MATHEMATICS-IMATHEMATICS-I

CONTENTSCONTENTS Ordinary Differential Equations of First Order and First DegreeOrdinary Differential Equations of First Order and First Degree Linear Differential Equations of Second and Higher OrderLinear Differential Equations of Second and Higher Order Mean Value TheoremsMean Value Theorems Functions of Several VariablesFunctions of Several Variables Curvature, Evolutes and EnvelopesCurvature, Evolutes and Envelopes Curve TracingCurve Tracing Applications of IntegrationApplications of Integration Multiple IntegralsMultiple Integrals Series and SequencesSeries and Sequences Vector Differentiation and Vector OperatorsVector Differentiation and Vector Operators Vector IntegrationVector Integration Vector Integral TheoremsVector Integral Theorems Laplace transformsLaplace transforms

TEXT BOOKSTEXT BOOKS

A text book of Engineering Mathematics, Vol-I A text book of Engineering Mathematics, Vol-I T.K.V.Iyengar, B.Krishna Gandhi and Others, T.K.V.Iyengar, B.Krishna Gandhi and Others, S.Chand & CompanyS.Chand & Company

A text book of Engineering Mathematics, A text book of Engineering Mathematics, C.Sankaraiah, V.G.S.Book LinksC.Sankaraiah, V.G.S.Book Links

A text book of Engineering Mathematics, Shahnaz A A text book of Engineering Mathematics, Shahnaz A Bathul, Right PublishersBathul, Right Publishers

A text book of Engineering Mathematics, A text book of Engineering Mathematics, P.Nageshwara Rao, Y.Narasimhulu & N.Prabhakar P.Nageshwara Rao, Y.Narasimhulu & N.Prabhakar Rao, Deepthi PublicationsRao, Deepthi Publications

REFERENCESREFERENCES

A text book of Engineering Mathematics, A text book of Engineering Mathematics, B.V.Raman, Tata Mc Graw HillB.V.Raman, Tata Mc Graw Hill

Advanced Engineering Mathematics, Irvin Advanced Engineering Mathematics, Irvin Kreyszig, Wiley India Pvt. Ltd.Kreyszig, Wiley India Pvt. Ltd.

A text Book of Engineering Mathematics, A text Book of Engineering Mathematics, Thamson Book collectionThamson Book collection

UNIT-IIIUNIT-III

CHAPTER-I:MEAN VALUE CHAPTER-I:MEAN VALUE THEOREMSTHEOREMS

CHAPTER-II:FUNCTIONS OF CHAPTER-II:FUNCTIONS OF SEVERAL VARIABLESSEVERAL VARIABLES

UNIT HEADERUNIT HEADER

Name of the Course: B.TechName of the Course: B.Tech

Code No:07A1BS02Code No:07A1BS02

Year/Branch: I Year Year/Branch: I Year CSE,IT,ECE,EEE,ME,CIVIL,AEROCSE,IT,ECE,EEE,ME,CIVIL,AERO

Unit No: IIIUnit No: III

No. of slides:17No. of slides:17

S. No.S. No. ModuleModule LectureLecture

No. No.

PPT Slide No.PPT Slide No.

11 Introduction, Mean Introduction, Mean value theoremsvalue theorems

L1-4L1-4 8-118-11

22 Taylor’s theorem, Taylor’s theorem, Functions of several Functions of several variables, Jacobianvariables, Jacobian

L5-9L5-9 12-1612-16

33 Maxima & Minima, Maxima & Minima, Lagrange’s method of Lagrange’s method of undetermined undetermined multipliersmultipliers

L10-12L10-12 17-1917-19

UNIT INDEXUNIT INDEXUNIT-III UNIT-III

Lecture-1Lecture-1INTRODUCTIONINTRODUCTION

Here we study about Mean value theorems.Here we study about Mean value theorems. Continuous function: If limit of f(x) as x tends Continuous function: If limit of f(x) as x tends

c is f(c) then the function f(x) is known as c is f(c) then the function f(x) is known as continuous function. Otherwise the function is continuous function. Otherwise the function is known as discontinuous function.known as discontinuous function.

Example: If f(x) is a polynomial function then Example: If f(x) is a polynomial function then it is continuous.it is continuous.

