Download M CC 160 Calculus for Physical Scientists I

MATH 160 Calculus for Physical Scientists I
Spring, 2008
Calculator Laboratory
Name: ___________________________
Section: ______________________
Date due: ________________________
Calculator: ___________________
Solving Equations by Newton’s Method
What can you do when you can’t solve an equation algebraically?
Overview In previous work you often faced the problem of solving equations f( x) = 0 where the
function f( x) is a polynomial, a trigonometric function, or, at worst, a root function. To solve optimization
problems you often need to solve equations like this where the function f( x) is the derivative of some
other function. The derivative of a function is usually more complicated than the original function. As a
result, to solve a realistic optimization problem you may well have to solve an equation f( x ) = 0 where the
function f( x) is quite complicated. (For example, see problem 26 on page 306 of the textbook.) In these
cases, tried-and-true algebraic techniques let you down. You need a numerical method. Newton’s Method
is one of the most powerful and commonly used numerical method for solving equations f( x) = 0.
One of the most useful skills you can learn in university science and mathematics courses is how to read
technical writing. In this lab we want you to practice reading mathematical writing by reading about
Newton’s Method. Reading technical writing is much different than reading news writing or popular
fiction. It requires diligence and attention to detail. Work at reading the discussion of Newton’s Method in
the textbook. Read with pencil and paper at hand. Draw your own pictures. Do calculations. If you need
some coaching to read and understand this material, talk with your instructor.
The investigations in this lab require a calculator that can produce traceable graphs, DRAW lines tangent
to a graph, and ZOOM IN on a graph. While many makes and models of calculators have these capabilities,
the authors used Texas Instrument calculators as they wrote this lab. The lab does not include instructions
for using a calculator. Use the manual for your calculator to learn how to perform the tasks in this lab
efficiently and accurately. Manuals for Texas Instrument calculators can be read from the Texas
Instrument web site. Go to http://education.ti.com/us/global/guides.html. Search for manuals for other
makes and models of calculators at the manufacturer’s web site.
The calculator skills you develop doing this lab will serve you well throughout this and other courses. If
you encounter difficulties, take your calculator and manual to your instructor and discuss the problem with
him/her. Classmates may be able to help out, too
The following factors will be considered in scoring your lab report:
• Completeness. Each investigation must be completed entirely, recorded fully, and explained or
interpreted thoroughly.
• Mathematical and computational accuracy.
• Clarity and readability. Tables and graphs must be presented in a clear, readable format. Explanations
must be written in complete sentences with correct spelling, capitalization and punctuation.
Handwriting must be legible.
Space for writing your report is provided within the lab. However, if you wish to word process your lab
report, your instructor will e-mail you a copy of this lab as an attached MS Word document. Space for
writing your report is provided within the lab. However, if you wish to word process your lab report, your
instructor will e-mail you a copy of this lab as an attached MS Word document. Submit your final lab
report as a printed document.
PLE A S E K E EP A C OPY O F Y OU R C OMP LE T E D L A B R EP OR T.
You may need to refer to the work you did on this lab before it is graded and returned.
© 2006 Kenneth F. Klopfenstein, Fort Collins, CO
Calculator Lab: Solving Equations by Newton’s Method
Investigation I. How Does Newton’s Method Work?
Study Section 4.7 of the textbook up to “Convergence of Newton’s Method” (pages 299 – 302). After you have
studied this section, you should be able to explain in terms of the graph of y = f( x) how Newton’s Method for
solving equations f( x) = 0 works and how equation (1) in the box on page 300 comes about. In this investigation
we will examine the details of Newton’s Method by solving x 4 – 2 x 3 – x 2 – 2x + 2 = 0 .
I.1 First we will use a graph to find the first two Newton approximations (x1 and x2 ) to the smaller of the two
solutions to x 4 – 2 x 3 – x 2 – 2x + 2 = 0 without using the formula for Newton approximations on page 300.
(a) On the graph below, draw the tangent line used to find x1 from the starting approximation xo = 0. Label
this tangent line “line 1”. Label the point x1 on the x-axis. x1 is approximately: _________________
(b) Draw the tangent line used to find x2 from x1. Label this tangent line “line 2”.
Label the point x2 on the x-axis.
x2 is approximately: _________________.
