1.1 Lines AP Calculus

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1 . Lines AP Clculus. LINES Notecrds from Section.: Rules for Rounding Round or Truncte ll finl nswers to 3 deciml plces. Do NOT round before ou rech our finl nswer. Much of Clculus focuses on the concept of locl linerit, mening tht even if function curves, if ou were to pick point nd st ver close (locl) to tht point, the function behves ver much like tht of line. Emple : Grph the functions = sin nd = on our clcultor. Obviousl these re not the sme function. However, if ou were to st close to the point (0, 0), these two functions re ver close. To see this, use the feture of our clcultor, nd zoom in on (0, 0). Tr zooming in more thn once. We cn s tht s long s we st close to (0, 0), the functions = sin nd = re lmost the sme thing. Now, the concept of close is more complicted thn it might sound, but more on tht in chpter. For now, we focus on lines. As stted in the sllbus, clculus hs to do with chnge. For nottionl purposes, we use the cpitl Greek letter delt,. Slope The slope of non-verticl line is given b A verticl line hs, nd horizontl line hs. Prllel Lines hve slopes tht re. Perpendiculr Lines hve slopes tht re. IMPORTANT : You will be best served in clculus if ou think of slope s. The slope between two points will be referred to s. Equtions of Line The first eqution of line ou used in lgebr ws probbl the slope intercept form: The slope is, nd the -intercept is. In clculus, it is ctull esier to write the eqution of line in point slope form: The point is, nd the slope is. : To write n eqution of line, ll ou need is nd the. Another formt used to write the eqution of line is clled stndrd (generl) form: All vribles re on the sme side (usull in lphbeticl order). Emple : Which of the equtions bove hs " written s function of "? Emple 3: The point-slope form is written s if ou wnt " written s function of " -

2 . Lines AP Clculus Emple 4: Find the equtions of the lines pssing through (, 4) nd hving the following chrcteristics: ) Slope of 7 6 b) Prllel to the line 5 3 = 3 c) Pssing through the origin d) Prllel to the is. Emple 5: Find the equtions of the lines pssing through (, 3) nd hving the following chrcteristics: ) Slope of 3 b) Perpendiculr to the line + = 0 c) Pssing through the point (, 4) d) Prllel to the is. Regression Anlsis is process of finding curve to fit set of dt. The bsic process involves plotting the points nd finding function tht best fits those points. The curve ou find is clled the regression curve. For the purposes of this section, our curve is liner, but it could be prbol or other power function, logrithmic function, trigonometric function, or n eponentil function. Emple 6: The medin price of eisting single-fmil homes hs incresed consistentl during the pst few ers. However, the dt in the tble below show tht there hve been differences in vrious prts of the countr. Yer South ($) 45,900 48,00 55,400 63,400 68,00 West ($) 73,700 96,400 3,600 38,500 60,900 ) Find the liner regression eqution for home cost in the South. b) Wht does the slope of the regression line represent? c) Find the liner regression eqution for home cost in the West. d) Where is the medin price incresing more rpidl, in the South or the West? Eplin. e) Using our regression eqution in prt c, predict the cost of home in the West in the er 0. -

3 . Functions nd Grphs AP Clculus. FUNCTIONS AND GRAPHS Notecrds from Section.: Odd/Even Functions Functions In the lst section we discussed lines nd when we needed to write " s function of ". But wht is function? In Algebr, we defined function s rule tht ssigned one nd onl one ( unique) output for ever input. We clled the input the domin nd the output the rnge. Definition: A function from set D to set R is rule tht ssigns unique element in R to ech element in D. The verticl line test is the grphicl interprettion of this definition. Emple : There re 3 domin restrictions ou MUST continue to be wre of throughout this course. Wht re the? Intervls In this course we would like not onl to know wht the domin nd rnge re, but how to describe them with the correct nottion. The domin nd rnge of function could be ll rel numbers, or we m need to limit the domin nd/or the rnge using intervls tht re either open or closed. Open nd closed intervls hve endpoints. If the endpoint is included, then we s the intervl is closed t tht point, nd if the endpoint is not included, then we s the intervl is open t tht point. We use prenthesis to indicte open nd brcket to indicte closed. Emple : Use intervl nottion AND inequlit nottion to describe ech intervl on the es below. b b b b : While it is mn times possible to just look for the restrictions to the domin, the rnge of function is esier to determine if ou hve grph or ou know wht the grph looks like. You should know wht the following bsic functions look like without hving to use our clcultor. Cn ou drw n ccurte sketch t lest 3 points on ech? 3 b, 0b b, b log b, b > sin cos tn Do ou know the domin nd rnge of ech of these functions? - 3

