1 Limits and Continuity

Transcription

1 1 Limits and Continuity 1.0 Tangent Lines, Velocities, Growt In tion 0.2, we estimated te slope of a line tangent to te grap of a function at a point. At te end of tion 0.3, we constructed a new function tat was te slope of te line tangent to te grap of a function at eac point. In bot cases, before we could calculate a slope, we ad to estimate te tangent line from te grap of te function, a metod tat required an accurate grap and good estimating. In tis tion we will start to look at a more precise metod of finding te slope of a tangent line tat does not require a grap or any estimation by us. We will begin wit a non-applied problem and ten look at two applications of te same idea. Te Slope of a Line Tangent to a Function at a Point Our goal is to find a way of exactly determining te slope of te line tat is tangent to a function (to te grap of te function) at a point in a way tat does not require us to actually ave te grap of te function. Let s start wit te problem of finding te slope of te line L (see margin figure), wic is tangent to f (x) x 2 at te point (2, 4). We could estimate te slope of L from te grap, but we won t. Instead, we can see tat te line troug (2, 4) and (3, 9) on te grap of f is an approximation of te slope of te tangent line, and we can calculate tat slope exactly: m y x But m 5 is only an estimate of te slope of te tangent line and not a very good estimate. It s too big. We can get a better estimate by picking a ond point on te grap of f closer to (2, 4) te point (2, 4) is fixed and it must be one of te two points we use. From te figure in te margin, we can see tat te slope of te line troug te points (2, 4) and (2.5, 6.25) is a better approximation of te slope of te

2 54 contemporary calculus tangent line at (2, 4): m y x Tis is a better estimate, but still an approximation. We can continue picking points closer and closer to (2, 4) on te grap of f, and ten calculating te slopes of te lines troug eac of tese points (x, y) and te point (2, 4): points to te left of (2, 4) points to te rigt of (2, 4) x y x 2 slope x y x 2 slope Te only ting special about te x-values we picked is tat tey are numbers close and very close to x 2. Someone else migt ave picked oter nearby values for x. As te points we pick get closer and closer to te point (2, 4) on te grap of y x 2, te slopes of te lines troug te points and (2, 4) are better approximations of te slope of te tangent line, and tese slopes are getting closer and closer to 4. Practice 1. Wat is te slope of te line troug (2, 4) and (x, y) for y x 2 and x 1.994? For x ? We can bypass muc of te calculating by not picking te points one at a time: let s look at a general point near (2, 4). Define x 2 + so is te increment from 2 to x (see margin figure). If is small, ten x 2 + is close to 2 and te point (2 +, f (2 + )) ( 2 +, (2 + ) 2) is close to (2, 4). Te slope m of te line troug te points (2, 4) and ( 2 +, (2 + ) 2 ) is a good approximation of te slope of te tangent line at te point (2, 4): m y x (2 + )2 4 (2 + ) 2 ( ) (4 + ) 4 + If is very small, ten m 4 + is a very good approximation to te slope of te tangent line, and m 4 + also appens to be very close to te value 4. Te value m 4 + is called te slope of te ant line troug te two points (2, 4) and ( 2 +, (2 + ) 2). Te limiting value 4 of m 4 + as gets smaller and smaller is called te slope of te tangent line to te grap of f at (2, 4). Example 1. Find te slope of te line tangent to f (x) x 2 at te point (1, 1) by evaluating te slope of te ant line troug (1, 1) and (1 +, f (1 + )) and ten determining wat appens as gets very small (see margin figure).

3 limits and continuity 55 Solution. Te slope of te ant line troug te points (1, 1) and (1 +, f (1 + )) is: m f (1 + ) 1 (1 + ) 1 (1 + )2 1 ( ) (2 + ) 2 + As gets very small, te value of m approaces te value 2, te slope of tangent line at te point (1, 1). Practice 2. Find te slope of te line tangent to te grap of y f (x) x 2 at te point ( 1, 1) by finding te slope of te ant line, m, troug te points ( 1, 1) and ( 1 +, f ( 1 + )) and ten determining wat appens to m as gets very small. Falling Tomato Suppose we drop a tomato from te top of a 100-foot building (see margin figure) and record its position at various times during its fall: time () eigt (ft) Some questions are easy to answer directly from te table: (a) How long did it take for te tomato to drop 100 feet? (2.5 onds) (b) How far did te tomato fall during te first ond? ( feet) (c) How far did te tomato fall during te last ond? ( feet) (d) How far did te tomato fall between t 0.5 and t 1? ( feet) Oter questions require a little calculation: (e) Wat was te average velocity of te tomato during its fall? average velocity distance fallen total time position 100 ft 2.5 s (f) Wat was te average velocity between t 1 and t 2 onds? average velocity position 36 ft 84 ft 2 s 1 s 48 ft 1 s 40 ft 48 ft

