##### Document Text Contents

Page 1

APPLICATIONS OF DERIVATIVES1

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APPLICATIONS OF DERIVATIVES

4.1 Derivative as a rate measure

If y = f(x) is a function of x, then

dx

dy or f ‘(x) represents the rate-measure of y with respect to x.

(a) represents)x('for

dx

dy

0

xx 0=

÷

ø

ö

ç

è

æ the rate of change of y w.r.t. x at x = x0.

(b) If y increases as x increases, then

dx

dy is positive and if y decreases as x increases, then

dx

dy is negative.

(c) Marginal Cost (MC) is the instantaneous rate of change of total cost with respect to the number of items

produced at an instant.

(d) Marginal Revenue (MR) is the instantaneous rate of change of total revenue with respect to the number

of items sold at an instant.

4.2 Tangents and normal

(a) To find the equation of the tangent to the curve y = f(x) at the given point P(x1, y1), proceed as under :

(i) Find

dx

dy from the given equation y = f(x).

(ii) Find the value of

dx

dy at the given point P (x1, y1), let m =

1

1

yy

xxdx

dy

=

=

÷

ø

ö

ç

è

æ

.

(iii) The equation of the required tangent is y – y1 = m (x – x1).

In particular, if =

1

1

yy

xxdx

dy

=

=

÷

ø

ö

ç

è

æ

does not exist, then the equation of the tangent is x = x1.

(b) To find the equation of the normal to the curve y = f(x) at the given point P(x1, y1), proceed as under :

(i) Find

dx

dy from the given equation y = f(x).

(ii) Find the value of

dx

dy at the given point P(x1, y1).

(iii) If m is the slope of the normal to the given curve at P, then m = –

1

1

yy

xxdx

dy

1

=

=

÷

ø

ö

ç

è

æ

.

(iv) The equation of the required normal is y – y1 = m (x – x1).

In particular if,

1

1

yy

xxdx

dy

=

=

÷

ø

ö

ç

è

æ

= 0, then the equation of the normal at P is x = x1; and if =

1

1

yy

xxdx

dy

=

=

÷

ø

ö

ç

è

æ

does not exist, then the equation of the normal at P is y = y1.

Angle of intersection of two curves

The angle of intersection of two curves is the angle between the tangents to the two curves at their point

of intersection.

If m1 and m2 are the slopes of the tangents to the given curves at their point of intersection P(x1, y1), then

the (acute) angle q between the curves is given by

Page 7

APPLICATIONS OF DERIVATIVES7

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Ex.6 Find the equation of the normal at the point (am2, am3) for the curve ay2 = x3.

Sol. Slope of the tangent to the curve ay2 = x3 at the point (am2, am3) is given by

)am,am( 32dx

dy

ú

û

ù

Given ay2 = x3

We have, 2ay

dx

dy = 3x2

Þ

dx

dy =

ay2

x3 2

)am.am( 32dx

dy

ú

û

ù

=

a2

3

3

42

am

ma

=

2

m3

\ Slope of the normal to the curve at (am2, am3) is –

m3

2 .

Hence, the equation of the normal at the point (am2, am3) to the curve ay2 = x3 is

y – am3 = –

m3

2 (x – am2)

i.e., 3my – 3am4 = –2x + 2am2

i.e., 2x + 3my – am2 (2 + 3m2) = 0

Ex.7 A circular metal plate expands under heating so that its radius increases by 2%. Find the

approximate increase in the area of the plate if the radius of the plate before heating is 10 cm.

Sol. Let r cm be the radius of the circular plate and A be its area. Then, A = pr2

Þ

dr

dA = 2pr

Given r = 10 cm (before heating)

Let Dr be the increase in the radius of the circular plate (after heating) and DA be the corresponding

increase in area.

Then, Dr = 2% of 10 cm

=

100

2 × 10 cm = 0.2 cm

For, r = 10 cm, Dr = 0.2 cm

DA =

dr

dA .Dr

= 2pr . (0.2) sq cm

= 2p (10) . (0.2) sq cm

= 4p sq cm

Ex.8 Find the approximate change in the total surface area of a right circular cone, when the

radius r remains constant while the altitude ‘h’ changes by a.

Sol. Let s denote the total surface area of the right circular cone. Then,

222 r)hr(rs 2

1

p++p=

Since r remains constant, s can be regarded as a function of h only.

