The Concept of Limits- A Review Girija Nair-Hart

I will discuss Early Calculus Early Calculus Early Calculus Early Calculus Modern Developments Modern Developments Modern Developments Modern Developments Why Calculus? Why Calculus? Why Calculus? Why Calculus? Why Limits? Why Limits? Why Limits? Why Limits? The Limits The Limits The Limits The Limits Educational Research Educational Research Educational Research Educational Research Implications Implications Implications

Ancient and Early Calculus

INDIA a few names… Aryabhata (476) Aryabhata (476) Bramhagupta (665) Bramhagupta (665) Bhaskara II (1100’s) Bhaskara II (1100’s) Madhava ( ) Madhava ( )

Liu Hui (3rd century) Liu Hui (3rd century) Zu Chongzhi (5th century) Zu Chongzhi (5th century) CHINA a few names…

GREECE a few names… Eudoxus (410 BC) Eudoxus (410 BC) Euclid (325 BC) Euclid (325 BC) Archimedes (287 BC– 212 BC) Archimedes (287 BC– 212 BC)

PERSIA a few names… Ibn Alhazen (AD 1000) Ibn Alhazen (AD 1000) Sharaf al-Din al-Tusi (1200) Sharaf al-Din al-Tusi (1200)

Isaac Newton (1642–1727, English) Gottfried Wilhelm Leibniz (1646–1716, German) Laid foundations to Differential and Integral Calculus A limit was described as a quantity which a variable approached but never exceeded No formal definition was given

Isaac Newton (1642–1727, English) Isaac Newton (1642–1727, English) Gottfried Wilhelm Leibniz (1646–1716, German) Gottfried Wilhelm Leibniz (1646–1716, German) Joseph-Louis Lagrange (1736 – 1813, French) Joseph-Louis Lagrange (1736 – 1813, French) Joseph-Louis Lagrange (1736 – 1813, French) Joseph-Louis Lagrange (1736 – 1813, French) Augustin Louis Cauchy (1789 – 1857, French) Augustin Louis Cauchy (1789 – 1857, French) Augustin Louis Cauchy (1789 – 1857, French) Augustin Louis Cauchy (1789 – 1857, French) Karl Weierstrass (1815–1897, German) Karl Weierstrass (1815–1897, German) Karl Weierstrass (1815–1897, German) Karl Weierstrass (1815–1897, German)

Lagrange A definition of derivatives without limits using only algebraic analysis A definition of derivatives without limits using only algebraic analysis

Cauchy Emphasized more rigor, continuity was explained analytically rather than geometrically Emphasized more rigor, continuity was explained analytically rather than geometrically

Weierstrass A formal definition for Limits A formal definition for Limits

Why Calculus? Calculus provides a gateway to higher mathematics Calculus provides a gateway to higher mathematics Calculus have many real life applications in Accounting, Astronomy, Engineering, Meteorology, Medicine, etc. Calculus have many real life applications in Accounting, Astronomy, Engineering, Meteorology, Medicine, etc. Many problems that cannot be solved algebraically could be solved using calculus Many problems that cannot be solved algebraically could be solved using calculus

Why Limits? The concept of limit is the underlying factor for differential and integral calculus The concept of limit is the underlying factor for differential and integral calculus Practical applications Practical applications Economics: Marginal Cost Medicine: Average heart-beat Rate of cell growth and more… and more…

Differential Calculus Tangent and Velocity Problems Velocity: rate of change of position Velocity: rate of change of position Velocity = displacement/time (Average) Velocity could be found in a given time interval (Average) Velocity could be found in a given time interval But (instantaneous) Velocity cannot be found at any particular time using the above formula But (instantaneous) Velocity cannot be found at any particular time using the above formula

Find the instantaneous velocity at 5 seconds if The position function is To find the instantaneous velocity after 5 seconds, we will pick a time interval say, [5, 6] and then compute the average velocity over successively smaller periods

Time intervalAverage velocity = change in position/time elapsed (S(t)-S(5))/(t – 5) [5, 6]53.9 [5, 5.1]49.49 [5, 5.05] [5, 5.01] [5, 5.001] [5, ] As t approaches 5, the ave. velocity approaches 49 m/s The instantaneous velocity after t = 5 will thus be approximately 49 m/s