4 56 contemporary calculus Some questions are more difficult. (g) How fast was te tomato falling 1 ond after it was dropped? Tis question is significantly different from te previous two questions about average velocity. Here we want te instantaneous velocity, te velocity at an instant in time. Unfortunately, te tomato is not equipped wit a speedometer, so we will ave to give an approximate answer. One crude approximation of te instantaneous velocity after 1 ond is simply te average velocity during te entire fall, 40 ft. But te tomato fell slowly at te beginning and rapidly near te end, so tis estimate may or may not be a good answer. We can get a better approximation of te instantaneous velocity at t 1 by calculating te average velocities over a sort time interval near t 1. Te average velocity between t 0.5 and t 1 is: 12 feet ft and te average velocity between t 1 and t 1.5 is 20 feet ft so we can be reasonably sure tat te instantaneous velocity is between 24 ft and 40 ft. In general, te sorter te time interval over wic we calculate te average velocity, te better te average velocity will approximate te instantaneous velocity. Te average velocity over a time interval is: position wic is te slope of te ant line troug two points on te grap of eigt versus time (see margin figure). average velocity position slope of te ant line troug two points Te instantaneous velocity at a particular time and eigt is te slope of te tangent line to te grap at te point given by tat time and eigt. instantaneous velocity slope of te line tangent to te grap Practice 3. Estimate te instantaneous velocity of te tomato 2 onds after it was dropped.

5 limits and continuity 57 Growing Bacteria Suppose we set up a macine to count te number of bacteria growing on a Petri plate (see margin figure). At first tere are few bacteria, so te population grows slowly. Ten tere are more bacteria to divide, so te population grows more quickly. Later, tere are more bacteria and less room and nutrients available for te expanding population, so te population grows slowly again. Finally, te bacteria ave used up most of te nutrients and te population declines as bacteria die. Te population grap can be used to answer a number of questions: (a) Wat is te bacteria population at time t 3 days? (about 500 bacteria) (b) Wat is te population increment from t 3 to t 10 days? (about 4, 000 bacteria) (c) Wat is te rate of population growt from t 3 to t 10 days? To answer tis last question, we compute te average cange in population during tat time: average cange in population population cange in population cange in time 4000 bacteria 7 days 570 bacteria day Tis is te slope of te ant line troug (3, 500) and (10, 4500). average population growt rate population slope of te ant line troug two points Now for a more difficult question: (d) Wat is te rate of population growt on te tird day, at t 3? Tis question asks for te instantaneous rate of population cange, te slope of te line tangent to te population curve at (3, 500). If we sketc a line approximately tangent to te curve at (3, 500) and pick two points near te ends of te tangent line segment (see margin figure), we can estimate tat te instantaneous rate of population growt is approximately 320 bacteria day. instantaneous population growt rate slope of te line tangent to te grap

6 58 contemporary calculus Practice 4. Find approximate values for: (a) te average cange in population between t 9 and t 13. (b) te rate of population growt at t 9 days. Te tangent line problem, te instantaneous velocity problem and te instantaneous growt rate problem are all similar. In eac problem we wanted to know ow rapidly someting was canging at an instant in time, and eac problem turned out to involve finding te slope of a tangent line. Te approac in eac problem was also te same: find an approximate solution and ten examine wat appens to te approximate solution over sorter and sorter intervals. We will often use tis approac of finding a limiting value, but before we can use it effectively we need to describe te concept of a limit wit more precision. 1.0 Problems 1. (a) Wat is te slope of te line troug (3, 9) and (x, y) for y x 2 wen: i. x 2.97? ii. x 3.001? iii. x 3 +? (b) Wat appens to tis last slope wen is very small (close to 0)? (c) Sketc te grap of y x 2 for x near (a) Wat is te slope of te line troug ( 2, 4) and (x, y) for y x 2 wen: i. x 1.98? ii. x 2.03? iii. x 2 +? (b) Wat appens to tis last slope wen is very small (close to 0)? (c) Sketc te grap of y x 2 for x near (a) Wat is te slope of te line troug (2, 4) and (x, y) for y x 2 + x 2 wen: i. x 1.99? ii. x 2.004? iii. x 2 +? (b) Wat appens to tis last slope wen is very small (close to 0)? (c) Sketc te grap of y x 2 + x 2 for x near (a) Wat is te slope of te line troug ( 1, 2) and (x, y) for y x 2 + x 2 wen: i. x 0.98? ii. x 1.03? iii. x 1 +? (b) Wat appens to tis last slope wen is very small (close to 0)? (c) Sketc te grap of y x 2 + x 2 for x near Te figure below sows te temperature during a day in Ames. (a) Wat was te average cange in temperature from 9 a.m. to 1 p.m.? (b) Estimate ow fast te temperature was rising at 10 a.m. and at 7 p.m.

