1. AB Calculus Introduction

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1 1. AB Calculus Introduction Before we get into wat calculus is, ere are several eamples of wat you could do BC (before calculus) and wat you will be able to do at te end of tis course. Eample 1: On April 15, many people mail in teir taes to te Internal Revenue Service. Te town of Newton monitors te number of ta forms tat are mailed from teir post office. Te total number of ta forms tat are mailed from te 2500 Newton post office on April 15 is modeled by te function: M ( t) = 1 +13e 0.25t were t is te number of ours from 12 midnigt on April 1 troug 12 midnigt on April 15 and M ( t) is te total number of letters mailed tat day from te Newton post office. Wat you can do wit precalculus Find te number of ta forms mailed by 9 PM on April 15. Answer: M 21 2,30 Wat you will be able to do wit calculus Find te average rate of ta forms mailed between 6 PM and 9 PM. Answer: 51.9 forms per our Find te rate tat te ta forms are coming into te post office at 9 PM. Answer: 37.3 forms/our. Find te time of day wen te letters are coming into te post office at te fastest rate and wat is tat rate? Answer: Approimately 10:15 AM at te rate of forms/our. Wat is te total number of ours tat all te forms sit in te post office? Answer: 33,939 ours. Eample 2: A new car called te Seus as its plant net to its only dealersip. Cars are sold directly from te plant. Suppose at te start of te mont of May tere are 100 cars on te lot waiting to be sold. Cars come off te assembly line at te rate of A( t) = 20t( 2 0.2t ) and cars are sold at te rate of S( t) = cos( 0.2t)were t represents te day of te mont (for May 1, t = 0 and for May 31, t = 30). Wat you can do wit precalculus Find te rate of cars tat are produced on May 15. Answer: A 1 = 0.2 cars/day Find te rate of cars tat are sold on May 15. Answer: S( 1) =15.9 cars/day Wat you will be able to do wit calculus Find te total number of cars available to be sold in te mont of May. Answer: Appro. 1,057 cars Find te total numbers of cars sold in May. Answer: 879 cars. Find te average number of cars produced per day in May. Answer: 31.9 cars/day. On wat day will tere be a maimum number of cars waiting to be sold and approimately wat will tat number be? Answer: May 22 and 355 cars will be on te lot. On wat day will tere be a minimum number of cars waiting to be sold and approimately wat will tat number be? Answer: May and 16 cars will be on te lot Illegal to post on Internet

2 All te mat courses you ave ever taken ave been, in a sense, precalculus. In tese courses, you are analyzing numbers and epressions and determining teir current state. (e: Find te value o 2 wen = ). In calculus, you analyze ow tings cange. Wen real-life quantities cange, tey will eiter cange in a positive direction, or in a negative direction. Your first assignment is more an Englis one tan mat. You are to find words tat are used to denote positive cange, negative cange, or no cange. I ave started you off wit one. Find as many as you can. Wen stuck, tink of real life areas were cange occurs suc as academics, sports, economics, biology, music, etc. Get your dictionaries out! Positive Cange Negative Cange No Cange 1. Increasing 1. Decreasing 1. Constant 2. Rise 2. Reduce 2. Steady 3. Epand 3. Diminis 3. Stable. Intensify. Lessen. Invariable 5. Augment 5. Reduce 5. Fied 6. Strengten 6. Decline 6. Unwavering 7. Enlarge 7. Dwindle 7. Permanent 8. Amplify 8. Srink 8. Uncanging 9. Enance 9. Fall off 9. Static 10. Improve 10. Cut 10. Stationary 11. Swell 11. Weaken 12. Etend 12. Worsen 13. Escalate 13. Contract 1. Ascend 1. Deflate 15. Grow 15. Descend Since calculus is te study of cange, let s talk about cange around us. Give several tings tat are canging about YOU rigt now. 1. Heigt is canging 2. Fingernails are growing 3. Hair is growing. Knowledge is increasing Coose one. How do we know te cange is occurring? Is te cange a constant cange? Not necessarily. A kid's eigt can stay constant for awile and ten just soot up quickly. Wit knowledge, a student can cram for a test and increase te knowdge quickly. In calculus, we study four topics: 1) its, 2) derivatives, 3) integrals (one kind) and ) integrals (anoter kind). All of tese topics are related to te concept of cange. Everyting we do in tis course will be related to tese concepts. Altoug we will be involved in many details, everyting comes down to tese concepts. Your job in tis course will be to answer te question wic of tese topics does te problem I am attempting to solve apply to. Altoug we will deal wit many little details, we always need to see te big picture. Te introductory lessons, found in te Non-Essentials section will focus on te derivative and te definite integral and meant to give you an idea wat tis fascinating course is all about. 1a Illegal to post on Internet

