Math 131 Lab 3 The Value of Euler's number e and more 4 Sep.
The objectives of lab 3 are to use the rule of 4, to approximate the value of Euler's number e, to improve your understanding of the concept of limit and to compare exponential and power functions for large x. The rule of 4 investigates a concept with numbers, algebra, graphs and sentences.
Part 1. Find the value of Euler's number e graphically and numerically. n
1
. They admit their definition has shortcomings. The n
n →∞
lab attempts to make some sense of this definition. The first part of the lab concerns the graph of the function f( x ) = ( 1 + 1/x ) x for x > 0. A mathematician knows that this function f( x ) = ( 1 + 1/x )x has a domain ( ­1 )  ( 0,  and a range of (1, e )  ( e, . Its graph has a horizontal asymptote of y = e and a vertical asymptote of x = ­1.
( )
On page 44 of the text, the authors define Euler's number e with e = lim 1 +
WolframAlpha provided the following graphs for f in the intervals [1000, 10000 ], [ 2 ∙ 10 7, 3 ∙ 10 7 ], [ 2.4 ∙ 10 7, 2.41 ∙ 10 7 ]. The last two graphs are not accurate and indicate that the computer cannot compute the y value of the function f precisely for large values of x. For
x > 1, the graph of the function f increases, is concave down and has y = e as a horizontal asymptote.
a. Estimate the number e graphically. Graph the function f( x ) = ( 1 + 1/x )x. Use the Y= screen to enter f as the function y1. You will investigate the behavior of f for large x and thus you will find an approximation for e. Use your TI to plot the graph on the interval [ 1000, 10000 ]. Then sketch the graph accurately onto graph paper. Use TABLE to identify several points on the graph. Next use your TI to find an interval with large values of xmin and xmax for which the TI displays an inaccurate graph for f. Identify the interval and sketch the graph quickly on graph paper. Identify several points on the graph. Describe the graph in English. Can you explain
why the calculator displays such a weird graph? Your reply should use some of the ideas in section 1.7 of the text.
Then find an interval with xmax as large as possible for which the TI displays an accurate graph. Identify the interval and sketch the graph
x
1
.
accurately on graph paper. Identify several points on the graph. Use this graph to provide a graphical estimate for lim 1+
x
x→∞
( )
b. Estimate Euler's number e numerically. Construct a table of data values for ( x, f( x ) ), where f( x ) = ( 1 + 1/x )x as in part a. Use MODE to set the number of decimals to its maximal value. Use TblSet to set Independent to ASK. Use TABLE to estimate e. Complete a table with two columns labeled x and f( x ) = ( 1 + 1/x )x. Underline the number of correct decimals in f for the approximation to e.
Use TABLE to find the smallest positive value x for which the TI reports that f( x ) = 1. Use TABLE to find the largest positive value of x for which the calculator computes f(x) accurately. Enter these values in your table.
x
1
.
Use this numerical evidence to specify lim 1+
x
x→∞
As in part a, beware of calculator "noise". If you move the cursor to the column of y values, the full y value of the current x value appears
in the bottom row. ( )
The value of e accurate to 25 decimal places is 2.71828 18284 59045 23536 02875.
c. Compare the graphical and numerical estimates for e of parts a and b. Which do you think is more accurate? Explain your answer. d. Other limits for f
Collect graphical and numerical evidence to find the limit of f( x ) = ( 1 + 1/x ) x as x approaches 0 from the right, as x approaches ­1 from the left and as x approaches ­.
x
x
x
1
1
1
,
lim 1+
. Use TABLE for numeric evidence. Use graph paper for graphical
That is, evaluate lim 1 +
and lim 1+
x
x
x
x →−∞
x →0
x →−1
evidence.
+
( )
−
( )
( )
Use TABLE to find the smallest negative value v for which the TI reports f( v ) = 1. Explain your work.
Use TABLE to find the largest negative value v for which the TI computes f( v ) accurately. Explain your work.
Find the largest value w near ­1 for which f( w ) = 131. Explain your work.
Part 2. A summation formula for Euler's number e Calculus 2 will provide the following formula for Euler's number: e = ( 1/n!, n, 0,  ). That is, e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + … = 1 + 1 + ½ + 1/6 + 1/24 + 1/120 + ...
where n! is n factorial. By definition, n! equals the product of the first n positive integers and 0! equals 1. To have the TI compute e, enter the sum as y2 in the Y= screen as y2 = ( 1/t!, t, 0, x ) On the TI 89,  requires the key strokes CATALOG, s and then the down arrow key to reach the line labeled sum and ! requires the key strokes 2nd MATH, Probability, 1. n
1
y2( x ) increases to e as x increases or lim ∑
e.
n →∞ t=0 t !
Use TABLE on your calculator to complete a table with two columns labeled x and ( 1/t!, t, 0, x ) for integers x = 5, 6, ... until all decimals on the TI are correct.
Compare the estimates for e in part 1 and part 2. Which is most accurate? Explain your answer.
Part 3. For large x, an exponential function dominates a polynomial function. An increasing exponential function explodes and dominates a polynomial function. To use the language of limits, the limit of the quotient of a polynomial function over an increasing exponential function is zero as x approaches infinity. You will illustrate this fact with the function of g( x ) = 131x2 / e^(0.0131x). The polynomial function is 131x2 and the exponential function is e^(0.0131x). The exponential function has a handicap of a small growth rate of 0.0131. But the quotient still has a limit of zero
as x approaches infinity. Enter the quotient g in the Y= screen. Use TABLE to verify that g increases for small positive x and but then eventually decreases. Record your results in a table with two columns labeled x and g( x ) = 131x2/e^(0.0131x).
Next use the results of table to find a window that illustrates that g increases to some maximum and then decreases slowly to zero. Draw the graph of g accurately on graph paper. Identify the maximum point and several other key points.
Verify numerically and graphically that lim
x→∞
(
2
131 x
0.0131 x
e
)
= 0.
Now your team will make an example to illustrate the dominance of an increasing exponential function over a polynomial function. Your polynomial must have degree at least three and must not be a power function. Write down your quotient on paper. Then provide numerical evidence with a table and graphical evidence to illustrate that the limit of your quotient is zero as x approaches 
Group Lab Report On Monday 14 Sep. submit a group report containing the results of parts 1 ­ 3. Use a word processor or a text editor to write a report for Lab 3. Discuss your observations and your results. Discuss the accuracy and reliability of the TI. Also reflect on what you learned from doing this lab. Feel free to send me email at [email protected], to see me in my office in CW 300 or see the tutors in the Math Center in CW 321 for assistance.