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INTRODUCTION TO GEOGEBRA
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In order to play with these interactive GeoGebra modules you will need to make sure that you have the Java Plugin installed. If you don't have the plugin installed you will not see the interactive module below. If you don't see the interactive module below download the Java Plugin here. Once the Java plugin is installed restart your browser and come back to this page and will see the GeoGebra module below.
PLEASE READ THE FOLLOWING:
To see some of the features that you can control in a GeoGebra module see my video here. Even though the video deals with the toolkit functions there is plenty to learn about how to interact with any GeoGebra module.
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PREREQUISITE IDEAS
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IDEA 1: CLOSED INTERVAL
You must understand how to use "closed interval" notation. See the example below:
IDEA 2: PARTITION AND SUBINTERVAL
A "partition" is a closed interval that is "broken up" into a number of "parts" which are closed intervals. These parts DO NOT have to be equal. Each part of the partition is called a "subinterval." Mathematicians say a closed interval can be "partitioned into subintervals."
See examples below.
IDEA 3: LEFT END, MIDPOINT AND RIGHT END OF A SUBINTERVAL
You must be able to name the left end of a subinterval, midpoint of a subinterval and right end of a subinterval. See examples below.
IDEA 4: EVALUATION POINT"
Given a subinterval, an "evaluation point" is defined as a number that you choose within the given subinterval; the left endpoint or the right endpoint can also be an evaluation point.
There are an infinite number of evaluation points in a subinterval. Also evaluation points are often symbolized by an asterisk as a subscript. See example below.
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WHAT IS A RIEMANN SUM?
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A Riemann Sum is a mathematical construction that has a definition and two major applications which the AP Calculus student is expected to understand. It is absolutely essential that you understand how to construct a Riemann Sum. Here is what you are expected to know:
- DEFINITION: Use the defintion with a given function.
- APPLICATION: Be able to determine the area of 2 dimensional and a volume of a 3 dimensional figures.
- APPLICATION: Be able to determine the accumulation of a quantity when given changing rates over a given period of time. For example, if a car is constantly changing its velocity(a rate) over a period of time a Riemann Sum enables you to determine the distance(a quantity) the car has traveled over that period of time. Think odometer in your car!
| DEFINITION OF A RIEMANN SUM |
Given a continuous function f on a CLOSED INTERVAL [a, c] in the domain of f
a) Partition [a, c] into n subintervals; the subintervals DO NOT have to be equal. See example below.
b) You will NAME each subinterval using the endpoints of each subinterval. The convention is to name the endpoints in Subinterval 1 x0 and x1 where x0 < x1 ; Subinterval 2' s endpoints are named x1 and x2 where x1 < x2 ; Subinterval 3's endpoints are named x2 and x3 where x2 < x3 , so on and so forth.
c) Choose an "evaluation point" from each subintervals. The convention is to call the evaluation point from the first subinterval x1* , the second subinterval x2* , x3* , etc.
d) Determine the length of each subinterval by the formula (c - a)/n IF the subintervals are equal in length; otherwise you will have to subtract the left hand endpoint of a subinterval from the right hand endpoint of the subinterval, xi* - xi-1* , to get the length of the subinterval.
e) Determine f(x1* ), f(x2* ), f(x3* ), etc. where x1* , x2* , x3* , etc. are the evalualtion points in each subinterval and f(x) is the given function.
f) The following SUM is called a RIEMANN SUM:
f(x1* )( x1 - x0 ) + f(x2* )( x2 - x1 ) + f(x3* )( x3 - x2 ) + f(x4* )( x4 - x3 ) + ... + f(xn* )( xn - xn-1 )
NOW YOUR JOB IS TO FIGURE OUT HOW TO CONSTRUCT THE ABOVE MATHEMATCIAL
STATEMENT CALLED A RIEMANN SUM USING THE GEOGEBRA MODULE BELOW |
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HOW TO USE THIS MODULE....under construction
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Step 1)
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Step 8) Once you finish, click the reset button,  , in the upper right hand of the GeoGebra module and the module will be set to the original state. Now you can choose new numbers for your variables and go through the process once again.
Comment: Make sure you number each problem that you do. The first problem that you do is to be labeled 1, the next problem 2, so on and so forth.
MODULE BELOW:
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