I.
Tangents and Normals
Let Px 1 , y1  be a point on the curve of y =
f (x)
Then, the slope of the tangent at Px 1 , y1  =
dy
at x 1 , y1 
dx
Let m =
dy
at x 1 , y1 
dx
Y
P (x, y)
X
Equation of the tangent at x 1 , y1  is y - y1   m x - x 1 
Equation of the normal at x 1 , y1  is y - y1  
-1
x - x 1 
m
Angle of intersection


Angle of intersection of two curves is the angle between the two tangents at the point of
intersection.
If the angle of intersection is a right angle, then we say that the curves cut orthogonally.
1.
Find the equation of the tangent and the normal to the curve of y = 2x3- 3x2 -3x – 1 at ( 1, -5 ).
2.
Find the equation of the tangent and the normal to the curve of y 
3.
Find the equation of the tangent and the normal to the curve y 
4.
Find the equation of the tangent and the normal to the curve x2+y2-2x-6y+9=0 at x= 2.
5.
Find the equation of the tangent and the normal to the curve y = 2sinx+sin2x at x=
6.
Find the equation of the tangent and the normal to the curve y = sinx+cosx+2 at x=0
2 x  1  3 x  1 at x= 5
x
1 at x=1
3x  1

6
Find the equation of the tangent and the normal to the curve of x = a sin3 and y = a cos3 
7.
at θ =

4
8.
Find the point on the curve of y = 2x2-3x-2 at which the tangent is parallel to the line
5x-y=3.Also find the equation of the tangent and the normal to the curve at that point.
9.
Find the point on the curve of y  2 x  1  x at which the tangent is parallel to the line
2x-y =3.Also find the equation of the tangent and the normal to the curve at that point.
10.
Find the point on the curve of y2 = 3x2-8 at which the normal is parallel to the line x+3y=4.
Also find the equation of the tangent and the normal to the curve at that point.
11.
Find the point on the curve of 4x2+9y2=1 at which the tangent is perpendicular to the line
2x+y=3.Also find the equation of the tangent and the normal to the curve at that point.
12.
Find the equation of the tangent to the curve of y 
4x-2y +3=0
13.
Prove that the line
5 x  3  2 which is parallel to the line
x
x y
  1 touches the curve y  be a at the point where the curve cuts
a b
the y-axis.
Find the angle of intersection of the following curves
14.
x=y2 and y=x2 other than at (0,0)
15.
xy=4 and x2+y2=8
16.
x2+4y=8 and x2=4y in the first quadrant
17.
x2+y2 =4x and x2+y2=8 in the fourth quadrant
18.
y=2x and x3+y3=6xy other than at (0,0)
19.
Prove that the curves 4x2+9y2=45 and x2-4y2=5 cut orthogonally in the second quadrant.
20.
Prove that the curves x=y2 and xy = p cut orthogonally if 8p2=1.
II.
Approximations
Note that
dy y

dx x
 dy 
 x .
 dx 
Hence y  
This called the approximation operator.
By using this we find the approximate values of n x where n is any integer and x is any real number.
This operator is also used in estimation of errors also.
By using the differentials find the approximate value of
21.
36.72
22.
0.019
23.
3
26.19
24.
4
64.64
25.
5
31.04
26.
4
0.255
27.
Solve the equation x5 – 242 = 0 for a real root of x correct to two places of decimals.
28.
Solve the equation x2 – 0.037 = 0 for a real root x correct to two place of decimals.
29.
A circular metal plate expands so that its radius increases by 3%. Find the approximate increase
in the area of the plate if its original radius is 30 cm.
30.
An error of 2% has occurred in measuring the radius of a spherical object. Find the percentage
error in its volume and total surface area if its original radius is 10 cm.
Rate Measures
Set I
1. The radius of a balloon is increasing at the rate of 10cm/sec. At what rate is the surface area of the
balloon is increasing when its radius is 15cm?
2. An edge of a variable cube is increasing at the rate of 3cm/sec. How fast is the volume of the cube
is increasing when the edge is 10cm long?
3. A particle moves along the curve 6y = x3+2. Find points, on the curve at which the y coordinate is
changing 8 times as fast as the x-coordinate?
4. Water is running out of conical funnel at the rate of 5cm3/sec. If the radius of the base of the
funnel is 10 cm and the altitude is 2 cm, find the rate at which the water level is dropping when it
is 5 cm from the top.
5. Sand is pouring from a pipe at the rate of 12cc.sec. 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 cone is increasing when the height is 4cm.
6. A ladder 16 m long is learning against a wall. The bottom of the ladder is pulled along the ground,
away from the wall, at the rate of 2m/sec. how fast is its height decrease when the foot of the
ladder is 4m. from the wall ?
