Application of Calculus in Real World

Application of…

…to the real world.

Introduction &

History of

Calculus-Divyarajsinh

C A L C U L U SWhat does it mean ?

Who invented it?

What we study in Calculus?

What was the need to invent it?

C A L C U L U S

Latin Word Small stones used for counting

Who is the first to invent Calculus?

Newton Leibnitz

Who is the first to invent Calculus?Brahmagupta

“Yuktibhasha” is considered to be the first book on Calculus…!!

BhaskracharyaHe used principle of differential calculus in problems on Astronomy.

He is pioneer of some principles of differential calculus.

He stated Rolle’s Mean Value Theorem in his book “Siddhant Shiromani”…!!!!

What we study in Calculus?

Geometry Algebra

Calculus is study of ‘Change’

What was the need to invent it?

We can find the area of above shapes with the help of Geometrical tools.

But What about these shapes…!!???

ContinuousDiscrete

ContinuousDiscrete

1+1+1

10 Drops ?

“We need a continuous summation tool.”

This idea leads to the invention of Calculus.

Calculus

Integration Differentiation

Integration

Origin from the word ‘to integrate’ or ‘to merge’.

In 18th century the calculation of area and volume are done using integration.

Differentiation

Differentiate means ‘to separate’.

In calculus derivative is a measure of how a function changes as its input changes.

dvdt = a

v = velocity,a = acceleration

Uses of Calculus

& Mathematical Modeling

-Milan Patel

1) Save money on experiments.

2) Perform impossible experiments.

3) Predict the future…!!

Assume that you are a General Manager of a company whichproduces open top boxes for fruitmarket…

To make open top boxes for fruit market, using square sheet of card board.

To maximize the volume of box in order to increase the profit of the company.

STEPS TO SOLVE THIS PROBLEM…

Create Mathematical Model

Solve it mathematically

Justify the answer

How to create a mathematical model?

Understand

Consider a card of 60cm × 60cm60

60

(60 -2x)

(60 -2x)

(60

-2x)

(60 -2x)

Volume of the box,

V = L × B × H

= (60-2x) × (60-2x) × (x)

= 4x – 240x + 3600x

3 2

Now, We will use…

= 4x – 240x + 3600x

3 2V

dV = 12x - 480x + 3600 = 02

dx x - 40x + 300 = 02

(x-30) (x-10) = 0

x = 30cm & x = 10cm

Substitute x=10cm to find volume :-

= 4x – 240x + 3600x

3 2V = 4(10)– 240(10) + 3600(10)

3 2

= 16000 cm

3

Justify the Answer :- Application of second derivative

dV = 12x - 480x + 3600dx

d V = 24x - 480dx2

2

2

d V = 24(10) – 480 = -240 < 0dx2

2

x=10

Cut the square of 10 cm X 10 cm from the corner in order to maximize the volume of the box.

Useful Applications

of Calculus

-Saumil Patel

Average Value

Area b/w Curves

Length of Arc

AVERAGE VALUE

5 6 3 2 4+ + + +5

= 4

AVERAGE VALUE

1

2

3

4

5

6

7

1 2 3 5 64 7

f(x)

5 5 5 5 5

a b

AVERAGE VALUE

Average value of f(x) in given interval

5 5 5 5 5+ + + +5

= 5

=

-1

0

1

AVERAGE VALUE

f(x)

a b

AVERAGE VALUE

favg = 1b-a

f(x) dxa

b

Application to the real world

Average growth of tree in given time period


Average growth of bacteria


Average amount of water falling from the water fall

AREA UNDER THE CURVE

We know that the integral… f(x) dxa

b

denotes the area bounded by the curve y=f(x) from x=a to x=b.

x=a x=b

y=f(x)

AREA BETWEEN CURVES

x=a x=b

f(x)

g(x)

Area b/w curves = [Area under f(x)]

- [(Area under g(x)]

AREA BETWEEN CURVES

= -f(x) dxa

bg(x) dx

a

b

= [f(x) - g(x)]dxa

b


LENGTH OF ARC

a b

Length of arc = b-a

f(x)

LENGTH OF ARC

a

b

Length of ab = b-a

LENGTH OF ARC

a

b

c

(Length of ac) + (Length of cb)

LENGTH OF ARC

a

b

dc

e

(Length of ad) + (Length of dc) + (Length of ce) + (Length of eb)

LENGTH OF ARC

a

b

Length of arc =

1+[f’(x)] dx 2



Newton’s Law of cooling

&

It’s Applications-Richa Raval

“Rate of change of the temperature of an object is proportional to the difference between its own temperature and the temperature of its surroundings.”

“Newton’s law of cooling”

Applying Calculus…

dT (T-Te)α dt

dT dt

= -k(T-Te) (‘k’ is a +ve constant)

dT (T-Te) = -k.dt

Integrating on both sides we get…

ln(T-Te)+C = -ktAt time t=0, temperature T=To…

C = -ln(To-Te)

…………………(1)

Substitute the value of ‘C’ in (1)…

ln = -ktT-TeTo-Te

= eT-TeTo-Te

-kt

T-Te = (To-Te) e-kt

T = Te + (To-Te) e-kt

…………………(2)

Application of “NEWTON’S

LAW OF COOLING”In

Crime Investigation

Detective came at 10:23 a.m.Temperature of body :- 26.7 CTemperature of room :- 20 C

After an hour…Temperature of body :- 25.8 CAssume that body temperature was normal i.e. 37 CWhat is time of death ?


Let the time of death be ‘x’ hour before the arrival of detective.Substitute given values in equation (2)…

T(x) = 26.7 = 20 + (37-20) e -kx

T(x+1) = 25.8 = 20 + (37-20) e -k(x+1)

Solving above two equations…0.394 = e -kx

0.341 = e -k(x+1)

Taking log on both sides of above two equations…ln(0.394) = -kx ln(0.341) = -k(x+1)

…………………(3)…………………(4)

Divide equation (3) by (4)…

ln(0.394) -kx ln(0.341) -k(x+1)

=

=0.8657 x

(x+1)

x = 7 hour

Murder took place 7 hour before arrival of detective.

i.e. 3:23 p.m.

Computer Manufacturing


27 = 20 + (50-20) e-0.5k

K=2.9

Some Important

Applications of

Calculus…

Growth of bacteria

Construction Technology

THANK YOU…

Education

Application of Calculus in Real World