MATH 160 Calculus for Physical Scientists I Name:Spring, 2008Calculator Laboratory Section:Date due:Calculator:The Differentiation Theorems GraphicallyMATH 160 Calculus for Physical Scientists I Name: _____________________________Spring, 2008Calculator Laboratory Section: _______________________Date due:______________________Calculator:_____________________The Differentiation Theorems GraphicallyC o n n e c t i n g d i f f e r e n t i a t i o n f o r m u l a s a n d t a n g e n t l i n e s .The investigations in this lab require a calculator that can produce traceable graphs and generate tables of values of elementary functions. 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 comprehensive 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 manufacturers’ web sites. The Texas Instrumentweb site will also point you to tutorials for various models of TI calculators.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 accurate and 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, you may ask your instructor to e-mail you a copy of this lab as an attached MS Word document. Submit your final lab report as a printed document. PLEASE KEEP A C OPY OF YOUR COMPLETED LAB REPOR T.You may need to refer to the work you did on this lab before it is graded and returned. © 2007 Kenneth F. Klopfenstein, Fort Collins, COOverviewFrom previous work, we know the derivative of a function y = f (x) at a point x = c can be interpreted graphically as the slope of the line tangent to the graph of y = f(x) at the point ( c, f(c) ). We also have sound intuitive understanding of the tangent line – it is the line the matches the position and the direction/alignment of the graph of y = f( x) at the point of tangency. So, knowing the tangent line is equivalent to knowing the derivative. In fact, in more advanced settings the derivative is defined in terms of the tangent line or tangent plane. This connection between derivatives and tangent lines can be used to discover the formulas for the derivatives of sums, products, reciprocals, and compositions of differentiable functions. Here’s the idea.Suppose the two functions y = f ( x ) and y = g(x ) are both differentiable at x = c. From their derivatives f (c) and g(c) we can write equations for the lines tangent to the graphs of the two functions y = f ( x ) and y = g(x) at x = c. We can add the equations for these two tangent lines to get an equation for the line tangent to the sum function s(x ) = f (x) + g(x ) at x = c. Then the derivative of the sum of y = f (x) and y = g(x ) at the point x = c is just the slope of this tangent line. An equation for the line tangent to the product function p(x) = f(x) g ( x ) at the point x = c can be found by multiplying the equations for these two tangent lines and using a little algebraic ingenuity. The slope of this tangent line is, of course, the derivative of the product. The differentiation formulas for reciprocals and compositions can also be discovered in a similar way.In this lab, you will carry out the idea just described to find the formulas for the derivatives of sums and products of differentiable function. The idea works with any differentiable functions, we will use specific functions and the point c = 0 to make it easier to see how the idea works.© 2007 Kenneth F. Klopfenstein, Fort Collins, COPreliminariesP.1 Graph the function f (x ) = sin(x) + 0.5 x – 1 in the decimal window. Be sure your calculator is in RADIAN mode.(On the TI-83® and TI-84® calculators , enter this function as Y1, then press [Zoom] and choose 4:ZDecimal. )(a) Describe what you see in the graph that tells you the function f(x) is differentiable at x = 0.Use the Draw Tangent command on your calculator to add the tangent line at (0 , f(0)) to the graph and to find an equation for the tangent line at (0, f ( 0 ) ). (On the TI-83® and TI-84® calculators, press [2nd] [DRAW] [5]. Position the cursor at the point on the graph of y = f ( x) where x = 0. Then press [ENTER]. The tangent line willbe added to the graph and an equation for the tangent line displayed at the bottom of the screen.)(b) The tangent line is the graph of a function. Denote this function by t1(x) (to indicate “tangent to the first function”). Enter the function t1(x) as Y2. (Round the coefficient of x to one decimal place.)An expression/equation for this function is t1(x) = _________________________ .(c) Sketch the graphs of the functions y = f ( x ) and y = t1(x) on the coordinate grid given. (Draw a very accurate graph. Use a ruler to mark units on the axes. Use a sharp pencil and draw the graph as a thin, but clearly visible, line.)_________________________________________________P.2 Turn off Y1 and Y2 by moving the cursor over the equal sign and pressing [ENTER] so the equal sign is not highlighted. Enter the function g(x) = 2 – 0.5 x – x2 as Y3. Press [GRAPH].(a) Describe what you see in the graph that tells you the function g(x) is differentiable at x = 0.Use the Draw Tangent command on your calculator to add the tangent line at (0 , g( 0 ) ) to the graph and to find an equation for the tangent line at (0, g(0) ). (On the TI-83® and TI-84® calculators, press [2nd]