Lecture-2Lecture-2ROLLE’SROLLE’S MEAN VALUE THEOREM MEAN VALUE THEOREM

Let f(x) be a function such thatLet f(x) be a function such that

1) it is continuous in closed interval [a, b];1) it is continuous in closed interval [a, b];

2) it is differentiable in open interval (a, b) and2) it is differentiable in open interval (a, b) and

3) f(a)=f(b)3) f(a)=f(b)

Then there exists at least one point c in open Then there exists at least one point c in open interval (a, b) such that f interval (a, b) such that f ''(c)=0(c)=0

ExampleExample: f(x)=(x+2): f(x)=(x+2)33(x-3)(x-3)44 in [-2,3]. Here in [-2,3]. Here c=-2 or 3 or 1/7c=-2 or 3 or 1/7

Lecture-3Lecture-3LAGRANGE’S MEAN VALUE LAGRANGE’S MEAN VALUE

THEOREMTHEOREM Let f(x) be a function such thatLet f(x) be a function such that

1) it is continuous in closed interval [a, b] and1) it is continuous in closed interval [a, b] and

2) it is differentiable in open interval (a, b)2) it is differentiable in open interval (a, b)

Then there exists at least one point c in open Then there exists at least one point c in open interval (a, b) such that f interval (a, b) such that f ''(c)=[f(b)-f(a)]/[b-a](c)=[f(b)-f(a)]/[b-a]

ExampleExample: f(x)=x: f(x)=x33-x-x22-5x+3 in [0,4]. Here -5x+3 in [0,4]. Here c=1+c=1+√37/3√37/3

Lecture-4Lecture-4CAUCHY’S MEAN VALUE CAUCHY’S MEAN VALUE

THEOREMTHEOREM If f:[a, b] If f:[a, b] →→R, g:[a, b] →R are such that R, g:[a, b] →R are such that

1)f,g are continuous on [a, b] 1)f,g are continuous on [a, b] 2)f,g are differentiable on (a, b) and 2)f,g are differentiable on (a, b) and 3)g 3)g ''(x)≠0 for all x(x)≠0 for all xЄЄ(a, b) then there exists (a, b) then there exists ccЄЄ(a, b) such that (a, b) such that [f(b)-f(a)]/[g(b)-g(a)]=f ′(c)/g′(c)[f(b)-f(a)]/[g(b)-g(a)]=f ′(c)/g′(c)

ExampleExample: f(x)=√x, g(x)=1/√x in [a,b]. Here : f(x)=√x, g(x)=1/√x in [a,b]. Here c=√abc=√ab

Lecture-5Lecture-5TAYLOR’S THEOREMTAYLOR’S THEOREM

If f:[a, b]If f:[a, b]→→R is such that R is such that 1)f 1)f (n-1)(n-1) is continuous on [a, b] is continuous on [a, b] 2)f2)f(n-1)(n-1) is derivable on (a, b) then there exists a is derivable on (a, b) then there exists a point cpoint cЄЄ(a, b) such that (a, b) such that f(b)=f(a)+(b-a)/1!f f(b)=f(a)+(b-a)/1!f ''(a)+(b-a)(a)+(b-a)22/2!f /2!f ""(a)+…..(a)+…..

ExampleExample: f(x)=e: f(x)=exx. Here Taylor’s expansion at . Here Taylor’s expansion at x=0 is 1+x+xx=0 is 1+x+x22/2!+…../2!+…..

Lecture-6Lecture-6MACLAURIN’S THEOREMMACLAURIN’S THEOREM

If f:[0,x]If f:[0,x]→→R is such that R is such that 1)f1)f(n-1)(n-1) is continuous on [0,x] is continuous on [0,x] 2)f2)f(n-1) (n-1) is derivable on (0,x) then there exists a is derivable on (0,x) then there exists a real number real number θθЄЄ(0,1) such that (0,1) such that f(x)=f(0) + xf f(x)=f(0) + xf ''(0) +x(0) +x22/2! f /2! f ""(0) +………(0) +………

ExampleExample: f(x)=Cosx. Here Maclaurin’s : f(x)=Cosx. Here Maclaurin’s expansion is 1-xexpansion is 1-x22/2!+x/2!+x44/4!-…./4!-….

Lecture-7Lecture-7FUNCTIONS OF SEVERAL FUNCTIONS OF SEVERAL

VARIABLESVARIABLES We have already studied the notion of limit, We have already studied the notion of limit,

continuity and differentiation in relation of continuity and differentiation in relation of functions of a single variable. In this chapter functions of a single variable. In this chapter we introduce the notion of a function of we introduce the notion of a function of several variables i.e., function of two or more several variables i.e., function of two or more variables.variables.