3 2. Tangent Lines Classwork P In plane geometry, we say tat a line is tangent to a circle if it intersects te circle at one point. In te figure to te left, te line is tangent to te circle at point P. However, for more general curves, we need a better definition. Te idea of tangent lines is crucial to your understanding of differential calculus so we must ave an accurate idea of its meaning. Tere are many roug ideas of wat a tangent line is tat are subtly incorrect. Witout looking at te rest of te page, write below wat your definition of a tangent line is: A line is tangent to a curve if. P P Misconception 1: A line is tangent to a curve if it crosses te curve at one point. Tis is wrong. Te line crossed te curve at point P and is not a tangent line. Misconception 2: A tangent line to a curve must cross te curve only once. Tis is wrong. Te line is tangent to te curve at point P but it also crosses it at two oter points. P Misconception 3: A line is tangent to a curve if it touces te curve at one point but does not cross te curve. Tis is also wrong. Te line touces te curve at point P but it is clearly not a tangent line. P Misconception : A tangent line to a curve is a line tat just grazes te curve at a point but does not cross te curve. Tis is also wrong. Te line is tangent to te curve at point P but it does cross te curve as well. It will be some time before we get a clear definition of a tangent line. At tis point, we simply wis to give you some inkling of te difficulty of creating a definition. We ave to rely on our general knowledge to draw a tangent line Illegal to post on Internet

4 For eac equation, grap in an appropriate window and draw te tangent line at te indicated -value. 1) y = 2 ( = 1) 2) y = 1 2 ( = 0) 3) y = ( = 1) ) y = 2 ( = 1) 2a 5) y = 3 ( = 3) 6) y = e ( = 0) 7) y = sin ( = π ) 8) y = ( = ) Illegal to post on Internet

5 2. Tangent Lines Homework For eac equation, grap in an appropriate window and draw te tangent line at te indicated -value. 1. y = 2 ( = 2) 2. y = 3 ( = 2) 3. y = 3 ( = 0). y = 2 2 ( = 1) 5. y = ( = 0) 6. y = 2 ( = 0) 7. y = cos = π 2 8. y = ( = 0) Tangent line is te y-ais Illegal to post on Internet

6 10. y = 2 3 ( = 0) 3 9. y = = 0 Tangent line is te y-ais. Tangent line is te y-ais. 11. y = 2 ( = 2 ) 12. y = tan = π y = ln ( = e) 1. y = + 1 ( = 1) 15. y = 16 2 ( = 0) 16. y = 16 2 ( = ) Illegal to post on Internet

7 3. Slopes of Secants and Tangent Lines Classwork A line is drawn troug points P and Q, bot on f ( ). Tat line is called te secant line troug P and Q. A line is drawn tat touces f ( ) at only point P. Tat line is called te tangent line troug P. We draw te secant line troug PQ. Point Q moves along towards point P. Te closer tat Q gets to P, te more te secant line starts to look like te tangent line at P. We will say tat as Q gets closer and closer to P, tat te secant line PQ gets closer to te tangent line troug P and tus te slope of te scant line (called m sec ) approaces te slope of te tangent line (called m tan ). As an eample, let f ( ) = 2. Let us try and find te slope of te secant line between = 1 and a value o greater tan, but very close to 1. Complete te cart. Set your calculator to maimum decimal place accuracy f ( ) Rise Run m tan error Wy can t you use tis metod to find te slope of te tangent line at = 1? Division by zero Secant lines use ow many points? 2 Tangent lines use ow many points? 1 Using te fact tat te closer gets to 1, te slope of te secant line approaces te slope of te tangent line, wat is your best guess for te slope of te tangent line to at = 1 2? Let us use an analogy. You leave ome at 10 AM on a trip and arrive at 12 noon. Te trip is a total of 100 miles. How fast are you going at 11 AM? unknown We cannot find te actual velocity at 11 AM (called te instantaneous velocity). But we can find te average total distance velocity between 10 AM and any oter time, using te fact tat average velocity =. Te average total time velocity between 10 AM and 12 noon is 50 mp. Complete te cart on te net page Illegal to post on Internet

