dr. siham alfred raritan valley community college [email protected]

A STUDENT’S FIRST JOURNEY INTO CALCULUS

- THEMED SESSION - TI SESSION # TIF ROOM: PLATINUM 2

Dr. Siham AlfredRaritan Valley Community [email protected]

39th AMATYC Annual ConferenceAnaheim, California

October 31- November 3, 2013

mailto:[email protected]

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A Student’s First Journey Into Calculus

Overview

In this session, I will share some simple and some more involved activities I use with my students in Calculus to motivate:

• Instantaneous rate of change• Alternate proof of the derivative of power functions and• A progressive approach to the understanding of area under a curve of a

continuous function as the limit of a Riemann sum.

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I invite the students to get to know the inventors of Calculus:

• Isaac Newton (1642 - 1727)

• Gottfried Leibnitz (1646 - 1716)

Their ideas are alive and with us in the classroom As well as the ideas of subsequently many other mathematicians

Including Archimedes (287 – 212 BC) whose method of exhaustion preceded Differential Calculus

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What is Calculus? A body of knowledge and models of thought motivated by seeking to understand that the world around us is in motion and undergoing change.

Arnold Toynbee (1889 - 1975), the famous British Historian was offered the opportunity at the age of 14 to take calculus and he refused.

He later wrote that he had so much regretted his decision for he lacks the understanding of the world in motion. His view of it remained static.

Students in Calculus classes virtually carry out an intellectual conversation with the inventors of Calculus. For that they are very fortunate.

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CALCULUS

The Study of Change, rates of Change and Accumulated Change

DIFFERENTIATION

INTEGRATION

Definitions, Techniques, Methods

Definitions, Techniques, Methods

Theorems & Applications Theorems & Applications

Show the students the forest before the trees

There are two main processes in Calculus and a strong and well built bridge between them

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On their first journey into Calculus:

• Are students ready for the trip? What’s in their bags?

• Generally it is their background in Pre-calculus that is the impediment -- not the new ideas in calculus

1. A Diagnostic Pre-calculus Test is highly recommended

2. Test should contain Algebra, Analytic Geometry, Functions and Trigonometry

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• Curriculum is packed. There is no extra time to review.

• Incorporate review within each lesson, homework & lab

• See in handout some sample exercises:

On the definition of a function

On inverse trigonometric functions

You only have these students one moment in time

Make the most of it

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Knowledge is a network

Professor Robert Davis, of Rutgers University, asked us:

“ What comes to your mind when I say “Birthday Party?” He followed it by another question:

“What should come to your students’ mind when you say slope of the tangent line?”

Our challenge then is to provide students with activities that will generate new concepts and enable them to make connections among them in order to build a network of knowledge to achieve Conceptual Understanding

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A Student’s First Journey Into CalculusActivity 1: Write a short log of your commute to the college. Before you begin your commute, set your odometer at 0 and note the time and then record the mileage and time when you get to the college. Describe what actually happens. I share my daily commute: I drive 14 miles to get from my house to the college. It takes me half an hour to get there.

My average velocity is

Writing it symbolically, the

on the interval where denotes the distance at time .

tan ( ) tan ( ) 14 0 281 02

Dis ce at end time Dis ce at initial time mphend time initial time

0

0

( ) ( )S t S tAverageVelocity

t t

0( , )t t ( )S t t

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This activity for students to experience Average and Instantaneous Velocity opens for them the gates to Calculus (see handout pages 2 & 3)

Students find the average velocity of their daily commute to the college, however, they become aware of the following:

1. Students become aware of the difference between their average and their instantaneous velocity

2. They drive above and below their average velocity3. At some time during their commute they drive at their average velocity

When we study the Mean Value Theorem students remember that activity and are not surprised at the result of the theorem.

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In Group Discussion, students were interested in:

1. How is the instantaneous velocity calculated in their car as shown on the speedometer?

2. How is the distance calculated? 3. How is the time calculated?

Students referred back to the linear Speed and angular speed on a circle which they have learned previously, namely:

where is the arc length through the angle and is the radius of the circle and t is time.

Angular velocity is defined as the change of angle with respect to time.

s rvt t

s r

t

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( , )

Demystify the definition of a Limit: Why

Newton used the word limit. It is crucial to define it properly because a continuous function, the derivative and the integral are all defined in terms of it.

Cauchy used in his definition of a limit at a point. He used for “différence” French for difference, and

for “erreur” again French for error When teaching limits: Clearly identify the investigation phase from the proof phase. Students confuse the two.

• At least write one example in details. (see handout pages 5 & 6)• Begin with Infinite Limits.

