MATH 160 Calculus for Physical Scientists I Name:Spring, 2008Calculator Laboratory Section:Date due:Calculator:Solving Equations by Newton’s MethodMATH 160 Calculus for Physical Scientists I Name: _____________________________Spring, 2008Calculator Laboratory Section: _______________________Date due: __________________________Calculator:_____________________Solving Equations by Newton’s MethodW h a t c a n y o u d o w h e n y o u c a n ’t s o l v e a n e q u a t i o n a l g e b r a i c a l l y ?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 optimizationproblems 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 thefunction 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 includeinstructions for using a calculator. Use the manual for your calculator to learn how to perform the tasks inthis 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 withhim/her. Classmates may be able to help out, tooThe 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 ASE K EEP A COPY OF YOU R CO MPL ETE D L AB REPORT.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, COCalculator Lab: Solving Equations by Newton’s MethodInvestigation 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 x4 – 2x3 – x2 – 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 x4 – 2x3 – x2 – 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: _________________.Page total © 2006 Kenneth F. Klopfenstein, Ft. Collins, CO Page 1 of 7Calculator Lab: Solving Equations by Newton’s MethodI.2 Next, we will calculate the exact values of three Newton approximations (x1, x2, and x3 ) to the smaller of the two solutions to x4 – 2x3 – x2 – 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) = x4 – 2x3 – x2 – 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