4 . Functions nd Grphs AP Clculus Even nd Odd Functions Recognizing the behvior of functions is not limited to their domin nd rnge. Mn functions hve the smmetric propert of being odd or even. You need to be ble to recognize the grph of function s odd or even, AND ou need to understnd how to show/verif/prove tht function is even or odd lgebricll. Grphicl Recognition of Even nd Odd Functions An EVEN function is smmetricl bout the is. Emple: = cos An ODD function is smmetricl bout the origin. Emple: = sin Algebric Properties of Even nd Odd Functions An EVEN function hs the propert tht f f. Tht is, if ou plug in " into the function nd simplif, ou will obtin the originl function. An ODD function hs the propert tht f f. Tht is, if ou plug in " into the function nd simplif, ou will obtin the opposite of the originl function. Emple 3: Prove whether the following functions re even, odd, or neither. 3 ) g b) h cos Piecewise Functions Some functions re broken into pieces nd behve differentl depending on the restricted domin of ech piece. Such functions re clled piecewise functions. An emple of function tht cn be written s piecewise function is the bsolute vlue function f. Be sure to use correct domin restrictions. Emple 4: Sketch f, nd write n eqution for the two "pieces" using domin pproprite to ech piece. Emple 5: Write piecewise function for the grph t the right. - 4

5 . Functions nd Grphs AP Clculus Composite Functions When the rnge of one function is used s the domin of second function we cll the entire function composite function. Becuse of the rnge of the first function is used s the domin of the second, ou must not ssume the finl function hs the sme domin nd rnge s it would hve hd if written independentl. We use the nottion f g f g to describe composite functions. This is red s "f composed with g" or "f of g of ". Emple 6: If f nd g, find g f. Wht is the domin nd rnge of g f? Emple 7: If 3 f, nd g, find f g. Bsed on our nswer, how might f nd g be relted? 3-5

6 .3 Eponentil Functions AP Clculus.3 EXPONENTIAL FUNCTIONS So fr we've delt with liner functions, piecewise functions, nd composite functions. Net up, eponentil functions. Definition: Eponentil Function If b > 0 nd b, then f b is n eponentil function with bse b. Emple : Sketch the following grphs s ccurtel s possible on the grphs below: ) 3 b) 3 c) d) 3 3 Emple : Which of these grphs show growth? dec? Emple 3: How does the ffect the grph? Emple 4: Wht is the domin nd rnge of ll 4 grphs? - 6

7 .3 Eponentil Functions AP Clculus Eponentil Growth/Dec Model In the eponentil model b, b is the rte of growth if b >, nd b is the rte of dec if 0 < b <. In either cse, the initil vlue is. Emple 5: Suppose ou invest $,000 in n ccount tht erns ou 5% interest compounded monthl for 0 ers. ) Wht is the initil mount? b) Wht is the growth rte? c) How mn times does our mone grow in 0 ers? (How mn times is interest dded to our ccount?) d) How much mone will ou hve in 0 ers? The Number e Mn eponentil functions in the rel world (ones tht grow/dec on continuous bsis) re modeled using the bse of e. Just like 3.4, we s e.78. We cn lso define e using the function s follows: As, e Tr convincing ourself tht this function pproches e using the function of our clcultor. Emple 6: Suppose the interest in emple 5 ws compounded continuousl. How much would ou hve in 0 ers? Emple 7: Using our grphing clcultor, let Y = nd Y =. Grph both equtions in the sme window. ) Solve the eqution mn re there? using our grphing clcultor. Where re the solutions to this eqution nd how b) Cler the two grphs from the screen nd use the eqution 0. Solve for b grphing the left side of this eqution. Where re the solutions to this eqution nd how mn re there? c) Wht did ou lern from the lst two questions? - 7

