MATH 160 Calculus for Physical Scientists I Name:Spring Semester, 2008Calculator Laboratory Section:Date due:Calculator make & model:The Fundamental Theorem of CalculusCalculator Lab: The Fundamental Theorem of CalculusMATH 160 Calculus for Physical Scientists I Name: ____________________________________Spring Semester, 2008Calculator Laboratory Section: ___________________________________Date due:__________________________________Calculator make & model:___________________The Fundamental Theorem of CalculusOverview In precalculus mathematics most functions are defined by equations involving the algebraic operations of addition, subtraction, multiplication, division, raising to powers and extracting roots. The trigonometric functions, of course, are an exception. They are defined geometrically in terms of the unit circle and the geometry of triangles. Definite integration provides another way to define functions. These functions can be visualized geometrically as “accumulated area functions”. The Fundamental Theorem of Calculus tells us that accumulated area functions defined from continuous functions are differentiable. This lab illustrates how accumulated area functions are specified by definite integrals and why these functions are differentiable.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, 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.P L E A S E K E E P A C O P Y O F Y O U R C O M P L E T E D L A B R E P O R T.You may need to refer to the work you did on this lab before it is graded and returned. Page total ________© 2006 Kenneth F. Klopfenstein, Ft. Collins, CO Page 0 of 5Calculator Lab: The Fundamental Theorem of CalculusInvestigation I Defining a Function by an IntegralTo specify a function we must give a procedure for obtaining an output (usually numerical and often denoted y)for each input (usually numerical and often denoted x). The definite integral gives us a new way to specify a function. Begin with a continuous function y = f (x) defined on some interval [a, b]. Choose a number c in the interval [a, b]. For each number x between a and b, define F(x) to be the definite integral of f over the intervalfrom c to x. In symbols,F(x) = xcdttf )(For example, suppose f(x ) = 24 x, so the interval on which f is defined is [a, b] = [ -2, 2 ]. The graph of y = f(x ) is the top half of a circle with radius r = 2 centered at ( 0, 0 ). Choose c = 0 and define the function F(x) for each x between –2 and 2 byF(x) = xdttf0)( = xdtt024I.1. (a) Explain why the variable of integration in the definition of the function F(x) is t rather than x. (b) There are some values of x for which F(x) is negative. For what values of x is F(x) negative? (c) Explain how F(x ) can have negative values when the graph of the integrand lies entirely above the x-axis.I.2. (a) The graph of y = 24 x is a semicircle. Generate this graph in a square window (so it looks like a semicircle). Then use this graph and the familiar formula for the area of a sector of circle to find F(–2), F(0), and F (2). Record both the exact values (in terms of ) of the function at these three points and these function values rounded to four decimal places in the table below. (b) Use the definite integral command on your calculator to find values of the function F for the x-values in the table below. Record the function values rounded to four decimal places. (The definite integral command is under the CALC menu on the TI-83® and TI-84®, and under MATH on the GRAPH screen of the TI-86® and TI-89®. On some models the lower limit of integration must be smaller than the upper limit.x F(x)-2-1.5-1-0.500.511.52Page total ________© 2006 Kenneth F. Klopfenstein, Ft. Collins, CO Page 1 of 5Calculator Lab: The Fundamental Theorem of CalculusI.3. Plot the points from the table you created in I.2. Then sketch the graph of F(x) = xdtt024 manually by drawing a curve through these points. Since the integrand is continuous, the function F(x ) is differentiable. So be sure to make the graph smooth, with no breaks, corners, or cusps. (The graph is not a straight line!)Calculator Note: You aren’t asked to do so, but it’s possible to graph the function F (x) = xdtt024 on your calculator using the integration command. (This is the command fnInt( under the MATH menu on the TI-83® and TI-84® and under the CALC menu on the TI-86® or ∫ under the MATH – Calculus menu on the TI-89®. However, evaluating F(x ) requires numerical evaluation of an integral, so the graph is generated very, very slowly. Try it Page total ________© 2006 Kenneth F. Klopfenstein, Ft. Collins, CO Page 2 of