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L5 Partial Derivatives

Apr 03, 2018

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  • 7/28/2019 L5 Partial Derivatives

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    Chapter 2: Multivariable Calculus

    Lecture 2: Partial Derivatives

    byAssoc.Prof. Mai Duc

    Thanh

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    Rate of change of a function f(x,y)depends on the direction

    Begin by measuring the rate of change if

    we move parallel to thexoryaxes

    These are called thepartial derivatives ofthe function

    Partial Derivatives

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    Definitions

    Partial derivative with respect to x :

    fx (x,y)

    x

    f(x,y) limh0

    f(x h,y) f(x,y)

    h

    Partial derivative with respect to y :

    fy (x,y)

    yf(x,y) lim

    h0

    f(x,y h) f(x,y)

    h

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    f(x,y) = 4 - 2x2 - y2

    Cut the surface withplanes:

    x= 1 and y= -1

    All meet at (1, -1, 1)

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    The intersection of the graph off(x,y) with the planey = b is the graph of

    g(x) = f(x,b)

    Then:

    Is the slope of the tangent line atx = a.

    Partial with respect tox

    fx

    (a,b) d

    dxf(x,b)

    x a

    g(a)

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    g(x) = f(x,-1) = 3 - 2x2

    g(x) = -4x

    fx(1, -1) = g(1) = -4

    - 1.0 - 0.5 0.5 1.0x

    0.5

    1.0

    1.5

    2.0

    2.5

    3.0

    z

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    The intersection of the graph off(x,y) with theplanex = a is the graph of

    h(y) = f(a,y)

    Then:

    Is the slope of the tangent line at y = b.

    Partial with respect to y

    fy (a,b) d

    dyf(a,y)

    y b

    h (b)

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    h(y) = f(1, y) = 2 - y2

    h(y) = -2y

    fy(1, -1) = h(-1)

    = 2

    - 1.5 - 1.0 - 0.5 0.5 1.0 1.5

    y

    0.5

    1.0

    1.5

    2.0

    z

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    The partial derivativesare the slopes of the

    tangent lines parallelto thexz-plane andthe yz-plane.

    (lines shown in red)

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    To compute fx, treat yas a constant.

    To compute fy, treatxas a constant.

    Computing partial derivatives

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    Find partial derivatives of the function

    Solution.

    To find fx, treat yas a constant. So

    Similarly, to find fy, treatxas a constant. So

    Example

    2( , ) sin( )f x y xy x y

    ( , ) cos( ) 2xf x y y xy xy

    2

    ( , ) cos( )yf x y x xy x

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    Find partial derivatives of the function

    Solution.

    Note that fx, and fy are functions of two variables xand y

    Example

    2( , ) sin( )f x y xy x y

    ( , ) cos( ) 2 ( , )xf x y y xy xy g x y

    2

    ( , ) cos( ) ( , )yf x y x xy x h x y

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    The second derivative of a function of onevariable is very useful in determiningrelative maxima and minima

    Second-order partial derivatives (partialderivatives of a partial derivative) are usedin a similar way for functions of two or

    more variables

    Higher Derivatives

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    Second-order partial Derivatives

    2

    2

    2

    2

    2

    2

    For a function ( , ) :

    ( , )

    ( , )

    ( , )

    ( , )

    xx xx

    yy yy

    xy xy

    yx yx

    z f x y

    z zf x y z

    x x x

    z zf x y z

    y y y

    z zf x y z

    y x y x

    z zf x y z

    x y x y

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    Example

    f(x,y) 3x 2y 2sin(xy) y 3

    fx (x,y) 6xy 2cos(xy)y

    fy (x,y) 3x2 2cos(xy)x 3y 2

    fxx(x,y) 6y 2sin(xy)y2

    fxy(x,y)

    y

    fx (x,y) 6x 2sin(xy)xy 2cos(xy)

    fyx(x,y)

    xfy (x,y) 6x 2sin(xy)xy 2cos(xy)

    fyy(x,y) 2sin(xy)x2

    6y

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    Clairauts Theorem

    If (a,b) is in a disk D

    and are continuous on D

    then : ( , ) ( , )

    xy yx

    xy yx

    f f

    f a b f a b

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    Functions of three variables

    f(x,y,z) 3x 2yzz3y

    x f(x,y,z) fx (x,y,z) 6xyz

    y

    f(x,y,z) fy (x,y,z) 3x2zz3

    zf(x,y,z) fz(x,y,z) 3x

    2y 3z2y

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    Find all the first and the second partialderivatives of the function

    Exercise

    ( , ) yf x y x

    2

    ( , ) cos( )

    x

    yg x y t dt

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    Recall

    ( ) ( )

    x

    a

    df t dt f x

    dx

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    Example 1: The temperature of water at some point in ariver where a nuclear power plant discharges its hotwater is approximated by

    x = temperature of river water before it reaches the plant

    y= number of megawatts (in hundreds) of electricityproduced by the plant

    a) Find and interprets

    b) Find and interprets

    Partial Derivatives as Rates ofchanges

    ( , ) 2 5 40T x y x y xy

    (9,5)xT

    (9,5)yT

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    a) We have

    Interpretation: This means that if x changes 1

    degree from 9 to 9+1=10, then T approximatelychanges 7 degrees, while y remains constant=5

    b)

    Interpretation: This means that if y changes 1 unitfrom 5 to 5+1=6, then T approximately changes14 degrees, while x remains constant=9

    Solution

    ( , ) 2 5 40

    ( , ) 2 , (9,5) 7x x

    T x y x y xy

    T x y y T

    ( , ) 5 , (9,5) 14y yT x y x T

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    A company that manufactures computers hasdetermined that its production function isgiven by

    x=size of labor force (work-hours/week)

    y=amount of capital (units of $1000)

    Find the marginal productivity of labor andcapital when x=50 and y=20, and interpretthe results

    Example 2

    4

    2 3( , ) 500 800 3 4

    yP x y x y x y x

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    The marginal productivity of labor is given by

    Interpretation: This means that if x change 1 unit from 50 to

    51, then the production approximately changes 14000 units,while y remains constant at 20

    The marginal productivity of capital is given by

    Interpretation: This means that if y changes 1 unit from 20 to21, then the production approximately changes 300 units,while x remains constant at 50

    Solution4

    2 3( , ) 500 800 3

    4

    yP x y x y x y x

    2

    2

    ( , ) 500 6 3

    (50,20) 500 6(50)(20) 3(50 ) 14000

    x

    x

    P x y xy x

    P

    2 3

    2 3

    ( , ) 800 3

    (50,20) 800 3(50 ) 20 300

    y

    y

    P x y x y

    P