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Find the rate of change of the area of a circle with respect to its radius r when.
(a) r = 3 cm. (b) r = 4 cm
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The volume of cube is increasing at the rate of 8 cm3/s. How fast is the surface area increasing when the length of an edge is 12 cm ?
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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 10cm.
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An adge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the increasing when the edge is 10 cm long ?
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A stone is droped 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 ?
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The radius of a circle is increasing at the of 0.7 cm/s. What is the rate of increase of its circumference ?
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The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at 4 cm/minute. When x = 8 cm and y = 6 cm, find the rate of change of (a) the perimeter, and (b) the area of the rectangle.
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A balloon, which always remains sphereical, is being inflated by pumping in 900 cubic centimeters of gas per second. Find the rate at which the radius of the ballon increases when r = 15 cm.
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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.
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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 2 cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall ?
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A particle moves along the curve 6y = x3 + 2. Find the points on the curve at which the y-coordinate is changing 8 times at fast at the x-coordinate.
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The
radius of an air bubble is increasing at the rate of
At what
rate is the volume of the bubble increasing when the radius is 1 cm ?
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A
balloon, which always remains spherical, has a variable diameter
. Find the
rate of change of its volume w.r.t. x
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Sand is pouring from a pipe at the rate of 12 cm2/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 ?
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The total cost c(x) in rupees associated with the production of x untis of an item is given by C(x) = 0.007 x3 – 0.003 x2 + 15x +4000. Find the marginal cost when 17 units are produced.
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Total revenue in rupees received from the sale of x units of a product is given by R(x) = 13x2 + 26x + 15. Find the marginal revenue when x = 7.
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Show that f(x) = 3x + 17 is strictly increasing on R.
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Show that f(x) = e2x is strictly increasing on R.
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Show
that f(x) = sin x is
(a)
strictly increasing in
(b)
strictly decreasing in
(c)
neither increasing nor
decreasing in (0,
)
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Find the interval is which f(x) = 2x2 – 3x is
(a) strictly increasing (b) strictly decreasing.
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Find the intervals in which f(x) = 2x3 – 3x2 – 36x + 7 is
(a) strictly increasing
(b) strictly decreasin+
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Find the intervals in which the following functions are strictly increasing or strictly decreasing :
(a) x2 + 2x – 5
(b) 10 – 6x – 2x2
(c) –2x3 – 9x2 – 12x +1
(d) 6 – 9x – x2
(e) (x +1)3 (x – 3)3
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Show
that y = log (1 + x)
x
> ?1, is an increasing function of x through out its domain.
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Find the value of x for which = [x (x – 2)]2 is an increasing function.
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Prove
that
is an
increasing function of
in
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Prove
that logarithmic function is strictly increasing on
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Prove that f(x) = x2 – x + 1 is neither increasing nor decreasing on (–1, 1).
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Find the least value of a such that f(x) = x2 + ax + 1 is strictly increasing on (1, 2).
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Let
I be any interval disjoint from (?1, 1). Prove that the function f defined by
f(x) =
is
strictly increasing on I.
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Prove that f(x) = log sin x is strictly increasing on and
strictly decreasing on
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Prove that f(x) = log cos x is strictly decreasing on
and
strictly increasing on
.
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Prove that f(x) = [x3 – 3x2 + 3x – 100] is increasing on R.
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Find the slope of the tangent to curve
= 3x4 – 4x at x = 4.
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Find
the slope of the tangent to curve
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Find the slope of tangent to the curve = x3 – x + 1 at the point whose x-coordinate is 2.
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Find the slope of tangent to curve
= x3 – 3x + 2 at x = 3.
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Find
the slope of the normal to the curve
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Find the slope of normal to the curve x = 1 ? a sin
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Find points at which are tangent to the curve y = x3 – 3x2 – 9x + 7 is parallel to the x-axis.
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Find a point on the curve y = (x – 2)2 at which the tangent is parallel to the chord joining (2, 0) and (4, 4)
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Find the point on the curve y = x3 – 11x + 5 at which the tangent is y = x – 11.
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Find the equation of all lines having slope ?1 that are
tangents to the curve
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Find the equation of all lines
having slope 2 which are tangent to the curve
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Find equation of all lines having
slope 0 which are tangent to the curve
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Find the points on the
curve
at which
the tangents are (a) parallel to X-axis (b) parallel to Y-axis.
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Find the equations of the tangent and normal to the given
curves at the indicated points :
(I) y = x4
? 6x3 + 13x2 ? 10x + 5 at (0, 5)
(II) y = x4 ? 6x3 + 13x2
? 10x + 5 at (1, 3)
(III) y = x3 at (1, 1)
(IV) y = x2 at (0, 0)
(V) x = cos t, y = sin t at t
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Find the equation of tangent to the curve y = x2 – 2x + 7 which is
(a) Parallel to line 2x – y + 9 = 0
(b) Perpendicular to lien 5y – 15x = 13
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Show that tangent to the curve y = 7x3 + 11 at the points where x = 2 and x = –2 are parallel.
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Find the points on the curve y = x3 at which the slope of the tangent is equal to the y-coordinate of the point.
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For the curve y = 4x3 – 2x5, find all the points at which the tangent passes through the origin.
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Find the points on the curve x2 + y2 – 2x – 3 = 0 at which the tangent are parallel to X-axis.
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Find the equation of normal at (am2, am3) for the curve ay2 = x3.
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Find the equation of normal to the curve y = x3 + 2x + 6 which are parallel to the line x + 14y + 4 = 0.
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Find the equation of the tangent and normal to the parabola y2 = 4ax at (at2, 2at)
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Find the equation of the tangent and normal to the parabola y2 = 4ax at (at2, 2at)
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Prove that the curve x = y2 and xy = K cut at right angles if 8K2 = 1.
