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  • 7/29/2019 NM-2-2011-2012-02-Slides.pdf

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    Lesson-2:

    Fundamental of

    DifferentiationEAS 3102

    Surjatin WiriadidjajaDepartment of Aerospace EngineeringUniversiti Putra Malaysia

    Semester 2/2011-2012

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    NUMERICAL METHODS NM-2-2011/2012-02: Fundamental of Differentiation p2

    Introduction

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    NUMERICAL METHODS NM-2-2011/2012-02: Fundamental of Differentiation p3

    Objective

    understand the basics of differentiation,

    relate the slopes of the secant line and tangentline to the derivative of a function,

    find derivatives of polynomial, trigonometric andtranscendental functions,

    use rules of differentiation to differentiate functions,

    find maxima and minima of a function, and

    apply concepts of differentiation to real worldproblems.

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    NUMERICAL METHODS NM-2-2011/2012-02: Fundamental of Differentiation p4

    Differentiation

    The concepts of differentiation:

    the secant line, the slope of a tangent line as a background tosolving nonlinear equations using the Newton-Raphson method,finding maxima and minima of functions as a means ofoptimization, the use of the Taylor series to approximate functions,etc.

    The derivative of a function represents the rate of change of avariable with respect to another variable.

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    NUMERICAL METHODS NM-2-2011/2012-02: Fundamental of Differentiation p5

    Differentiation (cont)

    x

    Q

    P

    f(x)

    secant line

    tangent

    line

    Figure 1 Function curve with tangent and secant lines.

    )(xf

    x

    P

    Q

    a a+h

    Figure 2 Calculation of the secant line. h

    afhafm

    hPQ

    )()(lim

    0tangent,

    Concepts of the secant lineand tangent line (Figure 1).

    As Q moves closer and closer to P, thelimiting portion is called the tangent line.The slope of the tangent line then is thelimiting value of as :

    aha

    afhafmPQ

    )(

    )()(secant,

    h

    afhaf )()(

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    NUMERICAL METHODS NM-2-2011/2012-02: Fundamental of Differentiation p6

    Differentiation (cont)

    Derivative of aFunction

    The derivative of a function at isdefined as

    hafhafaf

    h

    )()(lim)(0

    ax

    afxfaf

    ax

    )()(lim)(

    Other Notations ofDerivativesDerivates can be denoted in several ways.

    For the first derivative, the notations are

    For the second derivative, the notations are

    For the derivative, the notations are

    dx

    dyandyxf

    dx

    dxf ,),(),(

    2

    2

    2

    2

    ,),(),( dx

    yd

    andyxfdx

    d

    xf

    n

    nn

    n

    nn

    dx

    ydyxf

    dx

    dxf ,),(),(

    )()(

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    Differentiation (cont)

    Theorems of Differentiation

    1 The derivative of a constant is zero. If f(x)=k, where k is a constant.

    2 The derivative of f(x)=xn, where n0 is f(x)=nxn-1.

    3 The derivative of f(x)=kg(x), where k is a constant is f(x)=kg(x).

    4 The derivative of f(x0=u(x)v(x) is f(x)=u(x)v(x).

    5 The derivative of f(x)=u(x)v(x) is (Product Rule)

    6 The derivative of Is (Quotient Rule)

    )()()()()( xudx

    dxvxv

    dx

    dxuxf

    )(

    )()(

    xv

    xuxf

    2))((

    )()()()(

    )(xv

    xvdx

    dxuxu

    dx

    dxv

    xf

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    Differentiation (cont)

    Chain Rule of Differentiation

    Implicit Differentiation

    Higher order derivatives

    )())(())((( xgxgfxgfdx

    d

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    Differentiation (cont)

    Finding maximum and minimum of a functionLet f(x)be a function in domainD, then

    A.f(a)is the maximum of the function if f(x) f(a)for all values ofx in the domainD.

    B.f(a)is the minimum of the function iff(a) f(x)for all values ofxin the domainD.

    The minimum and maximum of a function are also the critical values of afunction. An extreme value can occur in the interval

    at end points

    a point in where .

    a point in where does not exist.

    These critical points can be the local maximas and minimas of thefunction

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    Differentiation (cont).

    f(x)

    x

    .

    Absolute Minimum

    Local Minimum

    LocalMinimu

    m

    Local Maximum

    (f (x) does not

    exist)

    Absolute Maximum

    Local

    Maximum

    c d

    maximum

    minimum

    xFigure 7 Graph illustrating the concepts of maximum and

    minimum.

    Domain = [c,d]

    c d

    xxf /1)( )(xf

    x

    Figure 10 Function that has no maximum or

    minimum.

    )(xf

    )(af

    ax

    ax

    x

    Figure 11 Graph demonstrates the concept of a singular point with

    discontinuous slope at

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    NUMERICAL METHODS NM-2-2011/2012-02: Fundamental of Differentiation p11

    Differentiation ofContinuous Functions

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    NUMERICAL METHODS NM-2-2011/2012-02: Fundamental of Differentiation p12

    Objective

    derive formulas for approximating the firstderivative of a function,

    derive formulas for approximating derivatives from

    Taylor series,

    derive finite difference approximations for higherorder derivatives, and

    use the developed formulas in examples to findderivatives of a function.

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    NUMERICAL METHODS NM-2-2011/2012-02: Fundamental of Differentiation p13

    Derivative

    The derivative of a function atxis defined as

    To be able to find a derivative numerically, one could make a

    finitexto give,

    There are 3 three such approximations.

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    NUMERICAL METHODS NM-2-2011/2012-02: Fundamental of Differentiation p14

    Forward Difference ApproximationThe First Derivative

    )(xf

    xx

    x

    x

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    NUMERICAL METHODS NM-2-2011/2012-02: Fundamental of Differentiation p15

    Backward Difference ApproximationThe First Derivative

    )(xf

    x

    xxx

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    NUMERICAL METHODS NM-2-2011/2012-02: Fundamental of Differentiation p16

    THE END


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