8 Prent Functions nd Trnsformtions AP Clculus PARENT FUNCTIONS AND TRANSFORMATIONS Notecrds from Prent Functions nd Trnsformtions: Trnsformtions (Prent Functions) Prent Functions One of the things we do in clculus is stud the behvior of functions. Some of the most bsic functions ou should be ble to recognize nd grph without the use of clcultor. You should be ble to sketch n ccurte grph (3 to 5 EXACT points) of the following prent functions: Check the notecrd checklist if ou ren t sure. 3 b, 0b b, b log sin cos tn Trnsformtions Not onl should ou be ble to grph the prent functions bove, but ou should be ble to grph the trnsformtions of these grphs. Without knowing ect points on the prent function, it will be difficult to trnsform the prent function! Emple : Suppose ou re given the function function is f bc d f. Wht effect do, b, c, nd d hve on originl function if our new Another w to look t this is with the following chrt: Inside Outside f c f c f d f d / f b f f f / - 8

9 Prent Functions nd Trnsformtions AP Clculus Emple : Let f be the grph given in the picture below. Grph the following trnsformtions of f. ) f b) f c) f d) f 3 e) f f) f g) f h) f 6 i) f j) f k) ( ) f

10 Introduction to Conics AP Clculus INTRODUCTION TO CONICS While there re mn relted ides nd topics tht cn be used with conics, we re onl ttempting to scrtch the surfce. We re onl interested in our bilit to identif/clssif conics b their eqution nd their grph s well s write n eqution of the following conic sections: Circles, Ellipses, nd Hperbols ou should dd these to our prent functions list. First, wht is conic section? A conic section is the cross-sectionl shpe formed b slicing double cone ( cone on top of nother cone). Agin, we re going to del with 3 of these circles, ellipses, nd hperbols. Got to the following website for n nimted view of these conic sections: Generl Form of Conic Section All conics cn be written in the form A B C D E F 0. The conics we will be focusing on will hve no term (unless tht is the onl term). If E 0, the conic will be rotted. Circles Consider circle centered t the origin, with rdius r. The Pthgoren Theorem, llows us to generte n eqution of the circle to be (, ) + = r. If we divide both sides of the eqution b r, we get r r r To grph the circle ourself, find the center, nd plot point tht is r spces up, down, left nd right of the center. Emple : Grph ech of the following equtions: ) 36 b) 3 5 Ellipses An ellipse is simpl circle tht hs been stretched more in one direction thn the other. Tht stretch cn be seen in the eqution of circle with the r, where both the horizontl stretch nd verticl stretch re r spces. If the horizontl nd verticl stretch re different, the vlues of ech denomintor will be different. Thus the generl form of n ellipse is, b where is the horizontl stretch from the center, nd b is the verticl stretch from the center. b To grph the ellipse ourself, find the center, nd plot point tht is spces left nd right of the center, nd point tht is b spces up nd down from the center. - 0

11 Introduction to Conics AP Clculus The mthemticl definition of n ellipse is the set of ll points whose distnces from two fied points (clled foci) hve constnt sum. Light (or sound) tht origintes t one focus point inside of the ellipse will reflect off the ellipse to the other focus point. Emple : Grph ech of the following equtions: ) b) Hperbols The mthemticl definition of hperbol is the set of ll point whose distnces from two fied points (clled foci) hve constnt difference. Thus, the eqution for hperbol is ver similr to n ellipse, ecept ou re subtrcting insted of dding. Since the order in which ou subtrct mtters, the hperbol opens in the direction of whichever is comes first. opens left nd right OR b opens up nd down b b b To grph hperbol ourself, find the center, crete rectngulr bo. Whtever the squre root of the denomintor is below, tht is the distnce ou move left nd right of the center. Similrl, the squre root of the denomintor below dicttes how fr ou move up nd down from the center. A hperbol hs slnted smptotes tht go through the corners of our rectngulr bo. If comes first (mening ou re subtrcting ), our hperbol will open left nd right, but if comes first then our hperbol will open up nd down. The onl points ou need to show will be the vertices which re ctull on the rectngulr bo. -

