After watching this ppt you will get answers of the questions like...
1) What does it mean?
2) What we study in calculus?
3) Who invented it?
4) What was the need to invent it?
and many more...
You will also learn about the basic difference between discrete and continuous.
And many real life and cool applications of calculus....
3. C a l c u l u s
What does it mean ?
Who invented it?
What we study in Calculus?
What was the need to invent it?
4. C a l c u l u s
Latin Word Small stones used for
counting
5. Who is the first to invent Calculus?
Newton Leibnitz
6. Who is the first to invent Calculus?
Brahmagupta
“Yuktibhasha” is considered to be
the first book on Calculus…!!
7. Bhaskracharya
He 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”…!!!!
8. What we study in Calculus?
Geometry Algebra
Calculus is study of ‘Change’
9. What was the need to invent it?
We can find the area of above shapes
with the help of Geometrical tools.
15. Integration
Origin from the word ‘to integrate’ or ‘to merge’.
In 18th century the calculation of area and volume
are done using integration.
16. Differentiation
Differentiate means ‘to separate’.
In calculus derivative is a measure of how a
function changes as its input changes.
dv
dt
= a
v = velocity,
a = acceleration
18. 1) Save money on experiments.
2) Perform impossible experiments.
3) Predict the future…!!
19. Assume that you are a General
Manager of a company which
produces open top boxes for fruit
market…
20. 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.
21. Steps to solve this problem…
Create Mathematical Model
Solve it mathematically
Justify the answer
52. “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”
53. 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 = -kt
At time t=0, temperature T=To…
C = -ln(To-Te)
…………………(1)
54. Substitute the value of ‘C’ in (1)…
ln = -kt
T-Te
To-
Te
= e
T-Te
To-
Te
-kt
T-Te = (To-Te) e
-kt
T = Te + (To-Te) e
-kt
…………………(2)
56. Detective came at 10:23 a.m.
Temperature of body :- 26.7 C
Temperature of room :- 20 C
After an hour…
Temperature of body :- 25.8 C
Assume that body temperature was normal i.e. 37 C
What is time of death ?
57. 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)
58. 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.