Tble of Contents INTRODUCTION 5 CROSS REFERENCE TABLE 13 1. Limits 1 1.1 The Forml Definition of Limit 1. The Squeeze Theorem 34 1.3 Are of Circle 43. Derivtives 53.1 Eploring Tngent Lines 54. Men Vlue Theorem 61.3 The Derivtive of Even Functions 70.4 The Derivtive of Odd Functions 78.5 Differentibility of Piecewise Function t Point 86.6 Derivtive of n Inverse Function 98 3. Applictions of Derivtives 109 3.1 Envelope of Prbol 110 3. Liner Approimtion 117 3.3 Newton s Method 130 3.4 Rectngle in Semicircle 143 3.5 Floting Log 150 3.6 Art Gllery 161 4. Integrls nd Their Applictions 169 4.1 Representtion of the Antiderivtive 170 4. The Fundmentl Theorem of Clculus 181 4.3 The Second Fundmentl Theorem of Clculus 190 4.4 Integrl of n Inverse Function 197 4.5 The Trpezoidl Method 08 4.6 Minimum Are 19 5. Differentil Equtions 7 5.1 Orthogonl Trjectory to Circle 8 5. Orthogonl Trjectory to Hyperbol 36 Clculus Eplortions with Geometry Epressions 3
6. Sequences nd Series 45 6.1 Infinite Stirs 46 6. The Snowmn Problem 58 6.3 Trigonometric Delight 67 6.4 Converging or Diverging? 80 7. Prmetric Equtions nd Polr Coordintes 91 7.1 Folium of Descrtes Using Prmetric Equtions 9 7. Folium of Descrtes in Polr Coordintes 30 Conclusions 311 Appendices 313 Appendi A: Geometry Epressions Keybord Shortcuts 314 Appendi B: More Geometry Epressions Books 316 4 Copyright Sltire Softwre, Inc. 010
Introduction Techers know the difficulties in motivting mny students to develop the hbits of mind nd criticl thinking skills necessry to thoroughly understnd the concepts of clculus. The purpose of this book is to use Geometry Epressions softwre in order to fcilitte nd enhnce the clculus syllbus by llowing students to ground clculus concepts in geometric wy. Mjor clculus concepts, such s derivtive nd integrl of function, hve cler geometric mening. This encourges students to visulize the concepts nd mke connections between its geometric nd lgebric representtions. For emple, function cn be represented geometriclly by its grph; the derivtive of the function is visully represented by the slope of the tngent line; the definite integrl of the function is n re under the grph of the function. These geometric representtions serve s bsis for conceptul introduction of the concepts of derivtive nd integrl. The forml definitions of the clculus concepts then led to clrifiction of these geometric ides. For emple Eplortion.1, Eploring Tngent Lines, introduces the conceptul ide of the derivtive. After the derivtive is formlly defined, it is possible to interpret the slope of the tngent line in terms of the derivtive. Similrly in Eplortion 4.5, The Trpezoid Method, n re of curviliner trpezoid is used to develop the concept of definite integrl. After n introduction of the Fundmentl Theorem of Clculus, the re cn be defined using definite integrl. Both of these emples re bsed on the concept of the differentil of function which hs cler geometric mening, s shown in Figure 1. Clculus Eplortions with Geometry Epressions 5
Problem Nme 1.1) The Forml Definition of Limit 1.) The Squeeze Theorem 1.3) The Are of Circle.1) Eploring Tngent Lines Pp. Pre-requisite knowledge Properties of bsolute vlue Concvity of monotonic functions 34 Bsic trigonometric rtios in unit circle Inverse trigonometric functions Are of circulr sector Limit of function t point 43 The re of regulr polygon inscribed in circle with given rdius A bounded incresing sequence converges sin( ) lim = 0 54 The slopes of tngent nd norml lines to the curve re opposite reciprocls. Point-slope eqution of line Derivtive formuls for bsic functions CROSS REFERENCE TABLE Key concept Level Clss Time ε-δ definition of limit sin lim = 1 0 Limits t infinity Horizontl symptote Eqution of tngent line Slope of the tngent line is equl to the derivtive of the function 3 60 90 min. AP Clculus AB nd BC* Topic *Limits of functions (including one-sided limits) 1 45 min. *Limits of functions (including one-sided limits) 1 45 min. Asymptotic nd unbounded behvior 1 45 min. Derivtive t point Clculus Eplortions with Geometry Epressions 13