8 Between 11 AM & Distance traveled Time duration (rs) Average velocity Instantaneous velocity at 11 AM 3a 11:30 11:15 11:10 11:05 11:01 11:00:30 11:00:01 11:00 2 miles 13 miles 10 miles 5.5 miles 0.8 miles 0.5 miles 80 feet mp 52 mp 60 mp 66 mp 8 mp 60 mp 5.5 mp unknown???????????????????????? Wy can t you actually use tis tecnique to find te instantaneous velocity at 11 AM. Division by zero Average velocity uses ow many times? 2 Instantaneous velocity uses ow many times? 1 As te time duration after 11:00 becomes small, te instantaneous velocity at 11 AM is muc more likely to be very close to wic of tese values? 5.5 mp Wy? almost no time for variation Let us now take te two points and give tem general coordinates P( 0, y 0 ) and Q( 1, y 1 ). We draw te rigt triangle below te curve (te rise and te run). Te lengt of te rise is y 1 y 0 and te lengt of te run is 1 0. We now find te slope of te secant line PQ and denote it as m sec = rise run = y 1 y 0. Since y = f ( ), we can also say tat 1 0 f ( 0 ) m sec = rise run = Concentrating on te tangent line at point P, we ave sown tat te above formula does not work? Wy not? Division by zero Define te variable as te orizontal distance between te two points P and Q: Since = 1 0 it follows tat 1 = 0 +. f ( 0 ) f ( 0 ) Since m sec = rise run = 1, it follows tat m sec = Te important step: We sowed tat te tangent line at P is defined as te secant line between P and Q and Q gets closer and closer to P. As Q gets closer to P, 1 gets closer to 0, so (te orizontal distance between P and Q) gets close to zero. So we ave te starting point for differential calculus: f ( 0 ) m tan = 0 + Division by zero as gets infinitely close to zero. Note tat cannot actually equal zero? Wy not? Illegal to post on Internet

9 Tree formulas you need to know: te slope of te secant line between 0, f ( 0 ) te slope of te tangent line at ( 0, f ( 0 )) : and ( 1, f ( 1 )) : m sec = 1 te point-slope equation of a line passing troug 0, f ( 0 ) f ( 0 ) 1 0 m tan = f ( 0 + ) f ( 0 ) as gets infinitely close to zero : y y 0 = m 0 Eample 1) 3b For te function f ( ) = 2 +1, find te following: Confirm c) on your calculator. a) te slope of te secant line b) 3c te slope of te tangent c) te equation of te tangent between = 1 and = 3 line at = 2 line at = 2 f ( 1) f = 2 f ( 2) f ( + ) = + as gets close to 0, m tan = y 5 = ( 2) y = 3 Eample 2) For te function f ( ) = 5 + 2, find te following: Confirm c) on your calculator. a) te slope of te secant line b) te slope of te tangent c) te equation of te tangent between = 1 and = line at = 2 line at = 2 f ( ) f ( 1) ( 3) = 5 5 f ( 2 + ) f ( 2) 5( 2 + ) = 5 = 5 as gets close to 0, m tan = 5 y 12 = 5( 2) y = Eample 3) For te function f ( ) = 2 + 1, find te following: Confirm c) on your calculator. a) te slope of te secant line b) te slope of te tangent c) te equation of te tangent between = 3 and = 1 line at = 2 line at = 2 f ( 3) f = 0 2 f ( 2) f 2 + ( 2) 2 + ( 2) 1+ 5 as gets close to 0, m tan = 0 = 2 = y + 5 = y = Illegal to post on Internet

