Calculus in real life (Differentiation and integration )

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1. Calculus In Real Life “nothing takes place in the world whose meaning is not that of some maximum or minimum.” -- leonhard euler 12/12/2015 2NDS 1

2. What is calculus ? 12/12/2015 2NDS 2 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 .

3. Calculus is everywhere The differentiation and integration of calculus have many real-world applications from sports to engineering to astronomy and space travel. 12/12/2015 2NDS 3

4. Types of Calculus 12/12/2015 2NDS 4 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.

5. Differential Calculus Newton’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   12/12/2015 2NDS 5

6. Applications on Newton’s Law of Cooling : 12/12/2015 2NDS 6

7. Calculate Time of Death 12/12/2015 2NDS 7 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?

8. Solution 12/12/2015 2NDS 8 T ( t ) = T e + ( T − T e ) 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 )

9. Result By dividing (1) by (2) Thus x≃7 hours Therefore the murder took place 7 hours before the arrival of the detective which is at 3:23 pm   12/12/2015 2NDS 9

10. Computer 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 ) = T e + ( T − T e ) 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) 12/12/2015 2NDS 10

11. It can be used to find an area bounded, in part, by a curve Integral Calculus 12/12/2015 2NDS 11

12. . . . give the boundaries of the area. The limits of integration . . . 1 x = 0 is the lower limit ( the left hand boundary ) x = 1 is the upper limit (the right hand boundary ) 1 e.g. gives the area shaded on the graph 12/12/2015 2NDS 12

13. 1 2 3 2 + = x y the shaded area equals 3 The units are usually unknown in this type of question Since Finding and Area 12/12/2015 2NDS 13

14. SUMMARY the curve 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 12/12/2015 2NDS 14

15. Business and politicians often conduct surveys with the help of calculus. I nvestment 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 12/12/2015 2NDS 15

16. THANK YOU ALL…!!! 12/12/2015 2NDS 16