CAREER POINT UNIVERSITYKOTA

C A L C U L U S AND ITS REAL WORLD APPLICATIONMAJOR ASSIGNMENT

SUBMITTED TO- SUBMITTED BY-DR. SONA RAJ ASHISH KUMAR JHA(K12593) RAJ PREMANI(K12629) MANU CHATURVEDI(K12780)

CONTENTS:

INTRODUCTION OF CALCULUS, TAYLOR’S THEOREM, MATRIXEXAMPLEUSE OF CALCULUS IN DIFFERENT ENGEINEERING BRANCHESUSE OF MATRIX IN DIFFERENT ENGEINEERING BRANCHESUSE OF TAYLOR’S THEOREM IN DIFFERENT ENGEINEERING BRANCHESREAL WORLD APPLICATION OF CALCULUSREAL WORLD APPLICATION OF MATRIXREAL WORLD APPLICATION OF TAYLOR’S THEOREM CONCLUSION REFERENCES

INTRODUCTION OF CALCULUS• Calculus is a Latin Word, generally means small stones used for counting• Brahmagupta is considered to be the first inventor of calculus.• “Yuktibhasha” is considered to be the first book on Calculus.• Calculus is study of ‘Change’• We can divide calculus into to major parts –

Integration• Origin from the word ‘to integrate’ or ‘to merge’. • In 18th century the calculation of area and volume are done using integration.

Differentiation• Differentiate means ‘to separate’.• In calculus derivative is a measure of how a function changes as its input changes.

INTRODUCTION TAYLOR’S THEOREM• In calculus, Taylor’s theorem gives an approximation of a k-times differentiable

functions around a given point by a k-th order Taylor polynomial.• Taylor’s theorem is taught in introductory level calculus courses and it is one of the

central elementary tools in mathematical analysis.• Taylor’s theorem is named after the mathematician brook Taylor, who started a version

of it in 1712.• For analytic functions the Taylor polynomials at a given points are finite order

functions of the Taylor's series, which completely determines the function which determine the some neighborhood points.

)( 01 xp)( 0xf

)( 0xf

Taylor’s theorem:

)(xf

x

y)(xfy

o

))(()( 000 xxxfxf

)(1 xp

Use a straight to approximate the curve0x

)(1 xp

)( 01 xp xLinear function

About matrixNAME DESCRIPTION EXAMPLE

Row matrix A matrix with only 1 row

Column matrix A matrix with only I column

Square matrix A matrix with same number of rows and columns

Zero matrix A matrix with all zero entries

3 2 1 4

23

2 4 1 7

0 00 0

Example of calculus • Example of integration

Ex: , C

rdaI jxyiyxa ˆ)(ˆ)( along the following paths:

(i) xy 2 from (1,1) to (4,2)

(ii)

,12 2 uux ,1 2uy )2,4()1,1(

(iii) line followed line ,1y ),1,4()1,1( ,4x )2,4()1,4(

Sol: (i)

3/34])(2)[(

])()[(

2

22

1

2

)2,4(

)1,1(

2

dyyyydyyy

dyxydxyxI

dxydyxy

(ii) ududyuy

duudxuux

21

)14(122

2

3/322)()14)(23(

])()[(1

0

1

0

22

)2,4(

)1,1(

uduuuduuuu

dyxydxyxI

(iii)

4

1

2

1

)2,4(

)1,1(

8)4()1(

])()[(

dyydxx

dyxydxyxI line y=1, dy=0

line x=4, dx=0

dv = a

v = velocity,a = acceleration dt

Example of Differentiate

Example of matrix

21 20 3 4 5

4 56 8 2 6

( 2)1 ( 2) 2( 2)0 ( 2)3( 2) 4 ( 2)5

4 56 8 2 6

2 40 68 10

4 56 8 2 6

6 96 146 4

Use of calculus in computer science & engendering

Robotics. For one, robotics has the notion of a Jacobian that is associated with the linear and angular motion of a mechanism, and that is also used to define the relationship between forces applied on a mechanism and the torques needed to support these forces. The study of dynamics, used to develop control structures for robot stability and performance, also uses calculus.