10 Eample ) For te function f ( ) = , find te following: Confirm c) on your calculator. a) te slope of te secant line b) te slope of te tangent c) te equation of te tangent between = 0 and = 1 line at = 2 line at = 2 f ( 0) f = 3 1 f ( 2) f ( 2 + ) ( 3 + ) = 3 + as gets close to 0, m tan = 3 y + 5 = 3( 2) y = 3 11 Eample 5) For te function f ( ) = , find te following: Confirm c) on your calculator. a) te slope of te secant line b) te slope of te tangent c) te equation of te tangent between = 1 and = 1 line at = 1 line at = 1 f ( 1) f = 1 2 f ( 1) f 1+ ( 1+ ) 3 ( 1+ ) ( 1+ ) 2 ( 1+ 1) = ( 1+ ) 2 as gets close to 0, m tan = 1 y 1 = 1( 1) y = Eample 6) For te function f ( ) = 2, find te following: Confirm c) on your calculator. +1 a) te slope of te secant line b) te slope of te tangent c) te equation of te tangent between = 1 and = line at = 2 3d line at = 2 f ( 1) f = 1 5 f ( 2) f ( 3 + ) 3( 3 + ) = 2 as gets close to 0, m tan = 2 9 y 2 3 = 2 ( 9 2) 9y 6 = 2 + y = Illegal to post on Internet

11 Suppose tat an object is traveling along a straigt line according to te formula s( t) = 2t + 3 were t is measured in seconds and s( t) is measured in feet. Complete te table. To calculate te average velocity between t = 0 and t =, we know average velocity = average velocity is 2 measured in ft sec. total distance total time. So te But if we wis to calculate te instantaneous velocity (eactly ow fast we are traveling) at t = 3 seconds, te above formula doesn t work. Using te analogy above, we now state two formulas tat allow us to find bot average and instantaneous velocity: Two formulas you need to know: Given s t as te distance traveled in time t, s( t 1 ) Average velocity between t 1 and t 2 Avg. vel. = s t 2 t 2 t 1 Instantaneous velocity at t 1 Inst. vel. = s t 1 + So in te eample above were s t s( t 1 ) as gets infinitely close to zero = 2t + 3, te instantaneous velocity at t = can be found: f ( ) Inst. vel. = s + = 2 + = 2 2 = 2 As gets infinitely close to zero, te instantaneous velocity is also 2 ft sec. t = 2t In tis case, te average velocity between t = 0 and t = is te same as te instantaneous velocity at t =. In a car, if your average velocity is te same as te instantaneous velocity, wat is tat called? cruise control Eample 7) If s( t) = 3t +1 is a measure of feet traveled wit t measured in seconds, find a) te average velocity between t = 0 and t = 3 b) te instantaneous velocity at t = 2 seconds s t s( 0) s = = 3 ft sec s( 2) s 2 + = 3( 2 + ) +1 7 = 3 = 3 ft sec Eample 8) If s( t) = t 2 2t is a measure of feet traveled wit t measured in seconds, find a) te average velocity between t = 0 and t = 2 b) te instantaneous velocity at t = 2 seconds s( 0) s = = 0 ft sec s( 2) s 2 + = ( 2 + ) = ( 2 + ) = 2 ft sec Eample 9) If s( t) = t 3 + t 2 t 1 is a measure of feet traveled wit t measured in seconds, find a) te average velocity between t = 1 and t = 2 b) te instantaneous velocity at t = 1 second s( 1+ ) s( 1) 3 + ( 1+ ) 2 1+ s( 1) s = = 9 ft sec Illegal to post on Internet = 1+ = ft sec 1 0

12 3. Slopes of Secants and Tangent Lines Homework 1. For te function f ( ) = 5 + 3, find te following: Confirm c) on your calculator. a. te slope of te secant line b. te slope of te tangent c. te equation of te tangent between = 1 and = 5 line at = 2 line at = 2 f ( 1) f = 5 f ( 2) f = 5 = 5( 2 + ) y 13 = 5( 2) y = For te function f ( ) = 2 3, find te following: Confirm c) on your calculator. a. te slope of te secant line b. te slope of te tangent c. te equation of te tangent between = 0 and = 3 line at = 1 line at = 1 f ( 3) f ( 0) ( 3) = 3 3 f ( 1+ ) f ( 1) = ( 1+ ) = 2 + m tan = 2 y + 2 = 2( 1) y = 2 3. For te function f ( ) = 2 5 +, find te following: Confirm c) on your calculator. a. te slope of te secant line b. te slope of te tangent c. te equation of te tangent between = 3 and = 6 line at = 3 line at = 3 f ( 6) f ( 2) ( 2) = 3 f ( 3 + ) f ( 3) = ( 3 + ) = 1+ m tan = 1 + ( 2) y + 2 = 1( 3) y = 5. For te function f ( ) = , find te following: Confirm c) on your calculator. a. te slope of te secant line b. te slope of te tangent c. te equation of te tangent between = 1 and = line at = 1 line at = 1 f ( 1) f = 1 5 f ( 1) f = 2( 1)2 7 1 = 2 11 m tan = y 17 = y = Illegal to post on Internet