© 2006 Kenneth F. Klopfenstein, Ft. Collins, CO
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Calculator Lab: Solving Equations by Newton’s Method
I.2 Next, we will calculate the exact values of three Newton approximations (x1, x2, and x3 ) to the smaller of the
two solutions to x 4 – 2 x 3 – x 2 – 2x + 2 = 0 algebraically (without using the formula on page 300).
(a) Write an exact equation for tangent line used to find x1 from the starting approximation xo = 0 (the
tangent line you sketched I.1(a)). The x-intercept of this tangent line is x1. Find the exact value of x1
algebraically. Show your calculations.
x1 = ________________
(b) Write an exact equation for tangent line used to find x2 from x1. The x-intercept of this tangent line is x2.
Find the exact value of x2 algebraically. Show your calculations.
x2 = ________________
(c) Write an exact equation for tangent line used to find x3 from x2. Find the exact value of x3 algebraically.
Show your calculations.
x3 = ________________
(d) Create the graph of f( x) = x 4 – 2 x 3 – x 2 – 2x + 2 on your calculator. Use the window suggested by the
graph in I.1. (The graph on the calculator screen will be compressed vertically.) Check your algebraic
work above by graphing the three tangent lines you found and verifying that these lines appear to be the
correct tangent lines and intersect the x-axis at the points you calculated as x1, x2, and x3.
© 2006 Kenneth F. Klopfenstein, Ft. Collins, CO
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Calculator Lab: Solving Equations by Newton’s Method
Investigation II. Implementing Newton’s Method on a Calculator
Texas Instrument calculators allow you to enter several commands on the same line of the screen by simply
inserting a colon between commands. We’ll use this feature to program the calculator to generate successive
Newton approximations to the solutions to x 4 – 2 x 3 – x 2 – 2x + 2 = 0 as we press ENTER repeatedly.
• Enter f( x) = x 4 – 2 x 3 – x 2 – 2x + 2 as Y1 and f′( x) = 4x 3 – 6 x 2 – 2x – 2 as Y2 on the Y= screen.
• N will count the number of successive approximations we have calculated. Store 0 as N.
• X will be the most recently calculated Newton approximation. Store the starting approximation as X. Above we
used 0 as our starting approximation for finding the smaller of the two solutions to x 4 – 2x 3 – x 2 – 2x + 2 = 0,
so this time store 0 as X.
• Finally, enter the “mini-program” N + 1 → N : X – Y 1 ( X ) / Y 2 ( X ) → X: { N , X , Y 1 ( X ) } . On the TI-83® and TI-84®,
enter Y 1 and Y 2 from the V A R S menu. Be sure to enter the colons.
II.1 Explain the purpose of each command in the mini-program that implements Newton’s method. That is, explain
what each command does and why it is included in the mini-program. Consult your calculator manual as
necessary.
(a) N + 1 → N
(b) X – Y 1 ( X ) / Y 2 ( X ) → X
(c) { N , X, Y 1 ( X ) }
II.2 After you have set up everything and entered the commands described above, press ENTER repeatedly to
generate successive Newton approximations to the smaller of the two solutions to x 4 – 2 x 3 – x 2 – 2x + 2 = 0.
In the table below, record the first five Newton approximations to the smaller of the two solutions calculated
from the starting approximation xo = 0. Also record the value of f( x) at each of these approximations.
Record table entries either exactly as a fraction or rounded to no less than 5 decimal places.
n
0
xn
0
Y1(xn)
2
1
2
© 2006 Kenneth F. Klopfenstein, Ft. Collins, CO
3
4
5
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Calculator Lab: Solving Equations by Newton’s Method
II.3 (a) Circle the starting approximation that could be used to find the larger of the two solutions
to x 4 – 2 x 3 –x 2 – 2x + 2 = 0 by Newton’s Method:
xo = 1.5
xo = 1.75
xo = 2.25
Explain what you see in the graph of f( x) = x 4 – 2 x 3 – x 2 – 2x + 2 that tells you approximations
calculated from this starting approximation should converge to the larger zero.
(b) Make a table showing the first five Newton approximations obtained by using the starting approximation
you chose in (a). Record the value of f( x) at each of these approximations. Record table entries to at
least 5 decimal places.
n
0
1
2
3
4
5
xn
Y1(xn)
© 2006 Kenneth F. Klopfenstein, Ft. Collins, CO
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Calculator Lab: Solving Equations by Newton’s Method
Investigation III. One Way Newton’s Method Can Go Wrong
Study pages 302 – 303 to “Fractal Basins and Newton’s Method”. You saw in Investigation I that it took very few
iterations to get excellent approximations to the solutions to the equation x 4 – 2 x 3 – x 2 – 2x + 2 = 0. Newton’s
Method works well when the function involved is “nice”. There are situations where Newton’s Method works well
in principle, but not in practice. In this investigation we examine one such situation.