12 Introduction to Conics AP Clculus For pplictions of these conic sections or more emples, see Appendi Section A5 in our tetbook strting on pge 578. Emple 3: Grph ech of the following equtions: ) b)

13 .4 Prmetric Equtions AP Clculus.4 PARAMETRIC EQUATIONS Notecrds from Section.4: Trigonometric Identities Up to this point ll the functions we hve been looking t hve used single eqution with two vribles, nd. In this section we use third vrible to represent the curve. This third vrible is clled prmeter. Emple : Using our grphing clcultor, set our window to Xmin 5, Xm 80, Xscl 5, Ymin 5, Ym 0, Yscl 5. Grph, which models the pth of n object thrown into the ir t 45 ngle t n initil velocit of 7 48 feet per second. (Just tke m word for it ) Use to locte few points on the grph. Wht do ou lern bout the object from this informtion? Emple : Now chnge our clcultor to prmetric mode nd plot 4 t. 6t 4 t Notice it is the sme grph more on tht lter. Use to locte few points on the grph. Wht do ou lern bout the object from this informtion tht ou did not know before? Definition: Prmetric Curve If nd re given s functions f t, gt over n intervl of t vlues, then the set of points, ft, gt defined b these equtions is prmetric curve. Grphing Prmetric Curves Without Clcultor Just like when ou lerned to grph for the first time bck in Algebr, we re going to mke tble of vlues. The difference is tht we now hve three vribles insted of two. Emple 3: Grph the prmetric curve t 4 t3 t Emple 4: Grph the prmetric curve 4t 4 t t 3 Emple 5: Compre nd contrst the two grphs bove. - 3

14 .4 Prmetric Equtions AP Clculus Chnging from Prmetric to Rectngulr (Crtesin) To chnge prmetric eqution bck into more fmilir rectngulr (Crtesin) eqution ou must eliminte the prmeter. The tpicl pproch to doing this is to solve for the prmeter in one of the equtions nd then simpl substitute tht solution into the other eqution. Emple 6: Chnge the prmetric eqution defined b t t into Crtesin eqution. The net two emples illustrte nother w to eliminte the prmeter b using trigonometric identities. Emple 7: Chnge the prmetric eqution 3cos into Crtesin eqution nd then grph. 5sin Emple 8: Chnge the prmetric eqution defined b 4sec 3tn into Crtesin eqution nd then grph. - 4

15 .4 Prmetric Equtions AP Clculus Writing Prmetric Eqution Chnging from Rectngulr to Prmetric mens ou get to crete prmeter. As the grphs on the previous pge indicted, our choice of prmeter should not chnge the shpe of the grph, onl the speed in which the grph is drwn. Emple 9: Find prmetriztion for the left hlf of the prbol. Emple 0: Find prmetriztion for the line segment with endpoints (, 3) nd (3, ). - 5

16 .5 Functions nd Logrithms AP Clculus.5 FUNCTIONS AND LOGARITHMS Notecrds from Section.5: Finding nd Proving Inverse Functions Inverse Functions In technicl jrgon, n inverse of function mps the elements of the rnge to the elements of the domin. In English, this mens tht the inverse of function reverses the domin nd rnge. Not ll grphs were defined s functions, nd we hd the verticl line test to determine whether grph ws or ws not function. Similrl, not ll functions hve n inverse tht is function, nd we hve the horizontl line test to determine whether or not given function hs n inverse function. Definition: One to One Function A function f () is one to one on domin D if f f b whenever b. A function tht is one to one hs n inverse. The definition bove cn be seen grphicll with the use of horizontl line test. If there re two vlues for n given vlue of function, then the function does NOT hve n inverse. Emple : Does 5 hve n inverse? Wh or wh not? Emple : Does 3 hve n inverse? Wh or wh not? Once we know whether function hs n inverse, our net tsk is to find n eqution nd/or grph for the inverse. Finding the Inverse Grphicll (two ws). Reflect the grph of the originl function over the line =.. Plot the reverse of the coordintes. Finding the Inverse Algebricll Switch the nd in the originl eqution, then solve the new eqution for in order to write s function of. Emple 3: Let 3 f. ) Grph the function on the grid to the right. b) Drw the line = c) Reflect the grph of f over the line =. d) Find the inverse of the function lgebricll. e) Use our grphing clcultor to verif our nswer to prt d. - 6

