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    Algebra

    Ahmad Syaiful Rizal 110210151001

    Amalia Martha Santosa 110210151004

    Brigita Febrianti 110210151019

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    GEO GEBRA

    GeoGebra is dynamic mathematics software that

    combines geometry, algebra, and calculus.

    This software is developed for teaching and

    learning mathematics in schools by Markus

    Hohenwarter at Florida Atlantic University.On the one hand, GeoGebra is a dynamic geometry

    system.

    You can do constructions with points, vectors,

    segments, lines, conic sections, as well as

    functions and change the outcome of the next

    construction. On the other hand, equations andcoordinates can be entered directly. .

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    APPLICATION OF GEOGEBRA

    LINES

    DERIVATIVES AND TANGENTLINE

    RIEMANN SUM AND INTEGRAL

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    LINES

    There are two types of equation commonly used todescribe a line:

    - Slope-intercept-Point-slope.

    Both forms are really both variations on the same idea. Inboth cases is needed to know the coordinates of one

    point, and the slope of the line. :

    -In the slope-intercept form, the given point is always onthe y-axis and you supply the y-coordinate of that point(Its x-coordinate is always zero).

    - In the point-slope form, you can use any point.

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    INTERSECTION OF TWO LINE

    The intersection of a line and aline can be the empty set, a point, or a

    line. Distinguishing these cases andfinding the intersection point have use,for example, in computer graphics,motion planning, and collision

    detection.

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    y = 2x+3 and y=6x+7

    FIND THE SLOPE ANDINTERSECTION OF TWO LINES ?

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    FOR FINDING A SLOPE :

    Slope[line]

    FOR FINDING AN INTERSECTION OF TWO

    LINES :

    Intersect[object,obejct]

    THE COMMAND

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    DERIVATIVE

    The definition of derivative of a function can be expressed by

    I think all of you have already known about this

    lim0

    + ()

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    DERIVATIVE

    Find the derivative of f(x) = 3x3 + 2x2 6x+ 5

    1) type the equation in Input Barand press Enter

    2) type this command (or choose from drop down list) in Input Barand press Enter

    Or,

    GeoGebra will determine the derivation in Algebra View and construct the curve f(x)

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    DERIVATIVE

    Find the derivative of f(x) = (3x2 5)4

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    TANGENT LINE

    A line that touches a curve at a point without crossing

    over. Formally, it is a line which intersects a differentiable

    curve at a point where the slope of the curve equals theslope of the line.

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    Construct a tangent line at the graph off(x) = 2x3 + 3x 2

    1) type the equation in Input Barand press Enter

    2) choose New Point from Construction Tools and click at the graph

    3) choose Tangents from toolbox and click at point and graph

    4) Drag the point at the graph

    TANGENT LINE

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    TANGENT LINE

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    RIEMANN SUM

    Let a closed interval be partitioned by points < 1 < 2 < < 1 < , where the lengths of the resulting intervals between thepoints are denoted x1, x2, ..., xn. Let be an arbitrary point in

    the th subinterval. Then the quantity (k)=

    is called a Riemann sum for a given function () and partition, andthe value max xis called the mesh size of the partition.

    . The shaded areas in the above plots show the lower and uppersums for a constant mesh size.

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    RIEMANN SUM

    What is it?

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    RIEMANN SUM

    Using geogebra

    Approxiate and demonstrate the riemann sum for f(x) = 2x3

    +8x2 + 4x 2 by taking the sampel point a = -3 and b = -1, and

    n = 8. sketch the graph of function and use the geogebra to

    determine the

    1) type the equation in Input Barand press Enter

    2) type this command (or choose from drop down list) in Input Barand press Enter

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    RIEMANN SUM

    Using geogebra

    Command script in geogebra

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    DEFINITE INTEGRAL

    Definite integral is the number that magnitude is determined bytaking the limit of riemann sum, which is associated with a normclosedinterval partition approach

    Determine the area under the graph of function

    f(x) = 2x3 + 3x 2 with the boundary x = -3 dan x = -2

    1) type the equation in Input Barand press Enter

    2) type this command (or choose from drop down list) in Input Barand press Enter

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    DEFINITE INTEGRAL

    Determine the area under the graph offunction

    4 3 + 1

    with the boundary x = -1 dan x = 1.9

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    THANKS FOR YOUR ATTENTION