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APPLICATIONS OF DERIVATIVES
(1 mark Questions)
1.
If the Rolle’s theorem is applicable to the function
f ( x ) = x 4 + 3x 3 + 3x 2 + 2x on [k,0], what is value of k?
2.
State with reason whether or not the Rolle’s theorem is applicable to the function
2/ 3
f ( x ) = ( x − 1)
3.
on [0, 2].
Find the point on the curve y = 3x 4 − 4x 3 + 6x 2 − 12x + 5 at which the
tangent is parallel to x-axis.
4.
Find the point on the curve y = x 2 + 5x + 3 at which the tangent is parallel to
the line y = 3x + 2.
5.
What is the slope of the tangent to the curve y = 10 + 3x − x 2 at the point
( -2, 0 )?
6.
What is the slope of the normal to the curve y = 2x 2 − 3x + 1 at the point ( 2, 3 )
7.
Find the point on the curve y = 3x 2 − 5x + 4 at which the tangent is parallel to
the cord joining A (1, 2 ) and B ( 3, 14 ) .
8.
What is the least value of the function f ( x ) = x 2 − 7x + 15.
9.
What is the least value of x at which f ( x ) = 2x 2 − 20x + 25 attains its least
value?
10.
What is the greatest possible value of the function f ( x ) = 20 + 4x − x 2 .
11.
What are the minimum and maximum values of the function
f ( x ) = 2 sin x + 3?
12.
What are the minimum and maximum values of the function 3 − 4 cos x ?
13.
What are the minimum and maximum values of f ( x ) = 3 sin x + 4 cos x.
14.
What is the value of x at which f ( x ) = sin x + cos x is maximum?
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15.
What are the minimum and maximum values of the function sin 4x + 3 ?
16.
What are the minimum and maximum values of the function 3 − 8 cos x .
17.
If the function f ( x ) = f ( x ) = x 2 − ax + b has a local minimum at
(2, −1 ). What are the values of a and b?
18.
What are the stationary points of the function f ( x ) = 2x 3 + 3x 2 − 36x − 60.
19.
What is the value for which he function f ( x ) = x 4 − 162x 2 + ax + 9 attains its
maximum value at x = 1?
20.
The diameter of a balloon is given by
1
( 4x + 3 ) where, x is a variable. Find the
2
rate of change of its surface area w.r.t. x.
21.
The radius of a cone of eight 15 cm is increasing at the rate of 0.5 cm/sec. What is
rate of increase of its volume when the radius is 8cm?
22.
If the side of a square is increasing at the rate of 0.3 cm/sec., what is the rate of
increase of its area when the rate of increase of its area when the side is 5 cm?
23.
If the edge of a variable cube is increasing at the rate of 0.3cm / sec, what is the
rate of increase of its surface area when the edge is 20 cm?
24.
If the edge of a cube is increasing at the rate of 0.5cm / sec, find the rate of
increase of its volume when the edge is 16 cm.
25.
State with reason whether or not the function f ( x ) = x 3 − 3x 2 + 3x + 1 is
increasing on R.
26.
Find the values of x for which the function f ( x ) = x 2 − 6x + 7 is strictly
decreasing.
27.
Find the interval in which the function f ( x ) = x 3 − 12x + 9 is strictly
decreasing.
28.
Find the interval in which f ( x ) = 17 + 12x − 2x
2
is strictly increasing.
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Mathematics Class 12
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2x + 1
29.
What is the domain of the function f ( x ) =
30.
Find the interval in which the function f ( x ) = log ( x − 1) is increasing.
16 − x 2
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ANSWERS
1.
k = −2
2.
No, since differentiability does not hold.
3.
(1, −2) .
4.
( −1, −1) .
5.
7
6.
−
7.
 11 59 
 ,