17 .5 Functions nd Logrithms AP Clculus Verifing/Proving Inverses It is one thing to find the inverse function (either grphicll or lgebricll), but it is nother to verif tht two functions re ctull inverses. Whenever ou re verifing nthing in mthemtics, ou must go bck nd use the definition. Definition: Inverse Function A function f hs n inverse f if nd onl if f f f f Emple 4: According to this definition, how mn composite functions must ou use to check whether or not two functions re inverses of ech other? Emple 5: Find f nd verif if 3 f. Logrithmic Functions How do Logrithms fit into this discussion? A logrithmic function is just the inverse of n eponentil function. Emple 6: Grph nd find the inverse of the function grphicll. The eqution of the inverse function is. - 7

18 .5 Functions nd Logrithms AP Clculus Properties of Logrithms Definition of logrithm: Inverse Properties of logrithm: log log log Logrithm of product: log log log Logrithm of power: log log Logrithm of quotient: log log log Chnge of Bse Formul: ln log ln Other properties: log log 0 Emple 7: Evlute the following without using our clcultor. log b) log 7 9 ) 8 Emple 8: Solve for in the following equtions. ) log 5 b) c) log 8 log - 8

19 .6 Trigonometric Functions AP Clculus.6 TRIGONOMETRIC FUNCTIONS There re two common mesures of ngles: degrees nd rdins. As ou'll see, lmost ll of clculus uses rdins. Before beginning n eercise with trigonometric functions, mke sure our clcultor is set in rdin mode. (ESPECIALLY THOSE YOU WHO HAD PHYSICS YESTERDAY!) Unless otherwise stted, the ngles in the tet re mesured in rdins. For emple, sin 3 mens the sine of 3 rdins, but sin 3 mens the sine of 3 degrees. Just for fun, ou should understnd ectl wht rdin is. Definition: Rdin An ngle of rdin is defined to be the ngle t the center of unit circle which spns n rc of length, mesured counterclockwise. Arc length = rdin : You should lso be VERY fmilir with the 6 trigonometric vlues of the ke points on the unit circle 6, 4, 3,, Grphs of Trigonometric Functions Grph one period of ll 6 trigonometric functions below. Lbel t lest 5 points (or smptotes) for ech grph. sin cos tn csc sec cot - 9

20 .6 Trigonometric Functions AP Clculus Trnsformtions of Trigonometric Functions Emple: For ech of the following emples, do three things.. Describe the trnsformtion.. Grph the function. 3. Where pproprite, give the mplitude nd the period. ) sin b) tn 3 π π π Inverse Trigonometric Functions Due to the periodic propert of trigonometric functions, ll 6 of the trig functions fil the horizontl line test. Use the grph of the originl function nd highlight the portion of the grph used to grph the inverse. Use this highlighted portion to determine the domin nd rng e of ech inverse function. = cos = sin Domin of Inverse: Rnge of Inverse: Domin of Inverse: Rnge of Inverse: - 0

21 .6 Trigonometric Functions AP Clculus = tn = cot Domin of Inverse: Rnge of Inverse: Domin of Inverse: Rnge of Inverse: = csc = sec Domin of Inverse: Rnge of Inverse: Domin of Inverse: Rnge of Inverse: Check our nswers with the informtion on pge 50 nd 5 of our tetbook. You cn lso find grphs of the inverse trig functions on these pges s well. The restricted domin of the inverse trig functions mens ou must p close ttention to our solutions. Your clcultor onl gives ou the solution for which the domin of the inverse function is defined. Your clcultor lso onl hs 3 of the si inverse functions. Emple: Find the domin nd rnge of the following functions: ) sin cos b) sec tn -

22 .6 Trigonometric Functions AP Clculus Emple: Evlute the epression WITHOUT clcultor. tn sin 3 Emple: Solve for : sec 3 where 0. -