( , )

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Students

• Learn to evaluate limits numerically, graphically and algebraically

• Define and compute the derivative numerically, graphically and algebraically

• Estimate the derivative graphically. (see handout pages 3 & 4)

Students are given a graph whose equation is unknown initially and a set of points on the graph.

For each point given, they are required to draw a tangent line and find another point on the tangent line to estimate the slope of thetangent line.

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Estimate the slope of the tangent line (Activity on handout pages 3 - 5)

Point of tangency

(-3,0) (-2, ) (1, ) (2, ) (3, )

Est. 2nd point (-2, 5)

Est. slope =MTAN 5

Est. Equation of tangent line

5 15y x

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Use Students Prior Knowledge to create new knowledge.

Two Examples (see handout pages 6 & 7):

1. Power Rule by Synthetic Division a) as an alternative to the Binomial Theorem

'( ) limn n

x a

x af ax a

1 2 2 3 2 1( )( ... )limn n n n n

x a

x a x ax a x a x ax a

1 2 2 3 2 1lim( ... )n n n n n

x ax ax a x a x a

1 2 2 3 2 1( ... )n n n n na aa a a a a a

1 1 1 1 1 1( ... )n n n n n na a a a a na

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2. Proof of the Product and the Quotient Rule for Derivatives Using Logarithmic Differentiation (see handout page 7)

Most calculus books use the formal definition of the derivative to prove the product and quotient rules. In some cases the product and quotient rules can be proved using logarithmic differentiation

The Product Rule

such that for all x in the domain of f, and u and v are differentiable functions of x, then . Taking the derivative of both sides yields:ln ( ) ln[ ( ) ( )] ln ( ) ln ( )f x u x v x u x v x

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The Quotient Rule (see handout page 7)

are defined as above,

Differentiating both sides,

Multiply both sides by to get

( )ln ( ) ln ln ( ) ln ( )( )u xThen f x u x v xv x

( ) '( ) ( ) '( )( ) ( )

'( ) '( ) '( )( ) ( ) ( )

v x u x u x v xu x v x

f x u x v xf x u x v x

( ) '( ) ( ) '( ) ( ) ( ) '( ) ( ) '( )( ) ( ) ( ) ( ) ( )

'( ) ( ) v x u x u x v x u x v x u x u x v xu x v x v x u x v x

f x f x

2( ) '( ) ( ) '( )'( ) [ ( )]v x u x u x v xf x v x

( )( ) ( ) ( )( )u xIf f x where u x and v xv x

Finding the derivative of a product and quotient functions under the conditions stated gives an alternative useful method for reinforcing the product and quotient rules for derivatives and provides good practice and incentive for using logarithmic differentiation.

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1st Lab: Four Existence Theorems in Differential Calculus (See handout 1st Lab)

• The Intermediate Value Theorem (IVT)• The Extreme Value Theorem (EVT)• Rolle’s Theorem (RT)• The Mean Value Theorem (MVT)

An effective way to teach these theorems is to teach them together by comparing and contrasting the condition of each theorem and which is a consequence of the other.

Students have better retention of them when they invest in that comparison.

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2nd Lab: Estimating and Computing Areas under Curves (see handout two pages)

To find areas under curves using the limit of a Riemann sum approach,

Begin with a finite number n for all three methods: (see handout questions 1 – 4) • Left hand on the interval [2,5] for n = 6• Right hand and• Midpoint Rule

( ) 1f x x

Method Value obtained

Percent Error

Left Hand Rule

Right Hand Rule

Trapezoid Rule

Midpoint Rule

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2nd Lab: Estimating and Computing Areas under Curves (see handout last two pages)

Have students derive the general term for each of the three methods.

For the Left Riemann SumFor the Right Riemann SumFor the Midpoint Sum

Painful but necessary

Proceed in a gradual approach Find limit of computed expression in rational function form as in questions 6Find the limit of a Riemann sum which is already set up as in question 8Set up and find the limit of a Riemann sum from the beginning as in questions 9 and 10.

ix

nL ( 1)ix a i x

nR ix a i x nM 1

2ix a i x

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References

1. Calculus: Early Transcendentals, Briggs & Cochran, 2010, Pearson Education

2. Single Variable Calculus: Early Transcendentals, James Stewart, 6 edition, 2007, Thompson Brooks /Cole 3. Jorge Sarmiento, MATYCNJ Presentation, April 2012, County College of Morris

Dr. Siham AlfredRaritan Valley Community [email protected]

39th AMATYC Annual ConferenceAnaheim, California

October 31- November 3, 2013

mailto:[email protected]

dr. siham alfred raritan valley community college [email protected]

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