Machine learning. The term "machine learning" is used very broadly (and often includes subfields like computer vision), but in general, ML usually involves solving for certain optimality conditions or iterating towards a solution with techniques like gradient descent or expectation-maximization. All of this involves calculus.

Network analysis. Any analysis regarding the dynamics of a network is likely to involve calculus. For example, to understand the outcome of an edge creation model like preferential attachment, which says that nodes connect to other nodes with probability proportional to their existing degrees, requires integration over time.

Use of calculus in civil engineering • Calculus, at least the concepts developed from calculus, are used all the time in civil

engineering.• Any time there is a rate of change of something then the derivative is an efficient

way to characterize it.• When we are designing any type of structure we will want to figure out that what will

happen to that structure when we will apply the weight (load) at various point.• Usually we know the rate of deformation as the function of stress/strain and to solve

for the total deformation we will need to solve the differential equations.

Use of calculus in electrical engineering

• Robotics. For one, robotics has the notion of a Jacobian that is associated with the linear and angular motion of a mechanism, and that is also used to define the relationship between forces applied on a mechanism and the torques needed to support these forces. The study of dynamics, used to develop control structures for robot stability and performance, also uses calculus.

• Also use to create electronic circuits .

Use of matrix in computer science & engineering • Matrix also has its application in c programming as a matrix multiplication problems

in c programming language.

Use of matrix in civil engineering • Matrices are maily used in 'Structural Analysis". Suppose u need to analyse a multi

storyed frame ( to get the stresses & strains consequent to different types of loads such as dead load, live load, wind load ,seismic load etc) we have to use " matrix method' of frame analysis. Of different Matrix method of analysis, direct stiffness method is used in computers

Use of matrix in electrical engineering • Use to find the value of different element in electronic circuit like resistance ,

current , voltage, etc by the help of matrix.

Use of in Taylor’s theorem in computer science & engineering • How do your calculator/computer calculate ππ, ee and sinsin and coscos values? I

would bet, mostly using Taylor polynomial approximation.• For example, integrating arctan (x)=11+x2=1−x2+x4−x6±…arctanx11x21x2x4x6, we ′

get that arctan(x)=x−x33+x55−x77±…arctanxxx33x55x77. One still needs to think over that it is legal, but if we put x=1x1 in it, we get

• π4=1−13+15−17±…

Use of in Taylor’s theorem in civil engineering Network of drains is an important constituent in citymunicipal constructive, and its service is also an importantcondition to realize the sustainable development of the wholecity . The optimum design toward network of drains is acomplicated system engineering, which has played animportant function of the normal decision-making . Thenew pipes and network will be designed and built fortreatment of investment cost, and considering the economicbearing capacity, which provides scientific principles for theplans and the decisions of long-term economic development

Use of in Taylor’s theorem in electrical engineering • Approximate calculation

• MatlabTaylor series are not applicable to Matlab. They must be computed manually and programmed.

real worldApplication of calculus

Average growth of bacteria

Average amount of water falling from the water fall

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)

AREA BETWEEN CURVES

x=a x=b

f(x)

g(x)

Area b/w curves = [Area under f(x)]

- [(Area under g(x)]

AREA BETWEEN CURVES

= -f(x) dxa

bg(x) dx

a

b

= [f(x) - g(x)]dxa

b

Newton’s Law of cooling

• “Rate of change of the temperature of an object is proportional to the difference between its own temperature and the temperature of its surroundings.”

(T-Te)

Reference

• DAS H.K(2015) “Advance engineering Mathematics” s.chand publication,1St edition Pp.6,46,171

• N.P.BALI(2006) “engineering mathematics” Lakshmi publication,9th Edition ,pp 423,558

• B.V RAMANNA(2007) “engineering mathematics” 2nd Edition Pp 8.22

• D.K.S.REWAR (2014) “Mathematics” College book house,New edition pp:84

Visited site:https://en.m.Wikipedia.orghttps://www.math.hmc.eduhttps://gowers.wordpress.comwww.dcs.warwick.ac.ukwww.math.csusb.edu

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