h31 Higher Order Derivatives Velocity and Acceleration

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  • 7/31/2019 h31 Higher Order Derivatives Velocity and Acceleration

    1/2

    Calculus and Vectors How to get an A+

    3.1 Higher Order Derivatives. Velocity and Acceleration2010 Iulia & Teodoru Gugoiu - Page 1 of 2

    3.1 Higher Order Derivatives. Velocity and Acceleration

    A Higher Order DerivativesLet consider the function )(xfy = .

    The firstderivative of for fprime is:

    dx

    dyyxf == ')('

    The secondderivative of f or fdouble prime is:

    2

    2

    )''(''))'('()(''dx

    yd

    dx

    dy

    dx

    dyyxfxf =

    ====

    The thirdderivative of f or f triple prime is:

    3

    3

    2

    2

    )'''('''))'(''()('''dx

    yd

    dx

    yd

    dx

    dyyxfxf =

    ====

    The fourth derivative of f or super 4 is:

    4

    4

    3

    3

    )4()4( )''''())'('''()(dx

    yddxyd

    dxdyyxfxf =

    ====

    The n-th derivative of f or super n is:

    n

    n

    n

    nnnnn

    dx

    yd

    dx

    yd

    dx

    dyyxfxf =

    ====

    1

    1)1()()1()()'())'(()(

    Notes.1. f is differentiable at x if )(' xf exists.

    2. f is double differentiable at x if )('' xf exists.

    3. fis n differentiable at x if )()( xf n exists.

    Ex 1. Find 'f , ''f , '''f , )4(f , and )5(f for

    13)( 24 ++= xxxxf .

    Ex 2. Find 'f , ''f , and '''f for1

    )(2

    3

    +=x

    xxf .

    Ex 3. Find )(nf for1

    )(+

    =x

    xxf . Ex 4. Show that 13

    3 ++= xxy satisfies 0'2''''' =+ yxyy .

  • 7/31/2019 h31 Higher Order Derivatives Velocity and Acceleration

    2/2

    Calculus and Vectors How to get an A+

    3.1 Higher Order Derivatives. Velocity and Acceleration2010 Iulia & Teodoru Gugoiu - Page 2 of 2

    B Velocity and AccelerationLet )(ts or )(th (height or altitude) be theposition

    function of a particle.

    mhsISIS

    == (meter)

    Displacementis the change in position over a time

    interval ],[ 21 tt :

    )()()()( 1212 ththhtstss ==

    Average Velocityis defined by:

    12

    12 )()(

    tt

    tsts

    t

    sAV

    =

    = )/( sm

    Velocityfunction is the derivative of the positionfunction:

    )(')(')( thtstv ==

    Speedfunction is the absolute value of velocity |)(| tv .

    Acceleration function is the derivative of velocity:

    )('')(')( tstvta == )/( 2sm

    Jerkfunction is the derivative of acceleration:)(''')('')(')( tstvtatj === )/( 3sm

    Notes:

    0)( =ts (particle is in the origin)

    0)( =th (particle is on the ground)

    0)( >ts (particle is on the positive side of the origin)

    0)( th (particle is above the ground)

    0)( tv (particle moves in the positive direction (right

    or up))0)( tvts .

    If || s is decreasing then the particle is moving towards

    the origin (MTO) and 0)()( tatv .

    If || v is decreasing then the particle is slowing down

    (SD) and 0)()(