7. A man is moving away from a tower 85ft high at a speed of 4 ft/sec. Find the rate at which the
angle of elevation of the top of the tower is changing, when it is at a distance of 60ft from the foot
of the tower. Assume the eye level of the man is 5ft from the ground.
8. A man 2m high walks at a uniform speed of 5km/hr. away from a lamp post 6m. high. Find the
rate at which the length of his shadow increases?
9. The side of the square sheet of metal is increasing at 3cm/min. At what rate is the area increasing
when its side is 10 cm. Long
10. A balloon, which always remains spherical, has a variable diameter 3/2(2x+3). Determine the rate
of change of its volume with respect to x.
11. The radius of an air bubble is increasing at the rate of ½cm/sec. At what rate is the volume of the
bubble increasing when the radius is 1cm?
12. The radius r cm. of a blot of ink is increasing at the rate of 1/5 mm/sec. Find the rate at which the
area A is increasing after 4sec.
13. The volume of a spherical balloon is increasing at the rate of 25cm/sec Find the rate of change of
its surface area at the instant when the radius is 5cm.
14. A stone dropped in quite lake and waves move is circles at the speed of 4cm/sec At the instant
when the radius of the circular wave is 10cm., how fast the enclosed area increasing
15. A balloon, which always remains spherical, is being inflated by pumping in 900cc of air per
second. Find the rate at which the radius of the balloon is increasing when the radius is 15cm.
16. A small stone thrown into a still pond produces circular disturbances on the surface of water,
whose radius is increasing at the rate of 2cm/sec. Find the rate at which the disturbed area is
increasing when its radius is 7cm.
17. The bottom of rectangular swimming pool is 25m by 40m. Water is pumped into the tank at the
rate of 500cubic meters per hour. Find the rate at which the level of the water in the pool is
increasing.
18. Water is dripping out from conical funnel at the uniform rate of 2cm3/sec through tiny hole at the
vertex at the bottom. When the slant height of the water is 4cm. find the rate of decrease of the
slant height of the water, given that the vertical angle of the funnel is 120o.
19. Water is poured into an inverted conical vessel of which the radius of the base is 6cm and height is
12cm. at the rate of 5.5cc/min. At what rate is the water level rising at the instant when the depth is
3.5cm
20. Hemisphere is constructed on a circular base. If the radius of the base is increasing at the rate of
0.5cm/sec. Find the rate at which the volume of the hemisphere is increasing when the radius is
10cm.
21. A ladder 5m long leans against a house. Find the rate at which the top of the ladder is moving
down wards if its foot is 3m from the house and moving always at the rate of ½ m/sec.
22. A ladder 5m long leaning against a wall, the bottom of the ladder is pulled along the ground away
from the wall at the rate of 2m/sec How fast is the height on the wall decreasing when the foot of
the ladder is 4m way from the wall.
23. Sand is being poured at the rate of 0.3m3/sec. into a conical pile. If the height of the conical pile is
thrice the radius of the base, find the rate of change of height when the pile is 5m high.
24. A man 160cm tall, walks away from a source of light situated at the top of a pole of 6m high, at
the rate of 1.1m/s. How fast is the length of the shadow increasing when he is 1m away from the
pole?
Rate Measures
Set II
1.
The radius of a circular plate increases at a rate of 0.2cm per second. Find the rate at which its
area increases when its radius is 20 cm.
2.
The volume of a spherical balloon is shrinking at the rate of 3cm3 per second. Find the rate at
which its surface area is changing when its radius is 10 cm.
3.
A ladder of length 13 m leans against a wall. Find the rate at which the top of the ladder is
moving down if the foot is 12 m from the wall and moving at the rate of 1.5 m per minute.
4.
Sand is pouring from a pipe at the rate of 18 cc per second. The falling sand forms a cone on
the ground in such a way that the ratio of its height and radius is in the ratio 2:3.How fast is the
radius of the cone increases when its radius is 6 cm.
5.
Water is poured into a conical vessel of depth 15 cm and radius 9m at a rate of 2.5 cc per
second. Find the rate at which the height of the water level increases when its depth is 5 cm.
6.
A man of height 1.6m walks away from a lamppost of height 4 m at a uniform speed of 5 km
per hour. Find the rate at which a). The length of his shadow increases and b). The tip of his
shadow moves.
7.
The edge of a cube is increasing at the rate of 15 cm per second. How fast is the volume and
the surface area is increasing when the edge is 9cm.
8.
From a cylindrical drum water is leaking at a rate of 10m/sec .If the radius of the drum is 10
cm and the height is 35cm, find the rate at which the level of the water is changing when the
water level is 15 cm.
9.
A particle moves along the curve x2=3(y+1). Find the point on the curve at which y coordinate
changes two times as fast the x coordinate.
10.
A boy is flying a kite at a height of 50 m. If the kite moves horizontally away from the boy at a
rate 6 m/sec, how fast is the string being paid out when the kite is 80 m from him?
