application of calculus in real world
TRANSCRIPT
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Application of…
…to the real world.
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Introduction &
History of
Calculus-Divyarajsinh
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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?
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C A L C U L U S
Latin Word Small stones used for counting
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Who is the first to invent Calculus?
Newton Leibnitz
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Who is the first to invent Calculus?Brahmagupta
“Yuktibhasha” is considered to be the first book on Calculus…!!
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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”…!!!!
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What we study in Calculus?
Geometry Algebra
Calculus is study of ‘Change’
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What was the need to invent it?
We can find the area of above shapes with the help of Geometrical tools.
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But What about these shapes…!!???
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ContinuousDiscrete
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ContinuousDiscrete
1+1+1
10 Drops ?
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“We need a continuous summation tool.”
This idea leads to the invention of Calculus.
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Calculus
Integration Differentiation
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Integration
Origin from the word ‘to integrate’ or ‘to merge’.
In 18th century the calculation of area and volume are done using integration.
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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
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Uses of Calculus
& Mathematical Modeling
-Milan Patel
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1) Save money on experiments.
2) Perform impossible experiments.
3) Predict the future…!!
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Assume that you are a General Manager of a company whichproduces open top boxes for fruitmarket…
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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.
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STEPS TO SOLVE THIS PROBLEM…
Create Mathematical Model
Solve it mathematically
Justify the answer
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How to create a mathematical model?
Understand
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Consider a card of 60cm × 60cm60
60
(60 -2x)
(60 -2x)
(60
-2x)
(60 -2x)
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Volume of the box,
V = L × B × H
= (60-2x) × (60-2x) × (x)
= 4x – 240x + 3600x
3 2
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Now, We will use…
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= 4x – 240x + 3600x
3 2V
dV = 12x - 480x + 3600 = 02
dx x - 40x + 300 = 02
(x-30) (x-10) = 0
x = 30cm & x = 10cm
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Substitute x=10cm to find volume :-
= 4x – 240x + 3600x
3 2V = 4(10)– 240(10) + 3600(10)
3 2
= 16000 cm
3
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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
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Cut the square of 10 cm X 10 cm from the corner in order to maximize the volume of the box.
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Useful Applications
of Calculus
-Saumil Patel
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Average Value
Area b/w Curves
Length of Arc
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AVERAGE VALUE
5 6 3 2 4+ + + +5
= 4
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AVERAGE VALUE
1
2
3
4
5
6
7
1 2 3 5 64 7
f(x)
5 5 5 5 5
a b
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AVERAGE VALUE
Average value of f(x) in given interval
5 5 5 5 5+ + + +5
= 5
=
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-1
0
1
AVERAGE VALUE
f(x)
a b
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AVERAGE VALUE
favg = 1b-a
f(x) dxa
b
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Application to the real world
Average growth of tree in given time period
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Application to the real world
Average growth of bacteria
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Application to the real world
Average amount of water falling from the water fall
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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)
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AREA BETWEEN CURVES
x=a x=b
f(x)
g(x)
Area b/w curves = [Area under f(x)]
- [(Area under g(x)]
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AREA BETWEEN CURVES
= -f(x) dxa
bg(x) dx
a
b
= [f(x) - g(x)]dxa
b
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Application to the real world
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LENGTH OF ARC
a b
Length of arc = b-a
f(x)
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LENGTH OF ARC
a
b
Length of ab = b-a
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LENGTH OF ARC
a
b
c
(Length of ac) + (Length of cb)
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LENGTH OF ARC
a
b
dc
e
(Length of ad) + (Length of dc) + (Length of ce) + (Length of eb)
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LENGTH OF ARC
a
b
Length of arc =
1+[f’(x)] dx 2
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Application to the real world
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Application to the real world
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Newton’s Law of cooling
&
It’s Applications-Richa Raval
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“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”
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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)
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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)
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Application of “NEWTON’S
LAW OF COOLING”In
Crime Investigation
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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 ?
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T = Te + (To-Te) e-kt
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)
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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.
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Computer Manufacturing
T = Te + (To-Te) e-kt
27 = 20 + (50-20) e-0.5k
K=2.9
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Some Important
Applications of
Calculus…
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Growth of bacteria
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Construction Technology
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THANK YOU…