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# Calculus in real life (Differentiation and integration )

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### Calculus in real life (Differentiation and integration )

1. 1. Calculus In Real Life “nothing takes place in the world whose meaning is not that of some maximum or minimum.” --leonhard euler 1
2. 2. What is calculus ? 12/23/20152NDS 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. 3. Calculus is everywhere The differentiation and integration of calculus have many real-world applications from sports to engineering to astronomy and space travel. 12/23/20152NDS 3
4. 4. Types of Calculus 12/23/20152NDS 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. 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: 𝑑𝑇 𝑑𝑡 = −𝑘(𝑇 − 𝑇𝑒) Where k is the positive proportionality constant 12/23/20152NDS 5
6. 6. Applications 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. 12/23/20152NDS 6
7. 7. Calculate Time of Death 12/23/20152NDS 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. 8. Solution 12/23/20152NDS 8 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)
9. 9. Result By dividing (1) by (2) ln(0.394) ln 0.341 = −𝑘𝑥 −𝑘 𝑥+1 0.8657 = 𝑥 𝑥+1 Thus x≃7 hours Therefore the murder took place 7 hours before the arrival of the detective which is at 3:23 pm 12/23/20152NDS 9
10. 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) = 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) 12/23/20152NDS 10
11. 11. It can be used to find an area bounded, in part, by a curve Integral Calculus 12/23/20152NDS 11
12. 12. . . . 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 12/23/20152NDS 12
13. 13. 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 3 1 0     xx 23  Finding and Area 12/23/20152NDS 13
14. 14. SUMMARY • 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 12/23/20152NDS 14
15. 15. • 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 12/23/20152NDS 15
16. 16. THANK YOU ALL…!!! 12/23/20152NDS 16