V. Monotonic Functions
a.
Increasing Function
Let f (x) be any real function defined in
the interval a, b such that whenever for
all x1 < x 2 in a, b if f  x1  < f  x 2  then
the function f (x) is said to be increasing in a, b .
b.
a
O
b
Decreasing Function
Let
f
interval
(x)
be
a, b
any
real
such
function
that
defined
whenever
x1 < x2 in a, b if f  x1  > f x2  the function
O
a
b
f (x) is said to be decreasing in a, b .
Working Rule to find the intervals of monotonicity
1. Put f ' x   0 and solve for x to get the critical points
2. Let the critical points in the given interval a, b be c1 and c2 such that c1 < c2
3. The intervals of monotonicity are a, c1  , c1 , c 2 and c2 , b
4. a). If f ' x   0 for all x in a, c1  then f (x) will be increasing in a, c1 
b). If f ' x   0 for all x in a, c1  then f (x) will be decreasing in a, c1 
This test will decide the nature of monotonicity of f (x) in each interval.
Practice Problems
Find the intervals in which the following functions are  or  .
1. f(x) = 3x2- 12x – 6
3. f(x) = 3x2- 12x –12
2. f(x) = 4x2+ 16x +7
4. f(x) =2x3- 9x2 -24x +3
5. f(x) =x3- 12x2 +36x +5
11. f(x) = 3x4-4x3+36x2-72x-1
6. f(x) =7-24x + 15x2 -2x3
12. f(x) = x4-6x2-8x+5
7. f(x) =8+60x + 9x2 -2x3
13.f(x) = x4-8x2-5
8. f(x) = 3x4+4x3-24x2-48x+5
14. f(x) = x4-2x2+5
9. f(x) = 3x4-8x3-6x2+24x+3
10. f(x) =7-24x + 14x –x
2
4
15.
f(x) 
x 3

3 x
in
for
the
all
16. f(x)  x 
1
x
x2
17. f(x) 
x 1
18. f(x)  x 
4
x
19. f(x) =2x3- 3x2 +36x +4
20. f(x) =2x3+15x2 +60x –7
21.f(x) =5-30x + 6x2 -x3
22. f(x) =5-12x + 3x2 -x3
23.
f(x) =2x + sinx
24. f(x) =2x –sinx
25.
f(x) =4x – cosx
25.
f(x) =cosx-x
26.
f(x) =cosx + sinx in 0  x  2π
26.
f(x) =cosx - sinx in 0  x  2π
27.
f(x) =x -
28.
f(x) =x + 2sinx in 0  x  2π
29.
f(x) =sinx +
1
cos2x in -   x  0
2
30.
f(x) =sinx +
1
cos2x in 0  x  π
2
2 sinx in 0  x  2π
31.
f(x) =sin2x - 2sinx in 
32.
f(x) =x+cos2x in 
π
π
x
2
2
π
π
x
2
2
VI. Maxima and Minima
a. Absolute Maximum and Absolute Minimum
Working
Rule
1. Put f x   0 and solve for x to get the critical points
'
2. Let the critical points in the given interval a, b be c1 and c2 such that c1, c2 and c3
3. The candidates for Absolute Maximum and Absolute Minimum in a, b  are a,c1,c2,c3 and b
4. Complete the following table
x
a
c1
c2
c3
b
f (x)
f (a)
f (c1)
f (c2)
f (c3)
f (b)
The Absolute Maximum is the greatest of f (a), f (c1), f (c2), f (c3) and f (b) and the Absolute
Minimum is the least of f (a), f (c1), f (c2), f (c3) and f (b)
Practice Problems.
Find the maximum and the minimum values of the following functions.
1.f(x) =2x3+ 9x2 –24x –1 in - 2  x  2
2.f(x) =x3+ 3x –1 in - 2  x  3
3.f(x) =2-12x - 9x2 -2x3 in - 2  x  0
4.f(x) = 3x4-8x3-6x2+24x-1 in - 1  x  2
5.f(x) = -x4+2x2+1 in - 2  x  1
6.f(x) = sinx - cosx in 0  x  2π
7.f(x) =cos4x + sin4x in 0  x  π
8.f(x) =sinx – ½ cos2x in -   x  0
b. Local Maxima and Minima
Working Rule
1. Put f x   0 and solve for x to get the critical points
'
2. Let the critical points in the given interval a, b be c1 and c2 such that c1, c2 and c3
3. The candidates for Local Maxima and Minima in a, b  are c1, c2 and c3
4. If f " c1   0 then c1 is a point of local minimum
If f " c1  0 then c1 is a point of local maximum
If f " c1   0 then c1 may be called as a point of inflexion.
Find local maxima and local minima of the following functions.
