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How to use ti calculators to perform discrete sums, specifically left- and right-hand sums, which are approximations of definite integrals. Creating and summing lists, using sequences to generate function values, and calculating sums of sequences to find left- and right-hand sums. The document also discusses the importance of these sums in calculus and how they can be used to approximate the area under a function's graph.
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The fundamental activity of integral calculus is adding. In the discrete case, we sum a set of values. In the continuous case, we use the integral to sum over an interval. In this chapter we will restrict our attention to finite discrete sums. You probably already know that these sums are approximations to the value of a definite integral. We will make that connection in Chapter 16, Riemann sums.
If you drive 50 miles per hour for 3 hours, then you will have traveled 50 + 50 + 50 = 150 miles. If you drive 20 mph for two hours, then 30 mph for two hours, and finally 40 mph for two hours, then in the six hours you will have traveled 20(2)+ 30(2) + 40(2) = 180 miles. We rarely travel a constant speed. Table 13.1 shows velocity readings at 6 different times. We do not know if we traveled mostly at 20 ft/sec or 30 ft/sec for the first two seconds. If we assume the velocity is constantly increasing, then these two numbers give us lower and upper bounds for the first two seconds.
We simply add the first five and multiply by 2, and then repeat this with the last five velocities to find a lower and upper bound on the distance traveled in ten seconds. We now introduce the technique of summing lists to do this job. The TI has 6 list variables, named L 1 , L 2 , L 3 , L 4 , L 5 , L 6. Lists can be created from the home screen using the set symbols { and } then storing ( STO > ) the list in one of the list variable names by pressing 2nd_1 to 2nd_.
After entering the first data list and storing it in L 1 (see Figure 13.1), we can use 2nd_ENTRY to edit and create the second list from the first. Delete the first entry, 20, insert 50 at the end of the list, and change L 1 to L 2. The sum command is under the MATH submenu of 2nd_LIST .Repeat for the upper sum.
Creating and summing lists from STAT EDIT There is an alternate way to enter lists, which we used in Chapters 9 and 10. It is particularly handy when the list is long and when you need to edit the data. Press STAT 1 :Edit and enter the data in the L 1 column. (Notice that data entered as L 1 and L 2 on the home screen will appear in the first two columns. To remove the entire first column of data, for instance, move the cursor to the heading that highlights L 1 , press CLEAR ENTER , and the previous data will be erased.) In Figure 13.2, we repeat the entry and computation above in this setting.
Enter and edit the data in lists by using the numeric and navigation keys.
Paste sum from the MATH submenu of 2nd_List ( *L 1 2 ) (or from the catalog) and enter
Highlight a list name. Press ENTER and define a list by a formula. Here L 1 is copied to L2.
The 20 at the top of the list has been deleted and 50 is placed at the end.
We place the sum of *L 2 2 in L 2 (6).
Figure 13.2 Summing a column using 2nd_LIST MATH 5:sum(.
The TI has a very handy feature called seq .Essentially, the sequence command makes a list of values using a given expression with the index taking values from a starting point to a stopping point, increasing by a given increment.
seq ( expression, index, start, stop, increment )
As an example, we can create a list of squares from 0 to 16. We first use the squaring function inside the sequence command, as shown in Figure 13.3, then we show that a Y 1 squaring function can alternatively used in the sequence command. The TI- default setting for the increment is 1 and this can be left off. However, entering the increment is mandatory for the TI-.
»- TI-82 Tip: You must enter an increment in all seq( ) commands. They may not always be shown in this book because they are optional for the TI-83. Use 1 as the increment.
sums as areas.
By increasing the number of partitions, the left- and right-hand sums may approach a limit which we interpret as the (signed) area under the function's graph. We write this as
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