7 limits and continuity Te figure below sows te distance of a car from a measuring position located on te edge of a straigt road. (a) Wat was te average velocity of te car from t 0 to t 30 onds? (b) Wat was te average velocity from t 10 to t 30 onds? (c) About ow fast was te car traveling at t 10 onds? At t 20? At t 30? (d) Wat does te orizontal part of te grap between t 15 and t 20 onds tell you? (e) Wat does te negative velocity at t 25 represent? 8. Te figure below sows te composite developmental skill level of cessmasters at different ages as determined by teir performance against oter cessmasters. (From Rating Systems for Human Abilities, by W.H. Batcelder and R.S. Simpson, UMAP Module 698.) (a) At wat age is te typical cessmaster playing te best cess? (b) At approximately wat age is te cessmaster s skill level increasing most rapidly? (c) Describe te development of te typical cessmaster s skill in words. (d) Sketc graps tat you tink would reasonably describe te performance levels versus age for an atlete, a classical pianist, a rock singer, a matematician and a professional in your major field. 7. Te figure below sows te distance of a car from a measuring position located on te edge of a straigt road. (a) Wat was te average velocity of te car from t 0 to t 20 onds? (b) Wat was te average velocity from t 10 to t 30 onds? (c) About ow fast was te car traveling at t 10 onds? At t 20? At t 30? 9. Define A(x) to be te area bounded by te t- (orizontal) and y-axes, te orizontal line y 3, and te vertical line at x (see figure below). For example, A(4) 12 is te area of te 4 3 rectangle. (a) Evaluate A(0), A(1), A(2), A(2.5) and A(3). (b) Wat area would A(4) A(1) represent? (c) Grap y A(x) for 0 x 4.

8 60 contemporary calculus 10. Define A(x) to be te area bounded by te t- (orizontal) and y-axes, te line y t + 1, and te vertical line at x (see figure). For example, A(4) 12. (a) Evaluate A(0), A(1), A(2), A(2.5) and A(3). (b) Wat area would A(3) A(1) represent in te figure? (c) Grap y A(x) for 0 x 4.

9 limits and continuity Practice Answers 1. If x 1.994, ten y , so te slope between (2, 4) and (x, y) is: 4 y 2 x If x , ten y , so te slope between (2, 4) and (x, y) is: 4 y 2 x Computing m: f ( 1 + ) (1) ( 1 + ) ( 1) ( 1 + ) ( 2 + ) 2 + As 0, m Te average velocity between t 1.5 and t 2.0 is: feet feet Te average velocity between t 2.0 and t 2.5 is: 0 36 feet feet Te velocity at t 2.0 is somewere between 56 feet probably around te middle of tis interval: and 72feet, ( 56) + ( 72) 2 64 feet 4. (a) Wen t 9 days, te population is approximately P 4, 200 bacteria. Wen t 13, P 5, 000. Te average cange in population is approximately: bacteria 13 9 days 800 bacteria 4 days 200 bacteria day (b) To find te rate of population growt at t 9 days, sketc te line tangent to te population curve at te point (9, 4200) and ten use (9, 4200) and anoter point on te tangent line to calculate te slope of te line. Using te approximate values (5, 2800) and (9, 4200), te slope of te tangent line at te point (9, 4200) is approximately: bacteria 9 5 days 1400 bacteria 4 days 350 bacteria day