13 5. For te function f ( ) = 3 + 2, find te following: Confirm c) on your calculator. a. te slope of te secant line b. te slope of te tangent c. te equation of te tangent between = 3 and = 3 line at = 2 line at = 2 f ( 3) f ( 3) ( 32) = 10 6 f ( 2) f 2 + = ( 2 + ) = m tan = 13 y 8 = 13( 2) y = For te function f ( ) = 5, find te following: Confirm c) on your calculator. 3 a. te slope of te secant line b. te slope of te tangent c. te equation of te tangent between = and = 6 line at = 1 line at = 1 f ( ) f = 5 3 f ( 1) f 1+ 2( 2) m tan = 5 = = ( 2) 2( 2) = 5 2( 2) y = 5 ( 1) y +10 = y = For te function f ( ) = 2, find te following: Confirm c) on your calculator. +1 a. te slope of te secant line b. te slope of te tangent c. te equation of te tangent between = 1 and = 5 line at = 3 line at = 3 f ( 1) f = 1 f ( 1) f 1+ ( + ) = = m tan = 3 16 ( + ) ( + ) y 1 = y = 3 9 y = Illegal to post on Internet

14 8. If s( t) = t +1 is a measure of feet traveled wit t measured in seconds, find a. te average velocity between t = 1 and t = 5 b. te instantaneous velocity at t = 2 seconds s( 1) s = 21 5 = ft sec s( 2) s 2 + = ( 2 + ) +1 9 = = ft sec 9. If s( t) = t 2 + is a measure of feet traveled wit t measured in seconds, find a. te average velocity between t = 0 and t = b. te instantaneous velocity at t = 1 second s( 0) s 0 = 20 = ft sec s( 1) s 1+ = ( 1+ )2 + 5 = ( 2 + ) = 2 ft sec 10. If s( t) = t 2 3t + 2 is a measure of miles traveled wit t measured in ours, find a. te average velocity between t = 0 and t = b. te instantaneous velocity at t = 1 our s( 0) s 0 = 6 2 = 1 mp s( 1+ ) s( 1) = ( 1+ )2 3( 1+ ) = ( 1) = 1 mp 11. If s( t) = t 3 + t 1 is a measure of meters traveled wit t measured in seconds, find a. te average velocity between t = 2 and t = 7 b. te instantaneous velocity at t = 2 seconds s( 2) s = = 68 m sec s( 2) s 2 + = ( 2 + ) = avg. vel. = 13 m sec ( ) 12. If s( t) = 6 is a measure of feet traveled wit t measured in seconds, find t + 2 a. te average velocity between t = 1 and t = 7 b. te instantaneous velocity at t = seconds s( 1) s = = 2 9 ft sec s( ) s + ( 6 + ) = = avg. vel. = 1 6 ft sec Illegal to post on Internet