1.
f(x) =2x3- 3x2–1
2.
f(x) =2x3- 9x2 +12x –1
3.
f(x) = x4-8x2+1
4.
f(x) = 2-x4+2x2
5.
f(x) = x4-8x3+22x2-24x+5
6.
f(x) = 3x4+8x3-18x2+5
7.
f(x) = sinx + cosx in 0  x  2π
8.
f(x) =x +
9.
f(x) =cos4x + sin4x in   x  2π
10.
f(x) = sinx(1+ cosx) in 0  x  π
2 cosx in -   x  
c. Applications of Maxima and Minima
51.
Find two positive numbers whose sum is 12 and the sum of whose squares is minimum.
52.
Find two positive numbers whose sum is 16 and whose square is maximum.
53.
Find two positive numbers whose sum is 8 and the sum of whose cubes is minimum.
54.
Divide 20 into two parts such that the product of the square of the first and cube of the
second is maximum.
55.
Find two positive numbers x and y such that x + y = 60 and xy3 is maximum.
56.
Of all rectangles of perimeter 20 m prove that the square has the maximum area.
57.
Of all rectangles of given area prove that the square has the least perimeter.
58.
Prove that the area of a right angled triangles of given hypotenuse is maximum when it is
isosceles.
59.
Prove that the perimeter of a right angled triangles of given hypotenuse is maximum
when it is isosceles.
60.
The perimeter of a triangle is 20cm and one of its sides is 8 cm. Find the other two sides
when its area is maximum.
61.
The total surface area of a closed cylinder is 36cm2. Find its radius and height when its
volume is maximum.
62.
The total surface area of an open cylinder is given. Prove that its radius is equal to its
height when its volume is maximum.
63.
Prove that the diameter of a closed of cylinder is equal to its height when its volume is
given and its total surface area is minimum.
64.
The volume of an open cylinder is 18m3. Prove that its radius is equal its height when its
total surface area is minimum.
65.
Prove that the semi vertical angle of a cone of given hypotenuse and maximum volume is
tan-1 2 .
66.
Prove that the semi vertical angle of a cone of given TSA and maximum volume is sin -1
1
.
3
67.
A window is in the shape of a rectangle surmounted by a semi circle. If the total
perimeter of the window is 20 m, find its total height and width when its area is
maximum.
68.
A window is in the shape of a rectangle surmounted by an isosceles triangle. If the total
perimeter of the window is 20 m, find its area when its area is maximum.
69.
A wire of length 16m is cut into two pieces. One of them is bent to form a square and the
other into a circle. Find the length of each piece if their combined area is minimum.
70.
A wire of length 12m is cut into two pieces. One of them is bent to form an equilateral
triangle and the other into a circle. Find the length of each piece if their combined area is
minimum.
71.
A rectangular sheet of metal is 30cm by 14cm.Equal squares are cut off at the four
corners and the remaining portion is folded up to form an open rectangular box. Find side
of the square removed so that the capacity of the box is maximum.
72.
A square sheet of metal is of side 18m.Equal squares are cut off at the four corners and
the remaining portion is folded up to form an open box. Find side of the square removed
so that the capacity of the box is maximum. Also find the maximum volume of the box.
73.
Find the equation of the line through the point (4.3), which cuts from the first quadrant a
triangle of minimum area.
74.
Find the cubic polynomial in x which has a maximum value 4 when x=1 and minimum
value –28 at x=5.
75.
Of all rectangles that can be inscribed in a circle prove that the square has the maximum
area.
76.
Prove that the triangle of maximum area that can be inscribed in a circle must be
equilateral.
77.
Prove that the height of the largest cone that can be inscribed in a given sphere of radius a
is
78.
4a
. Also find the volume of the cone.
3.
Prove that the volume of the largest cylinder which can be inscribed in a given cone of
height h and semi vertical angle α is
79.
4
πh 3 tan 2 α .
27
Prove that the height of the largest cylinder that can be inscribed in a given sphere of
radius r is
2r
. Also find the volume of the cylinder.
3.
80.
Find the area of the largest isosceles triangle that inscribed in a given ellipse having its
vertex coincident with one extremity of the major axis.
81.
Find the point on the curve y2=4x which is nearest to the point (2, -8).
82.
Find the point on the curve y=x2 which is nearest to the point (0,3).
83.
An open box with a square base is to be made out of a given sheet of metal of area c2.
c3
Show that the maximum volume of the box is
.
6 3.
84.
An open tank with a square base and vertical sides is to be constructed from a metal sheet
so as to hold a given quantity of water. Show that the cost of material will be the least
when the depth of the tank is half its width.
85.
The combined resistance R of two resistant R1>0 and R2>0 is given by
1
1
1


.
R. R 1 R 2
If R1 + R2 = c show that R is maximum when R1 = R2. where c is a constant