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s t P(5, 122.5) Velocity: limit of the slope of the tangent line

Area under a curve – Integral Calculus Example: Distance Problems Objective: To find the distance traveled by an object during a certain time period if the velocity of the object is known at all times. If the velocity remains constant then the distance could be easily computed using the formula: distance = velocity × time

But, if the velocity varies, the problem of finding distance will not very easy. Suppose you want to estimate the distance driven by your car over a 30 second time interval (your odometer is severely broken) We could take the speedometer readings every 5 seconds (or even every 2 seconds, or every second) and record them. Let’s see… Let’s see…

During the first 5 seconds, the velocity is approximately a constant. Assume 25 ft/sec. The approximate distance traveled in the first 5 seconds will therefore be, (25 ft/s)×(5 s) = 125 feet. TimeS Velocityft/s speedometer readings every 5 seconds

Finding the sumFinding the sum of distances Finding the sum By similar argument we will find the approximate distance traveled in each of the 6 intervals and consider the sum of those distances to be the approximate total distance traveled in 30 seconds.

Using the lower limit of each intervals as the average velocity corresponding to each interval, the sum of distances = 1135 feet Using the lower limit of each intervals as the average velocity corresponding to each interval, the sum of distances = 1135 feet Using the upper limit of each intervals as the average velocity, corresponding to each interval, the sum of distances = 1215 feet Using the upper limit of each intervals as the average velocity, corresponding to each interval, the sum of distances = 1215 feet Better yet, the approximate distance traveled in 30 seconds = the average of above 2 distances Better yet, the approximate distance traveled in 30 seconds = the average of above 2 distances Next….distance as area under the curve Next….distance as area under the curve Next….distance as area under the curve Next….distance as area under the curve

How would you estimate the area under this curve which represents the total distance traveled in 30 seconds?

Limits Consider the function Let us explore the Let us explore the behavior of this function as x approaches 2

Let’s explore numerically Using the language of Limit… Using Notation… Let’s explore numerically Using the language of Limit… Using Notation…

In that case…. Is this true? Is this true? Is this true? Is this true? How about? How about? How about? How about? What if x is approaching a number that is not in the domain of the function? What if x is approaching a number that is not in the domain of the function? What if x is approaching a number that is not in the domain of the function? What if x is approaching a number that is not in the domain of the function? The definition The definition The definition The definition No limit? No limit? No limit? No limit? Know limit?? Know limit?? Know limit?? Know limit?? Infinite limit?! Infinite limit?! Infinite limit?! Infinite limit?! Computing Limits Computing Limits Computing Limits Computing Limits

Function p in not defined at x = 5 Still as x approaches 5, p(x) approaches p

Computation of limit Graph Graph Table of values (numeric method) Table of values (numeric method) Intuition Intuition Algebra (why the rules work?) Algebra (why the rules work?) Limit at infinity Limit at infinity 0/0 form, K/0 form 0/0 form, K/0 form

Pitfalls Table of values Table of values Table of values Table of values Intuition 1 2 Intuition 1 2 1

Educational Research Tall (1978, 1981, 1986, 1992) Tall (1978, 1981, 1986, 1992) Szydlik (2000) Szydlik (2000) Roh (2005) Roh (2005) Juter (2006) Juter (2006)

Research -A Closer Look Language: both formal and informal Language: both formal and informal Language: both formal and informal Language: both formal and informal Conflicting concept images Conflicting concept images Conflicting concept images Conflicting concept images Common view of mathematics/pedagogy Common view of mathematics/pedagogy Inconsistency between intuitive and formal processes Inconsistency between intuitive and formal processes

Language ‘tends to ’ or ‘approaches ’ ‘tends to ’ or ‘approaches ’ ‘as close as we please’, ‘close enough’ ‘as close as we please’, ‘close enough’ ‘chord’ vs. ‘secant line’ ‘chord’ vs. ‘secant line’

Concept Images Concept of infinity, ‘infinite process’ Concept of infinity, ‘infinite process’ Dynamic motion: approached, but never reached Dynamic motion: approached, but never reached Limit of a function cannot be a finite number Limit of a function cannot be a finite number Limit of constant functions Limit of constant functions

Implications

The Concept of Limit - A Review Any Questions? References available on