15 . Grapical Approac to Limits Classwork To te rigt is te grap of f ( ) = 3 8. For all values o 2 not equal to 2, you can use standard curve sketcing tecniques. But te function is not defined at = 2. Tere is a ole in te grap. So let s get an idea of te beavior of te grap close to = 2. Set your calculator to -decimal place accuracy and complete te table f ( ) Error becomes closer and closer to 12. never actually equals 12, just gets closer to it. Tis is epressed wit te concept of a it. We It sould be obvious tat as gets closer and closer to 2, te value of Note tat say: te it of f ( ) as approaces 2 equals 12 and te notation we use is: f ( ) 3 8 = 12 or = 12. Te informal definition of a it answers te question: wat appens to y as gets close to a certain value. In order for a it to eist, we must be approacing te same y-values as approaces some number c from eiter te left or rigt side of c. If tis does not occur, we say tat te it does not eist (DNE) as we approac c. Te it of f ( ) as approaces some value of c from te left side (left-and it) is written: f ( ). c Te it of f ( ) as approaces some value of c from te rigt side (rigt-and it) is written: f ( ). In order for a it to eist at = c, c It is important to understand tat c For eac grap of f ( ), find te required information. must equal f ( ) and we ten say c + and f ( c) do not ave to equal eac oter. c = L. c DNE f ( 1) = = f ( ) = 1 DNE f ( 1) = Illegal to post on Internet

16 1 1 + = 1 f ( 1) DNE = 1 f ( 1) = 2 1 Wen tere is a vertical asymptote at = c, does not eist. If te function is c going up approacing te vertical asymptote, = wic is saying tat we say tat c tere is no it. If te function is going down approacing te vertical asymptote, we say tat c = wic again says tat tere is no it. + = 1 f ( 1) = 1 2 = f ( ) = DNE f ( 2) DNE 2 = f ( ) = = f ( 2) = Illegal to post on Internet

17 We also are interested te it of a function as approaces positive or negative infinity. Tis answers te question: wat appens to y as gets infinitely far to te rigt or to te left? Te terminology we use for tese its is: f ( ) and f ( ). Altoug we use te term as approaces infinity, realize tat cannot approac infinity as infinity does not eist. Te term approaces infinity is just a convenient way to epress te notion tat is getting infinitely far to te rigt of te y-ais. Note also tat it makes no sense to talk about f ( ) or f ( ) as we can + only get infinitely far to te rigt if we approac from te left, and we can only get infinitely far to te left if we approac from te rigt. and Tere are only 5 possibilities for Te curve can go up forever. In tis case, te it does not eist. For convenience s sake, we say tat f ( ). Te curve can go down forever. Again, in tat case, te it does not eist. For =. convenience sake, we say tat =. f ( ) = f ( ) = f ( ) = f ( ) = Te curve can become asymptotic to te curve can oscillate but te curve can oscillate a line. In tis case, te it is a get closer to a value. but not get closer to a value. We say f ( ) = L. Again, we say f ( ) = L. value. We say does not eist. a f ( ) = f ( ) = 2 f ( ) = 1 f ( ) = 0 f ( ) DNE f ( ) DNE Illegal to post on Internet

18 . Grapical Approac to Limits Homework a) 1 d) f 1 = 2 b) 1 + = 1 e) = 2 c) 1 = 1 f) f ( ) DNE f ( ) = 2 a) 2 d) f 2 = 3 b) 2 + = 3 e) = 0 c) 2 = 0 f) f ( ) DNE f ( ) = a) 3 d) f 3 = 1 b) 3 + = 1 e) = 1 c) 3 = f) f ( ) = 1 f ( ) = a) d) f 0 = 2 b) + = 2 e) = 2 c) = f) f ( ) = 2 f ( ) = a) 1 d) f 1 = 0 b) 1 + DNE e) = 0 c) 1 = f) f ( ) = 0 = a) 1 d) f 1 = 1 b) 1 + = 2 e) = 1 c) 1 = f) f ( ) = 1 f ( ) = Illegal to post on Internet

19 = b) a) d) f ( 1)DNE e) = c) 1 = 1 f) f ( ) DNE f ( ) = 2 = b) a) d) f ( 3)DNE e) = c) 3 = 2 f) f ( ) = f ( ) = 0 a) 3 d) f 3 = b) 3 + = 1 e) = c) 3 = 1 f) f ( ) = = 1 a) d) f 0 = b) + = 1 e) = 1 c) = f) f ( ) DNE f ( ) = a) d) f 0 = 0 b) + = 0 e) = 0 c) f ( ) = 0 DNE f) DNE a) d) f 0 = 0 b) + = 0 e) = 0 c) DNE f) f ( ) = 0 = Illegal to post on Internet