Techer Notes 4.3 The Second Fundmentl Theorem of Clculus Eplortion 4.3: Consider the definite integrl of function y = f() with vrible upper limit. Wht is the rte of chnge of the integrl nd how does tht relte to the integrnd f? SUMMARY Mthemtics Objective: Discover the nd d Fundmentl Theorem of Clculus: f ( ) d = f ( ) d. Estblish the fct tht differentition nd integrtion re inverse opertions. Vocbulry: Rte of chnge Antiderivtive Integrl Are s definite integrl Pre-requisites: Derivtive s slope of the tngent line. Bsic rules of differentition Antiderivtive Definite integrl The Fundmentl Theorem of Clculus Problem Notes: Students eplore the reltionship between the slope of the tngent line to the grph of the ntiderivtive of f() written s definite integrl with vrible upper limit nd the function f(). They use geometric concepts of re nd tngent line to estblish the reltionship between them. This reltionship is known s the second fundmentl theorem of clculus. While this theorem is often confusing to students, 190 Copyright Sltire Softwre, Inc. 010
Techer Notes the geometricl representtion cn help them understnd the mening of this theorem: when you pply two opertions tht re the inverse of ech other to given function, you will get the originl function. Students first eplore the definite integrl of specific function with vrible upper limit, lso known s the re function. They plot the grph of this function nd nlyze the rte of chnge of this function using the slope of the tngent line. Students then generlize their findings for generic function. Technology skills: Drw: function, polygon, rc, line segment Constrin: point proportionl long the curve Construct: tngent line Clculte: re, slope STEPS-BY-STEP INSTRUCTIONS THE DEFINITE INTEGRAL AS A FUNCTION 1. Drw function y =.. Use Toggle grid nd es to show the es without grid. b. Choose Drw Function. Select Type Crtesin. In the Y = prompt type ^.. Plot the region bounded by the grphs of y =, y = 0, =, nd vrible upper boundry.. Plot points A nd B on the -is nd points C nd D on the curve. b. Select point A nd -is, nd choose Constrin Point proportionl. Type in the edit bo. Select point C nd the curve, nd choose Constrin Point proportionl. Type in the edit bo. c. Select point B nd -is, nd choose Constrin Point proportionl. Type in the edit bo. Select point D nd y-is, nd choose Constrin Point proportionl. Type in the edit bo. d. Choose Drw Segment nd plot segments AB, AC nd BD. Choose Drw Arc, nd plot n rc CD. Clculus Eplortions with Geometry Epressions 191
Techer Notes e. Select points A, B, C, nd D nd choose Drw Polygon. Q1. Wht is the re of this plne region? A1: t dt 3 3 A( ) = =.The re is function of, = const. 3 3. Verify your nswer with the help of softwre. Select the polygon interior nd choose Clculte Are. D 6 Y=X C 4-3 + 3 3-4 - A B 4 6 THE RATE OF CHANGE OF THE AREA FUNCTION Q. The re of the region is function of. How does the re chnge s chnges? Wht is the derivtive of this function? A: As increses, the re increses. d d 3 3 3 =. 19 Copyright Sltire Softwre, Inc. 010
Techer Notes 3. To check the nswer to question, plot the grph of the re of the region bounded by the grphs of y =, y = 0, =, nd vrible upper boundry. Find the slope of this grph t point.. Click on the epression for the re nd choose Edit Copy As String. b. Choose Drw Function. Choose Crtesin for Type nd pste the re function in the Y= prompt using Ctrl-V. Delete bs in the epression of the function. (Or, before copying the string, set the Output Properties Use Assumptions Yes from the right-click contet menu. This gets rid of the bsolute vlue symbols.) c. Select the grph of the re function nd choose Construct Tngent to curve. d. Select the point of tngency (point E) nd the curve nd choose Constrin Point proportionl. In the open edit bo type. e. Select the tngent line nd choose Clculte Slope. D 6 E Y=X C 4-3 + 3 3-4 - A B 4 6 Y= X3-3 3 - Clculus Eplortions with Geometry Epressions 193
Techer Notes Q3. How does the slope of the tngent line to the re function compre to the originl function tht bounds the region? A3: They re equl. Q4. Will your nswer hold true for other functions? Eplin your nswer. A4: Answers will vry. 4. Verify your decision by choosing nother function nd repeting the work bove.. Double-click on the epression Y = ^ nd type different function. b. Click on the epression for the re nd choose Edit Copy As String. c. Choose Drw Function. Choose Crtesin for Type nd pste the re function in the Y= prompt using Ctrl-V. Delete bs in the epression of the function. Y=cos(X) C - A -1 1-1 E cos() Y=-sin()+sin(X) D 1 B 3 4 5 6 7 -sin()+sin() Q5. Write generl epression for the re of the plne region between the grph of the function y = f() nd -is on the intervl [, ]. A5: A( ) f ( t) dt = Note: to void the confusion of using in two different wys, the vrible of integrtion is switched to t. Q6. Write generl epression for the derivtive of the re function. 194 Copyright Sltire Softwre, Inc. 010