We have Ds =

dh

ds . Dh

APPLICATIONS OF DERIVATIVES1

www.thinkiit.in

APPLICATIONS OF DERIVATIVES

4.1 Derivative as a rate measure

If y = f(x) is a function of x, then

dx

dy or f ‘(x) represents the rate-measure of y with respect to x.

(a) represents)x('for

dx

dy

0

xx 0=

÷

ø

ö

ç

è

æ the rate of change of y w.r.t. x at x = x0.

(b) If y increases as x increases, then

dx

dy is positive and if y decreases as x increases, then

dx

dy is negative.

(c) Marginal Cost (MC) is the instantaneous rate of change of total cost with respect to the number of items

produced at an instant.

(d) Marginal Revenue (MR) is the instantaneous rate of change of total revenue with respect to the number

of items sold at an instant.

4.2 Tangents and normal

(a) To find the equation of the tangent to the curve y = f(x) at the given point P(x1, y1), proceed as under :

(i) Find

dx

dy from the given equation y = f(x).

(ii) Find the value of

dx

dy at the given point P (x1, y1), let m =

1

1

yy

xxdx

dy

=

=

÷

ø

ö

ç

è

æ

.

(iii) The equation of the required tangent is y – y1 = m (x – x1).

In particular, if =

1

1

yy

xxdx

dy

=

=

÷

ø

ö

ç

è

æ

does not exist, then the equation of the tangent is x = x1.

(b) To find the equation of the normal to the curve y = f(x) at the given point P(x1, y1), proceed as under :

(i) Find

dx

dy from the given equation y = f(x).

(ii) Find the value of

dx

dy at the given point P(x1, y1).

(iii) If m is the slope of the normal to the given curve at P, then m = –

1

1

yy

xxdx

dy

1

=

=

÷

ø

ö

ç

è

æ

.

(iv) The equation of the required normal is y – y1 = m (x – x1).

In particular if,

1

1

yy

xxdx

dy

=

=

÷

ø

ö

ç

è

æ

= 0, then the equation of the normal at P is x = x1; and if =

1

1

yy

xxdx

dy

=

=

÷

ø

ö

ç

è

æ

does not exist, then the equation of the normal at P is y = y1.

Angle of intersection of two curves

The angle of intersection of two curves is the angle between the tangents to the two curves at their point

of intersection.

If m1 and m2 are the slopes of the tangents to the given curves at their point of intersection P(x1, y1), then

the (acute) angle q between the curves is given by

Page 7

APPLICATIONS OF DERIVATIVES7

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Ex.6 Find the equation of the normal at the point (am2, am3) for the curve ay2 = x3.

Sol. Slope of the tangent to the curve ay2 = x3 at the point (am2, am3) is given by

)am,am( 32dx

dy

ú

û

ù

Given ay2 = x3

We have, 2ay

dx

dy = 3x2

Þ

dx

dy =

ay2

x3 2

)am.am( 32dx

dy

ú

û

ù

=

a2

3

3

42

am

ma

=

2

m3

\ Slope of the normal to the curve at (am2, am3) is –

m3

2 .

Hence, the equation of the normal at the point (am2, am3) to the curve ay2 = x3 is

y – am3 = –

m3

2 (x – am2)

i.e., 3my – 3am4 = –2x + 2am2

i.e., 2x + 3my – am2 (2 + 3m2) = 0

Ex.7 A circular metal plate expands under heating so that its radius increases by 2%. Find the

approximate increase in the area of the plate if the radius of the plate before heating is 10 cm.

Sol. Let r cm be the radius of the circular plate and A be its area. Then, A = pr2

Þ

dr

dA = 2pr

Given r = 10 cm (before heating)

Let Dr be the increase in the radius of the circular plate (after heating) and DA be the corresponding

increase in area.

Then, Dr = 2% of 10 cm

=

100

2 × 10 cm = 0.2 cm

For, r = 10 cm, Dr = 0.2 cm

DA =

dr

dA .Dr

= 2pr . (0.2) sq cm

= 2p (10) . (0.2) sq cm

= 4p sq cm

Ex.8 Find the approximate change in the total surface area of a right circular cone, when the

radius r remains constant while the altitude ‘h’ changes by a.

Sol. Let s denote the total surface area of the right circular cone. Then,

222 r)hr(rs 2

1

p++p=

Since r remains constant, s can be regarded as a function of h only.

We have Ds =

dh

ds . Dh