20 a) 1 d) f 1 = 3 b) 1 + = 2 e) = 2 c) 1 = 2 f) f ( ) DNE f ( ) = a) d) f 0 = 1 b) + = 1 e) = 0 c) f ( )DNE f) f ( )DNE f ( )DNE = b) a) d) f ( 1)DNE e) = c) 1 f ( )DNE f) f ( ) DNE f ( ) DNE = b) a) 1 d) f ( 1)DNE e) 1 + = c) 1 f ( )= 1 f) f ( ) DNE f ( ) = 2 a) d) f 0 = 3 b) + = 3 e) = 3 c) f ( )DNE f) f ( ) = 3 f ( )DNE a) d) f 0 = 0 b) + = 0 e) = 0 c) f ( )DNE f) f ( ) = 0 f ( ) = Illegal to post on Internet

21 5. Algebraic Approac to Limits Classwork Wile determining its based on graps is important, we don t want to spend time graping comple functions to determine its. Luckily, tere are purely algebraic tecniques tat don t depend on te grap. Tere are easy steps to find f ( ). Tey will be furter eplained below: a 1. Plug in a 2. Factor/cancel 3.,, or does not eist ) Indeterminate 1. Plug in a to f ( ) ; tat is find f ( a). If f ( a) eists, tat is te it. Eample 1a) find ( 2 +1) Eample 1b) find 2 2 ( 2 +1) = = = = cos Eample 1c) find π + 2 2cos = 2 π π + 2 π + 2 = 1 2. If, wen finding f a and/or denominator of 0 0, you get, a f ( ) may or may not eist. Factor bot numerator, do any cancellation and go back to step 1. Eample 2a) find = ( ) ( + 2) 2 2 ( + 2) 2 2 = 6 = 3 2 Eample 2b) find = ( 1) ( +1) = 5( 1) = Eample 2c) find = ( 1) ( 1) = 0 3 = 0 k k 3. Wen you find f ( a) and get or you get after factoring/canceling, a does not eist. Split your 0 0 problem into 2 its, f ( ) and f ( ). Eac of tese will be eiter + or. To determine wic, a a + plug in numbers close to a on te left and te rigt. We are not interested in te value, just te sign. Only if tese two its are te same (eiter + or ), can you go te etra step and say tat a = or =. If not, just say tat does not eist. Remember, if a a a a does not eist. But if a does not eist, tat does not necessarily mean tat = ±. is true tat a f ( ) = ±, it Illegal to post on Internet

22 a Eample 3a) Find = = 9 0 I = 1.9, = + I = 2.1, = Since , Eample 3c) Find = = 0 0 ( + 3) ( 1) = 0 ( + 3) = = = Since , = + = DNE does not eist 2 Eample 3b) Find 2 = 2 0 I = 0.1, = I = 0.1, 2 + = Since = 2 +, = 2 5b 2 Eample 3d) Find = = = 3 2 = ( 2) = ( 2) = ( 2) = ( 2), Since 2 2 =. Sometimes, wen trying to find a, you plug in, possible factor/cancel and still get 0. If no more 0 canceling can be done, you ave an indeterminate form. Tis means tat may or may not eist. a Tere are tecniques to determine wat te it is if it indeed eists, (and tey are covered in capter 2 of tis manual), for rigt now we simply call tem indeterminate (we cannot determine weter or not te it eists). sin Eample a) find Eample b) find e 1 2 Eample c) find +1 1 sin = 0 0 Since no factoring can be done, tis is indeterminate. Later, you sin will find tat = 1. e 1 2 = = 0 0 Since no factoring can be done, tis is indeterminate. Later, you will find tat e 1 2 = = = 0 0 Multiply top & bottom by te conjugate: = = Illegal to post on Internet