ExampleExample 1: Area A= ab 1: Area A= ab Example Example 2: Volume V= abh2: Volume V= abh

Lecture-8Lecture-8DEFINITIONSDEFINITIONS

Neighbourhood of a point(a,b): A set of points lying Neighbourhood of a point(a,b): A set of points lying within a circle of radius r centered at (ab) is called a within a circle of radius r centered at (ab) is called a neighbourhood of (a,b) surrounded by the circular neighbourhood of (a,b) surrounded by the circular region.region.

Limit of a function: A function f(x,y) is said to tend Limit of a function: A function f(x,y) is said to tend to the limit l as (x,y) tends to (a,b) if corresponding to to the limit l as (x,y) tends to (a,b) if corresponding to any given positive number p there exists a positive any given positive number p there exists a positive number q such that f(x,y)-lnumber q such that f(x,y)-l<p foa all points (x,y) <p foa all points (x,y) whenever x-a≤q and y-b≤qwhenever x-a≤q and y-b≤q

Lecture-9Lecture-9JACOBIANJACOBIAN

Let u=u(x,y), v=v(x,y). Then these two Let u=u(x,y), v=v(x,y). Then these two simultaneous relations constitute a simultaneous relations constitute a transformation from (x,y) to (u,v). Jacobian of transformation from (x,y) to (u,v). Jacobian of u,v w.r.t x,y is denoted by J[u,v]/[x,y] or u,v w.r.t x,y is denoted by J[u,v]/[x,y] or ∂(u,v)/∂(x,y)∂(u,v)/∂(x,y)

ExampleExample: x=r cos: x=r cosθθ,y=r sin,y=r sinθθ then ∂(x,y)/∂(r, then ∂(x,y)/∂(r,θθ) ) is r and ∂(r,is r and ∂(r,θθ)/∂(x,y)=1/r)/∂(x,y)=1/r

Lecture-10Lecture-10MAXIMUM AND MINIMUM OF MAXIMUM AND MINIMUM OF

FUNCTIONS OF TWO VARIABLESFUNCTIONS OF TWO VARIABLES Let f(x,y) be a function of two variables x and Let f(x,y) be a function of two variables x and

y. At x=a, y=b, f(x,y) is said to have maximum y. At x=a, y=b, f(x,y) is said to have maximum or minimum value, if f(a,b)or minimum value, if f(a,b)>f(a+h,b+k) or >f(a+h,b+k) or f(a,b)<f(a+h,b+k) respectively where h and k f(a,b)<f(a+h,b+k) respectively where h and k are small values.are small values.

ExampleExample: The maximum value of : The maximum value of f(x,y)=xf(x,y)=x33+3xy+3xy22-3y-3y22+4 is 36 and minimum +4 is 36 and minimum value is -36value is -36

Lecture-11Lecture-11EXTREME VALUEEXTREME VALUE

f(a,b) is said to be an extreme value of f if it is f(a,b) is said to be an extreme value of f if it is a maximum or minimum value.a maximum or minimum value.

Example Example 1: The extreme values of u=x1: The extreme values of u=x22yy22-5x-5x22--8xy-5y8xy-5y22 are -8 and -80 are -8 and -80

ExampleExample 2: The extreme value of x 2: The extreme value of x22+y+y22+6x+12 +6x+12 is 3is 3

Lecture-12Lecture-12LAGRANGE’S METHOD OF LAGRANGE’S METHOD OF

UNDETERMINED MULTIPLIERSUNDETERMINED MULTIPLIERS

Suppose it is required to find the extremum for the Suppose it is required to find the extremum for the function f(x,y,z)=0 subject to the condition function f(x,y,z)=0 subject to the condition фф(x,y,z)=0 (x,y,z)=0

1)Form Lagrangean function F=f+1)Form Lagrangean function F=f+λλфф

2)Obtain F2)Obtain Fxx=0,F=0,Fyy=0,F=0,Fzz=0=0

3)Solve the above 3 equations along with condition.3)Solve the above 3 equations along with condition. ExampleExample: The minimum value of x: The minimum value of x22+y+y22+z+z2 2 with with

xyz=axyz=a33 is 3a is 3a22