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Find the equation of the tangent and normal to the
hyperbola
at (x0,
y0).
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Find the equation of tangent to the curve which is
parallel to the line 4x ? 2y + 5 = 0.
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Find the approximate value of f(2.01) where f(x) = 4x2 + 5x + 2.
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Find the approximate value of f(5.001) where
f(x) = x3 – 7x2 + 15.
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Find the approximate change in the volume V of a cube of side x metres caused by increasing the side by 1%.
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Find the approximate change in the surface area of a cube of side x metres caused by decreasing the side by 1%.
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If the radius of a sphere is measured as 7m with an error of 0.02 m, find the approximate error in calculating its volume.
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If the radius of a sphere is measured at 9m with an error of 0.03 m, find the approximate error in calculating its surface area.
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Find the maximum and minimum values, if any of the following functions given by :
(i) f(x) = (2x – 1)2 + 3
(ii) f(x) = 9x2 + 12x + 2
(iii) f(x) = – (x – 1)2 + 10
(iv) g(x) = x3 + 1
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Find
the maximum and minimum values, if any of the following functions given by :
(I)
f(x) = |x
+ 2| ? 1
(II)
g(x) = ? |x + 1|
+ 3
(III)
h(x) = sin (2x) + 5
(IV)
f(x) = |sin 4x + 3|
(V)
h(x) = x + 1,
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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) = x2
(II)
g(x) = x3
? 3x
(III)
h(x) = sin x + cos x,
0 < x <
(IV)
f(x) = sin x ? cos x, 0 <
x <
(V)
f(x) = x3 ?
6x2 + 9x + 15
(VI)
g(x)
(VII)
(VII)
f(x) = x
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Prove that following functions do not have maximu or minima.
(I) f(x) = ex (II) g(x) = log x
(III) h(x) = x3 + x2 + x + 1
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Find the maximum profit that a company can make, if the profit function is given by:
p(x) = 41 + 24x – 18x2
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Find both the maximum and minimum values of f(x) = 3x4 – 8x3+ 12x2 – 48x + 25 on the interval [0, 3}.
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At what in the
interval [0,
, does the
function sin 2x attain its maximum value ?
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What is the maximum value of the function sin x + cos x?
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Find the maximum value of 2x3 – 24x + 107 in [1, 3]. Find the maximum value of the same function in [–3, –1].
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It is given that x = 1, the function x4 2 62x2 + ax + 9 attains its maiximum value, on the interval [0, 2]. Find the value of a.
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Find the maximum and
minimum values of x + sin 2x on [0,
]
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Find two numbers whose sum is 24 and whose product is as large as possible.
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Find two positive number x and y such that x + y = 60 and xy3 is maximum.
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Find two positive numbers x and y such that their sum is 35 and the product x2 y5 is a maximum.
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Find two positive numbers whose sum is 16 and sum of whose cubes is minimum.
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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 from the box. What should be the side of square to be cut off so that the volume at the box is the maximum possible ?
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A rectangular sheet of tin 45 cm × 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 to be cut off so that the volume of the box is maximum ?
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Show that of al the rectangles inscribed in a given fixed circle, the square has the maximum area.
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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.
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Of all the right circular cylindrical Cans, of given volume of 100cm3, find the dimensions of the Can which has the minimum surface area ?
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A wire of length 28m is to be cutr 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?
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Prove
that the volume of the largest cone that can be inscribed in a sphere of radius
R is
of the
volume of the sphere.
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Show
that the right circular cone of least curved surface and given volume has an
altitude equal to
times
the radius of the base.
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Show
that the semi-vertical angle of the cone of maximum volume and given slant
height is tan?1
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Show that semi-vertical angle of the cone of given surface
area and maximum volume is sin?1
.
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Using
differentials, find the approximate value of each of the following :
(a)
(b)
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Show that the function
given by f(x)
has
maximum at x = e.
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The two equation sides of an isosceles triangle with fixed base b are decreasing at the rate 3 cm per second. How fast is the area decreasing when the two equal sides are equal to the base ?
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Find the equation of the normal to the curve y2 = 4x at (1, 2).
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Show that normal at any point to the
curve x = a cos +
a sin , y = a sin
? a cos is at a
constant distance from the origin
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Find the intervals in which on function f given by f(x)
is (I)
increasing (II) decreasing.
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Find the maximum area of an isosceles triangle inscribed in
the ellipse
with its
vertex at one end of the major axis.
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A tank with rectangular base and sides, open at the top of two be constructed so that its depth is 2m. and volume is 8m3. If building of tank costs Rs.70 per m2 for the base and Rs.45 per m2 for sides. What is the cost of least expensive tank ?
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The sum of the perimeter of a circle and square is k, where k, where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.
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A
window is in the form of a rectangle surmounted be semi-circular opening. The
total perimeter of the window is 10 m. Find the dimensions of the window to
admit maximum light through the whole opening.
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A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle Show that minimum length of the hypotenuse is
.
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Find the points at which f(x) = (x – 2)4 (x + 1)3 has
(1) Local maxima (II) Local minima (III) Point of inflexion.
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Find the absolute maxima and minimum values of f(x) = cos2x
+ sin x,
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Show that altitude of the right circular cone of maximum
volume that can be inscribed in a sphere of radius r is
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Let f be a function defined on [a, b] such that Then
prove that f is an increasing function on (a, b).
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Show that the height of the cylinder of maximum volume that
can be inscribed in a sphere of radius
Also find
the maximum volume.
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Show
that the height of the cylinder of greatest volume which can be inscribed in a
right circular cone of height h and semi-vertical angle
is
one-third that of the cone and greatest volume of cylinder is
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