23 were a is te value canges. Tese problems are simple. You find and f ( ) a a + f ( ), does f ( ) eist. Wen taking its involving piecewise functions, you are usually asked to find a o for wic te formula for bot by te tecniques above. Only if f ( ) = a a + a E. 5a) If 3 +1, > 1 = 2, 1 find 1 E. 5b) If 2 + 3, < 2 =, = 2 find 2 3, > 2 E. 5c) If = find 3 + 3, , > f ( ) = 1 = 3 f ( ) = 3 +1 = 2 DNE f ( ) = + 6 = 6 2 = 8 2 = = 6. Tis sows tat is not related to f 2. f ( ) = 3 = f ( ) = = = 3 We need to eamine tecniques for finding its of functions as approaces ±. We first eamine te graps of = 1 (solid curve) and f ( ) = 1 2 (dased curve). We see tat bot tese functions are 1 asymptotic to te -ais. So ± = 0 and 1 a = 0. Consequently, we can say ± 2 ± = 0 n for any positive value of n > 1 were a is a constant. E. 6a) Find E. 6b) Find E. 6c) Find We first plug in and get. Tis is indeterminate (te it may or may not eist). We divide eac term by and take te it = = 8 = 1 2 We first plug in and get (indeterminate). Divide every term by 2 and take te it = = 0 1 = 0 We first plug in and get (indeterminate form). We divide every term by and take te it = = 0 = Illegal to post on Internet

24 We can make tis process easier by using te sortcut below for rational functions f ( ). To find f ( ) or f ( ), write f ( ) as a fraction. = 0 and 1) If te igest power o appears in te denominator (bottom-eavy), 2) If te igest power o appears in bot te numerator and denominator (powers-equal), = coefficient of numerator's igest power ± coefficient of denominator's igest power. 3) If te igest power o appears in te numerator (top-eavy), ± te it is + or, plug in a large number to f ( ) for small number to f ( ) for f ( ) and determine te sign. = ±. To determine f ( ) = 0. and determine te sign. Plug in a E. 7a) Find Bottom-eavy so it is 0. E. 7b) Find Powers-equal so it is 5. 2 E. 7c) Find Top-eavy so it is + + =. E. 7d) Find 1 E. 7e) Find E. 7f) Find Bottom-eavy so it is 0. Powers-equal so it is 2 = 1 2. Powers-equal so it is 2 = 1 2. If f ( ) is not a rational function and you wis to find or f ( ), use common sense. Attempt to see weter te numerator is growing faster tan te denominator or vice versa. Plug in large or small numbers to confirm your teory. 5c E. 8a) Find e Te denominator is growing faster tan te numerator. E: 10,100. So e e te it is 0. e + E. 8b) Find e Wen is large, te 's make little difference. Tis is powers-equal and te it is 1. sin E. 8c) Find sin will always be a number between 1 and 1 so tis is bottom-eavy and te it is Illegal to post on Internet

25 Find te following its Algebraic Approac to Limits Homework 2. 2π π = y 6y 3 2y y 1 ( 5) 1 = = 1 3( 1) 6( 1) 2( 1) = = ( 1) = = ( ) t 2 t t + 2 t + 2 = + = 8 t 2 t + 2 t 2 2t = + + = ( 2) ( 2) = ( + 5) +1 1 ( +1) = ( 2 ) +1 2 ( 2 + ) = DNE = = DNE = 5 + ( 1) = 2 ( 1) = = ( 1) = y 6 y + 6 y = y 6 y 6 y 6 + y 6 = y y 6 DNE 5 5 = = 5 = Illegal to post on Internet ( + 2) = 3 ( + 2) = 3 ( + 2) DNE 21. ( 3) = ( 3) = 2 2 =

26 , > 3 find 3. Sow work. = 1, 3 f ( ) = 3 1 = 2 3 = DNE 3 = π 1, > π find π. Sow work. π π + = cos sinπ, π π f ( ) = 1 0 = 1 f ( ) = π π 1 = 1 = 1 2. f ( ) = 3 +, 1 2, > 1 find f ( ). Sow work. 1 1 f ( ) = 1 1 = = 21 = 2 = f ( ) = 2 1, < 1, > 1 Sow work. 1 f ( ) = 1 = f ( ) = = = 28. If 1 find f ( ) 1 f ( ) = 2 2 3, 2 k 3, = 2 find k suc tat 2 3 = k 3 k = 0 = f ( 2) 26. Find = 2 2 = = f ( ) = = , 7 k 2 2, = 7 ( 7) Find cosπ cos0 = 0 0 ( ) cosπ ( ) k 2 2 = 1 k 2 = 16 k = ± + 2 cosπ = find k suc tat f ( ) = f ( 7) 7 = = = Find Find ( 11 2) = Find Find Find Find Find Illegal to post on Internet