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Summing Lists and Sequences: Left- and Right-Hand Sums in Calculus, Exams of Calculus

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.

Typology: Exams

Pre 2010

Uploaded on 08/19/2009

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13. LEFT- AND RIGHT- HAND SUMS
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.
Distance from the sum of the velocity data
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 L1,L2,L3,L4,L5,L6. 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_6 .
Creating and summing lists from the home screen:
1
{...}
L
After entering the first data list and storing it in L1(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 L1to L2. 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 L1
column. ( Notice that data entered as L1and L2on 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 L1, pr ess 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 (L1*2) (or from the catalog) and enter
Highlight a list name. Press ENTER and define a list by a formula. Here L1is 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 L2*2 in L2(6).
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13. LEFT- AND RIGHT- HAND SUMS

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.

Distance from the sum of the velocity data

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_.

Creating and summing lists from the home screen: {...}  L 1

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(.

Using sequences to create a list of function values: seq(...)

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.

We can use our long entry line to find left-hand sums for the function f ( ) x sin( x^2 )on the interval 0   x 2 and then on the

shorter interval 0   x . The graph in Figure 13.5 shows that the function is negative from to 2 . The fact that that

the sum on the longer interval 0   x 2 ; is less than the sum on the subinterval 0   x is understandable since we are

adding the negative values on the subinterval 0   x 2 . This shows that we will need to be careful when interpreting these

sums as areas.

Approximating area using the left- and right-hand sums

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

1

( ) lim ( )

b^ n a n i i

f x dx f x x

 

^ ^  

As an example, let's see if there is a limit to the left hand sums of sin( ) x over the interval 0   x  as the number of partitions

increases. The computations with N 10, 20, and 50 in Figure 13.6 suggest that the sums approach 2. We will later confirm that

0

sin( ) x dx 2

  by using the Fundamental Theorem of Calculus.