III.1 The only solution to (x – 1)40 = 0 is x = 1. Use the “mini-program” given above with starting approximation
xo = 1.9 to calculate successive Newton approximations for the solution to this equation. How many Newton
approximations must be calculated to get within 0.1 of the actual solution to (x – 1)40 = 0?
______________
To get within .05?
______________
III.2 Sketch the graph of f ( x) = (x – 1)40 and use it to explain why so many iterations are required. (Read the
suggested pages of the text before you answer.)
1–
___________________________________________________________________________________________________________
0
1
2
III.3 Computers can do millions of calculations in a second, so the large number of calculations needed to get close
to the solution to (x – 1)40 = 0 is not a problem. Nevertheless, the phenomenon you saw in III.2 might cause
difficulties when Newton’s Method is used to solve an equation. Explain what these difficulties might be.
© 2006 Kenneth F. Klopfenstein, Ft. Collins, CO
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Calculator Lab: Solving Equations by Newton’s Method
Investigation IV. Another Way Newton’s Method Can Lead You Astray
IV.1 Find all three roots (zeros) of f( x) = 4 x 4 – 4 x 2 algebraically (by factoring). Show details of the algebra.
IV.2 The Newton approximations to a root of f( x) = 4 x 4 – 4 x 2 calculated using starting value xo =
21
do not
7
21
21
and 
.
7
7
Verify by calculating ex a ct l y b y ha nd that the first Newton approximation x1 calculated using starting
converge to a root of f( x) = 4 x 4 – 4 x 2, but instead alternate between
21
21
is x1 = 
. Use exact square roots (not decimal approximations for the square roots)
7
7
in your calculation. Show details of the calculations clearly.
value xo =
A similar calculation that you are not asked to do will show x2 =
21
.
7
IV.3 Sketch an accurate graph of f ( x) = 4 x4 – 4 x2 and use it to explain why the phenomena in IV.2 occurs.
1–
______________________________________________________________________________________________________
0
1
© 2006 Kenneth F. Klopfenstein, Ft. Collins, CO
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Calculator Lab: Solving Equations by Newton’s Method
IV.4 Answer the following question looking at the graph but not doing any calculator calculations.
Which one of the three zeros of f( x) = 4x4 – 4x2 do you expect to find by using Newton’s Method
(a) with starting approximation xo = 0.6544?
________________
(b) with starting approximation xo = 0.6547?
(c) with starting approximation xo = 0.6550?
________________
________________
Still without doing any calculations, explain what you see in the graph that leads you to these conclusion.
(Don’t change your answers after you’ve done IV.4!)
IV.5 (a) Use Newton’s Method with starting value xo = 0.6544 to calculate a root of f( x) = 4 x 4 – 4 x 2.
Record the indicated Newton approximations and corresponding values of f( x) in the table below.
Note that the required entries are NOT successive. Record entries to at least 5 decimal places.
n
0
xn
0.6544
1
2
3
10
25
Y1(xn)
(b) Use Newton’s Method with starting value xo = 0.6547 to calculate a root of f( x) = 4 x 4 – 4 x 2.
Record the indicated Newton approximations and corresponding values of f( x) in the table below.
Note that the required entries are NOT successive. Record entries to at least 5 decimal places.
n
0
xn
0.6547
1
2
3
10
25
Y1(xn)
(c) Use Newton’s Method with starting value xo = 0.6550 to calculate a root of f( x) = 4x4 – 4x2.
Record the indicated Newton approximations and corresponding values of f( x) in the table below.
Note that the required entries are NOT successive. Record entries to at least 5 decimal places.
n
0
xn
0.6550
1
2
3
10
25
Y1(xn)
IV.6 (a) Even though the starting values used in IV.5 are very close together, they produce sequences of Newton
approximations that converge to different zeros. Do you think there are other starting values, much
different from these, where the same phenomenon occurs? Explain why or why not.
(b) Explain what difficulty the phenomenon you observed in IV.4 might cause when Newton’s Method is
used to solve an equation.
© 2006 Kenneth F. Klopfenstein, Ft. Collins, CO
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