 6 12 
8.
11 / 4
9.
5
10.
24
11.
Min. = 1, Max = 5
12.
Min v = −1, Max = 7
13.
Min = -5, max = 5
1
5
14.
π/4
15.
Min = 2, Max = 4
16.
Min = 0, Max = 11
17.
a = 4, b = 3
18.
x = −3 and x = 2
19.
a = 320
20.
8π ( 4x + 3 )
21.
40 cm3 / sec
22.
3 cm2 / sec
23.
72cm2 / sec
24.
384 cm3 / sec .
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25.
Increasing since f' ( x ) > 0 for all x ∈ R
26.
x<3
27.
( −2, 2)
28.
( −∞, 3 )
29.
( −4.4 )
30.
(1,∞ )
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Mathematics Class 12
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DERIVATIVES AS A RATE MEASURE
4 Marks and 6 Marks Questions
1.
An air-force plane is ascending vertically at the rate of 100km/hour. If the radius
the earth is r km, how fast in the area of the earth visible from plane increasing at
3 minutes after it starts ascending.
(Hint:Visible area A =
2.
2πr2 h
, where h is the height of the lane above the earth)
r+h
Water is dripping out from a conical funnel, at a uniform rate of
2 cm3 / sec through a tiny hole at the vertex of the bottom when the slant height
of the water is 4 cm. Find the rate of decrease of the slant height of water given
the vertical angle of the funnel is 1200.
3.
A man is moving away from a tower 41.6 high at the rate of 2m/sec. Find the rate
at which the angle of elevation of the tower is changing when he is at a distance of
30 m from the foot of the tower. Assume the eye level of the man is 1.6 m from
the ground.
4.
Water is running into a conical vessel, 15 cm deep and 5 cm in radius at the rate
0.1 cm3 / sec . W hen the water is 6 cm deep find at what rate is:
5.
(a)
the water level rising?
(b)
the water surface area decreasing?
(c)
the wetted surface of the vessel increasing.
Water is dripping out from a conical funnel of semi-vertical angle π / 4 at a
uniform rate of 2cm2 / sec through a tiny hole at the vertex in the bottom.
When the slant height of the water is 4 cm. Find the rate of decrease of the slant
height of water.
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Mathematics Class 12
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Rate of Change
1.
3. −
200 πr3
2
(r + 5)
sq km/h
2. −
1
cm / sec
3π
4
radian/sec
125
1
cm / sec
40π
4.
(a )
5.
2
cm / sec
4π
(b)
1
cm2 / sec
30
(c )
10
cm2 / sec
30
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Mathematics Class 12
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INCREASING AND DECREASING FUNCTIONS
1.
1 

Show that f  x =
 decreases in the interval [0, ∞ ]. and increases in the

1 + x2 
interval [ −∞,0]
2.
Show that f ( x ) = ( x − 1) e x + 15
3.
Show that f ( x ) = 2x + sin x + cos x increases for all x ∈ R.
4.
 π
Show that f ( x ) = sin 2 x increases on 0,  .
 4
5.
Find the intervals in which the function f ( x ) = x 4 − x 3 / 3 is strictly increasing
is an increasing function for all x > 0.
or strictly decreasing.
6.
Show that he function f ( x ) = tan −1 ( sin x + cos x ) , x > 0 is always a strictly
increasing function in ( 0, π / 4 )
7.
Find the intervals in which the given function is increasing in which the given
function
is
strictly
increasing
or
strictly
decreasing
f ( x ) = x 4 − 8x 3 + 22x 2 − 24x + 21.
8.
Separate
( 0, π / 2)
into sub-interval in which f ( x ) = sin 3x is strictly
increasing or strictly decreasing.
9.
Separate
( 0, π / 2)
into sub-intervals in which f ( x ) = sin 4 x + cos 4 x is
strictly increasing or strictly decreasing.
10.
Find the intervals in which f ( x ) =
x
is strictly increasing or strictly
log x
decreasing ( given x > 0 ) .
11.
Show that tan −1 x ≥
x − x3
for all x ≥ 0.
3
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ANSWERS
Increasing & Decreasing function
6.
1

