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Calculus In Real Life

“nothing takes place in the world whose meaningis not that of some maximum or minimum.”

--leonhard euler

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2What is calculus ?

The word Calculus comes from Latin meaning "small stone", Because it is like understanding something by looking at small pieces.

Derived from the Latin “calx” (counter) – ancient Babylonians would use pebbles to represent units, tens, hundreds, etc, on a primitive abacus.

Later, defined as measuring varying rates of change.

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Calculus is everywhereThe differentiation and integration of calculus have many real-world applications from sports to engineering to astronomy and space travel.

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4Types of Calculus

• Differential Calculus cuts something into small pieces to find how it changes. 

• Integral Calculus joins (integrates) the small pieces together to find how much there is.

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5Differential CalculusNewton’s Law of Cooling

Newton’s observations:He observed that observed that the temperature of the body is proportional to the difference between its own temperature and the temperature of the objects in contact with it . Formulating:First order separable DE Applying calculus:

Where k is the positive proportionality constant

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6Applications on Newton’s Law of Cooling:

Investigations.• It can be used to

determine the time of death.

Computer manufacturing.• Processors.• Cooling systems. solar water

heater. calculating the surface area of

an object.

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7Calculate Time of Death

The police came to a house at 10:23 am were a murder had taken place. The detective measured the temperature of the victim’s body and found that it was 26.7 . Then he used a ℃thermostat to measure the temperature of the room that was found to be 20 through the last three days. After an hour ℃he measured the temperature of the body again and found that the temperature was 25.8 . Assuming that the body ℃temperature was normal (37 ), what is the time of death?℃

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8Solution

T (t) = Te + (T0 − Te ) e – kt

Let the time at which the death took place be x hours before the arrival of the police men.Substitute by the given values T ( x ) = 26.7 = 20 + (37 − 20) e-kx

T ( x+1) = 25.8 = 20 + (37 − 20) e - k ( x + 1) Solve the 2 equations simultaneously 0.394= e-kx  0.341= e - k ( x + 1) By taking the logarithmic function ln (0.394)= -kx …(1) ln (0.341)= -k(x+1) …(2)

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Result

By dividing (1) by (2) Thus x≃7 hoursTherefore the murder took place 7 hours before the arrival of the detective which is at 3:23 pm

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10Computer Processor Manufacture

A global company such as Intel is willing to produce a new cooling system for their processors that can cool the processors from a temperature of 50 to 27 in just half ℃ ℃an hour when the temperature outside is 20 but they don’t know what kind of ℃materials they should use or what the surface area and the geometry of the shape are. So what should they do ?

Simply they have to use the general formula of Newton’s law of cooling

T (t) = Te + (T0 − Te ) e – k

And by substituting the numbers they get 27 = 20 + (50 − 20) e-0.5k

Solving for k we get k =2.9 so they need a material with k=2.9 (k is a constant that is related to the heat capacity ,

thermodynamics of the material and also the shape and the geometry of the material)

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It can be used to find an area bounded, in part, by a curve

Integral Calculus

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. . . give the boundaries of the area.

The limits of integration . . .

0 1

23 2 xy

x = 0 is the lower limit( the left hand

boundary )x = 1 is the upper limit(the right hand

boundary )

dxx 23 2

0

1

e.g.

gives the area shaded on the graph

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0 1

23 2 xy

the shaded area equals 3

The units are usually unknown in this type of question

1

0

2 23 dxxSince

31

0

xx 23

Finding and Area

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14SUMMARY

• the curve

),( xfy

• the lines x = a and x = b

• the x-axis and

PROVIDED that the curve lies on, or above, the x-axis between the values x = a and x = b

The definite integral or

gives the area between

b

a

dxxf )( b

a

dxy

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• Business and politicians often conduct surveys with the help of calculus.• Investment plans do not pass before mathematicians approves.• Doctors often use calculus in the estimation of the progression of the illness.• Global mapping is done with the help of calculus.• Calculus also used to solve paradoxes.

Calculus in other fields

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THANK YOU ALL…!!!