Materials: Whiteboard, TI-Nspire classroom set, quadratic tangents program, and a computer projector.

Transcription

1 Adam Clinc Lesson: Deriving te Derivative Grade Level: 12 t grade, Calculus I class Materials: Witeboard, TI-Nspire classroom set, quadratic tangents program, and a computer projector. Goals/Objectives: Formulate te general definition of te derivative function, use te definition to find te derivative of functions, and ceck our work using te quadratic tangents program on a TI-Nspire. Standards 1) Algebra a. Equations b. Aritmetic wit Polynomials and Rational Functions c. Creating Equations d. Reasoning wit Equations 2) Functions a. Interpreting Functions b. Building Functions c. Linear and Quadratic Functions Assessment: Use te derivative definition to find te derivative of tree functions and confirm your work using a TI-Nspire. Lesson Plan To begin today s discussion I would like to review wat we learned in te last section on limits wit a few examples. Find te limits. 1) lim2s 2 + 3s 1 = 13 s 2 2) lim 3) lim x 3 (3 + ) 2 9 x 2 x 6 x 3 = 6 =5 Now tat we ave refresed our minds on tat, let s begin our new topic.

2 I would like to begin by reviewing a topic. Say we are given two points: (2, -3) and (5,6). How do we find te slope of te line tat connects tose points? We use te fact tat slope = rise/run = 6-(-3)/5-2 = 9/3 = 3. We would ten say te line tat connects tose points two points as a slope of 3. Now wat if we were only given a function and te x coordinates of te two points, were tose two points satisfy te function. For instance, wat if we knew two points satisfied te function y = x and te x coordinates for tose two points were -1 and 2. Wat would te slope of te line be tat connects tose two points? We must first find te y coordinates of tose points. How do we do tat? We use te function and find tat te two points are terefore (-1, 1) and (2, 10). We ten go troug te same steps as we did before. Now we will use only tese tools and wat we ve recently learned about limits to find wat is known as te derivative, wic is noting more tan te slope of te tangent line. We begin wit an arbitrary curve.

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4 We ave derived our expression for finding te derivative function to any given function. Let s try it out wit a few examples. Starting wit someting we already known te answer to, let s consider te line y = 3x. Do we know te slope of tis line for all x values? Yes, it sould be 3. Let s ceck. 1) We know f(x) = 3x. Ten f(x+) = 3(x+). Plugging tis in we get f (x + ) f (x) f "(x) 3(x + ) 3x 3 3 = 3 Our derivative function is f (x) = 3, so it is just a flat line. Wat would appen if y = constant, say 2? Would f(x+)=f(x)? So wat is te slope of a flat line? Let s try anoter. 2) Let s find te derivative function of f(x) = x 2 -x. Wat is f(x+)? f (x + ) f (x) f "(x) [(x + ) 2 (x + )] [x 2 x] [(x 2 + 2x + 2 ) x )] x 2 + x 2x + 2 lim (2x + 1) (2x + 1) = 2x 1 If I plug te value x=2 into 2x-1 I get 3. Wat does tat 3 mean? Now tat we ave seen te general manner to calculate te derivative function for any given function let s get a better feel for wat is going on using our TI-Nspire.

5 Say we are given f(x) = x 2 we want to find f (x). We use our definition and obtain: f (x + ) f (x) f "(x) [(x + ) 2 ] [x 2 ] [(x 2 + 2x + 2 )] x 2 2x + 2 (2x + ) lim(2x + ) = 2x Now, turn on your TI-nSpire and go to My Documents. Select te file called quadratictangents. Page 1.1 sould look like te picture below. You will see tat te grap of y=x 2 is graped as well as te tangent line, T. Also note tat te slope of te tangent line is provided. In tis case te slope is 4.6 as m=4.6. Grab te point X and move it left and rigt. Describe in te space below ow te slope of te tangent line canges as you move X. Now go to page 1.2. It sould look like te picture below. We see again te function y=x 2 and te tangent line T are graped. Tere is anoter point of interest tis time and tat is point P. Notice tat P s x value is te x coordinate were it is located so 2.6 in tis case. However, it s y value is not it s y coordinate. It is instead te slope of te function y=x 2 at tat specific x we are at so 5.2 in tis case. I want you to grab point X and slide it to te rigt. You will notice tat as you

6 move X P moves because P depends on X. You will also see tat P traces out its pat. Terefore for eac x value P marks wat te value of te slope at tat X. Once you ave slid X over to at least -3 you will see a defined pat formed by P. It sould look like te picture below. Wat is te meaning of tis pat? It is indeed te derivative function of x 2. Let s ceck te fit by graping 2x. Go to te double arrow in te lower left corner and click on it. F2(x) sould appear. Now enter 2x and press enter. You will see te line appear. It sould look like te figure below.

7 How well does it fit our points? For omework tonigt I want you to repeat te same steps we did ere for f(x)=x 2 were we found te derivative function using our definition and ten used te TInSpire to ceck our work, for tree different functions. Te work you need to sow is te steps in te derivative definition, your derivative function and a sketc of your final grap tat verifies you are correct. Te tree functions I want you to do are: 1) f(x) = 3x + 2 2) f(x) = x ) f(x) = x 2 + x