1 
 −∞,  decreasing  , ∞  increasing
4

4 
7.
( −∞,1) ∪ (2,3) decreasing (1, 2) ∪ (3, ∞ )
8.
 π
π π
 0,  increasing  ,  decreasing
 6
6 2
9.
 π
 π π
 0,  decreasing  ,  increasing
 4
 4 2
10.
(e, ∞)increasing(0,e) decreasing
increasing
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TANGENTS AND NORMALS
1.
Find
the
slope
of
the
normal
to
the
x = a cos 3 θ, y
curve
y = a sin 3 θ at θ = π/4.
2.
Find the point at which tangent to the curve y =
3.
Prove that the curves x 2 − y 2 = 16 and xy = 25 cut each other at right angles.
4.
Prove that for all values of n, the line
n
4x − 3 − 1 has its slope 2/3.
x y
+ = 2 touches the curve
a b
n
x
y
  +   = 2 at the point ( a , b ) .
a
b
5.
Show that the line
x y
+ = 1 is tangent to the curve y = b.e x / a at the point
a b
where the curve crosses the axis of y.
6.
If the tangent to the curve y = x 3 + ax + b at p (1, −6 ) is parallel to the line
y − x = 5. find the values of a and b.
7.
Show that the curves 4x = y 2 and 4xy = k cut at right angles it k 2 = 512
8.
If the straight line x cos α + y sin α = p touches the ellipse
x2
a2
+
y2
b2
= 1 . Prove
that a 2 cos 2 α + b2 sin 2 = p2 .
9.
Find the coordinates of the points on the curve y = x 2 + 3x + 4 , the tangent at
which passes through the origin.
10.
Find the equation of a normal to the curve y = x 2 + 2x + 4 = 0 which is parallel
to the line x+14y+4=0.
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Tangent and Normal
(1)
1
( 2)
( 6 ) a = −2,
3/2
b = −5
( 9 )( 2,14 ) , ( −2, 2)
APPROXIMATION
1.
Using differential find approximate value of following:
(i)
0.037
( ii )
loge 4.01 given log e 4 = 1.3863
(iii)
( 0.007 )1/ 3
( iv )
cos 610 given that sin 600 = 0.8603 and 10 = 0.01745 radian
(v)
1
( 2.002)2
2.
Find the approximate value of f ( 2.01) where f(x)= 4x 2 5x + 2
3.
Find the approximate change in volume V of a cube of side x meters caused by
increasing the side by 2%.
4.
If radius of a sphere is measured as 9cm with an error of 0.03 cm. find the
approximate error in calculation of its volume.
5.
If y = x 2 . + 4 and x changes from 2 to 2.1. Find the approximate change in y.
6.
If y = x 4 − 10 and x changes from 2 to 1.97 what is the approximate change in
y.
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Approx.
(1)
(i)
(v)
0.2495
(2 )
0.1925
28.21
( 3)
( ii )
( iii )
1.3888
0.06 x 3 cum
(4)
( iv )
0.1916
9.72π cm3
( 5 ) 0.6
0.4849
(6)
− 0.96
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MAXIMA AND MINIMA
1.
Find the points of Local Maxima and Local Minima if any of the following
function
2.
(i)
f ( x ) = sin 2x − x where − π / 2 < x < π / 2
( ii )
f ( x ) = sin 4 x + cos 4 x where 0 < x < π/2
( iii )
f ( x ) = x 4 − 62x 2 + 120x + 9
Find maximum and minimum value of following function in the indicated
intervals
(i)
( ii )
3.
f ( x ) = cos 2 x + sin x
f ( x ) = x50 − x 20
[0, π]
[0,1]
Given the sum of the perimeters of square and a circle show that the sum of their
area is least when the side of the square is equal to the diameter of the circle.
4.
An open box with a square base is to be made out of a given iron sheet of area 27
sq. m. Show that the maximum volume of the box is 13.5 cu m.
5
A rectangular window is surmounted by an equilateral triangle. Given that the
perimeter is 16 m, find the width of the window so that the maximum amount of
light may enter.
6.
Show that the height of a right circular cylinder of a maximum total surface area
including the two ends that can be inscribed in a sphere of radius ‘r’ is given by

1 
h2 = 2r2 1 −

5

7.
A square tank of capacity of 250 cu. meters has to be dug out. he cost of the land
is Rs. 50 per sq. meter. The cost of digging increases with the depth and cost for
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2
the whole tank is 400 ( depth ) rupees. Find the dimensions of the tank for the
least cost.
8.
An open tank with a square base and vertical sides is to be constructed from a
meal sheet so as to hold a given quantity of water. Show that the cost of the
material will be least when the dept of the tank is half of its width.
9.
If the sum of side and hypotenuse of a right-angled triangle be given, show that
the area of the triangle will be maximum if the angle between the given side and
the hypotenuse be 600.
10.
Prove that the surface area of a solid cuboid, of square base and given volume is
minimum when it is a cube.
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Answers
Maxima & Minima
(1)
(i)
-
π
π
is pt of local minima &
is a point of local maxima
6
6
π
is a pt of L:ocal Minima
4
( ii )
( iii )
Local Maxima at x = 1 & Local mima at x = 5 − 6
( 2) ( i ) Maximum value
( ii )
= 5/4, Minimum value 1
Max. value = 0,
5/3
Minimuum value ( 2 / 5 )
− ( 2 .5 )
2/3
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