$ at the point

asked Nov 26, 2012 by thanvigandhi_1

1 answer

Choose the correct answer in the slope of the normal to the curve $ y = 2x^2 + 3 \sin\: x$ at $x = 0$ is

asked Nov 26, 2012 by thanvigandhi_1

1 answer

Find the equation of the tangent to the curve $y = \sqrt {3x - 2}$ which is parallel to the line $4x - 2y + 5 = 0 $.

asked Nov 24, 2012 by thanvigandhi_1

1 answer

Find the equations of the tangent and normal to the hyperbola $\large {\frac{x^2}{a^2}} -\large { \frac{y^2}{b^2}} =\normalsize 1$ at the $ (x_0, \: y_0)$.

asked Nov 24, 2012 by thanvigandhi_1

1 answer

Prove that the curves \(x = y^2\) and \(xy = k\) cut at right angles* if \(8k^2 = 1.\)

asked Nov 24, 2012 by thanvigandhi_1

1 answer

Find the equations of the tangent and normal to the parabola $y^2 = 4ax$ at the point $(at^2, 2at)$.

asked Nov 24, 2012 by thanvigandhi_1

1 answer

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