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A student’s first journey into Calculus Siham Alfred

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1. Fifty Word Summary of Presentation: Sharing some simple and some more involved activities I use with my students in Calculus to motivate instantaneous rate of change, the 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.___________________________________________________________________

2. Share with Students who invented Calculus

Students in a calculus class are privileged to learn the ideas of the inventors of Calculus: Isaac Newton (1642, 1727) and (Gottfried Leibnitz (1646, 1716) and the ideas of subsequently many others, beginning with the Greek Archimedes who employed the method of exhaustion. All these Mathematicians’ ideas are alive and we continue to study them. So in a sense, in the classroom, we are having a conversation with these great minds.

3. What is Calculus? The invention of a body of knowledge and models of thought to answer some great questions which lead to understanding and calculation instantaneous rates of change accumulated change and all their applications which help us understand and model the world in motion

Arnold Toynbee (1889- 1975), the famous British Historian was offered the opportunity at the age of 14 to take calculus. He refused and took a history course instead. He later regretted his decision. As he said, he lacked the understanding of the world in motion. His view of it remained static.

4. Give the students a brief overview of what Calculus is about.

Show them the forest before the trees. There are two main processes in Calculus Differentiation and Integration each of which is loaded with techniques, methods, theories and applications. (As the course progresses, it is important to point out to the students where we are at each stage)

CALCULUSThe Study of Change, rates of Change and Accumulated Change

DIFFERENTIATIONINTEGRATION

Definitions, Techniques, methods,Definitions, Techniques, methods

Theorems and ApplicationsTheorems and Applications

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5. On their first journey into Calculus, Are students ready for the Trip?

Generally it is their background in Precalculus that is the impediment to their success in the course not the new ideas in calculus. A diagnostic Precalculus test would be an efficient venue to get to know the students early on in the semester. The test should contain Algebra, Analytic Geometry, Functions and Trigonometry.

There is no extra time to review. Incorporate reviews within the lesson, homework & labsAttached are some sample exercises (1) on the definition of a function, (2) on making connections in trigonometric functions (3) on inverse trigonometric functions which I insert whenever possible for the students to review

6. Knowledge is a network

Professor Robert Davis, of Rutgers University, asked us, his graduate students, one day in class to list all words that came to mind when given the two words “birthday party”. Each one of us wrote some words. Then we shared what we had written: invitations cards, guests, food, birthday cake, drinks, entertainment, music, gifts, balloons and presents. He then asked us, how is it that these two words bring to our minds so many other words. By experience we have connected all these words because we experienced birthday parties and used or applied all these words as a result of our experiences. Then he told us that he was quite sure we all could explain how each of these words we mentioned was related to the word birthday party. He left us with the understanding that true knowledge is a network made out of mental connections brought about by experiences and not a series of abstract isolated words

Then he asked us, what we would like our students to think about when we say the word “derivative”. We came up with the words: instantaneous rate of change, slope of the tangent line, instantaneous velocity, acceleration, tool for optimization, linearization and related rates etc.

He challenged us to provide students with experiences to create opportunities for them to make connections and build a network of knowledge about the derivative or any other topic they set out to learn in order to achieve conceptual understanding.

7. A simple Activity to Engage the Students from Day 1. On the first day of class, ask each student to compute the average velocity of his or her commute to the college. Ask them to record the mileage and time just before they start their engines and again just before they turn off their engines. Then calculate their distance divided by their time in hours.

I share my daily Commute as an example: I drive 14 miles every day to get from my house to the college. It takes me half an hour to get there. My average velocity is 14/(1/2) = 28 mph - Pathetic speed !

Average Velocity =

tan ( ) tan ( ) 14 0 281 02

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

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Writing it symbolically, the on the interval where denotes the distance at time . More questions for the student follow:

a. Did you drive at that velocity from the moment you left your house to the moment you reached the college? Write a paragraph or two and describe what actually happened.

b. How is the instantaneous velocity as shown on the car’s speedometer calculated?

- Students learned how to compute the average velocity and became more aware of their instantaneous velocity. - Students learned that the average velocity is not the same as the instantaneous velocity. - Their instantaneous velocities varied above and below their average velocity throughout their commute.- Some realized that sometimes during their commute they drove at their average velocity.- Students were not surprised when they studied the Mean Value Theorem

In the group discussion about this activity, more questions were raised, some by the students themselves

c. How is the instantaneous velocity calculated in your car as shown on the speedometer? How is the distance calculated? How is the time calculated?

Students calculated the circumference of the wheels of their cars, referred back to the linear Speed and angular speed on a circle which they have learned previously, namely:

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

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

Therefore the linear speed = .

This activity created for them many connections and they referred to it many times during the

semester.

9. Estimating the Slope of the tangent line from a given graph.

Given the graph below, estimate the slope of the tangent line graphically at x =-3, -2, 1, 2, 3 by drawing a tangent line to the curve carefully with a ruler at the x values indicated. Write each point of tangency as an ordered pair, then find a second point on each tangent line. Using the two points estimate the slope of the tangent line

0

0

( ) ( )S t S tAverageVelocityt t

0( , )t t ( )S t

t

v s rvt t

s r

t

s rvt t

r

4

First point is done as an example

The slope of the tangent line at (-3, 0) can be estimated using the point of tangency (-3,0) and the second point (-2,5) or the second point (-4,-5).

Point of tangency

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

Esti.2nd point (-2, 5)Estimated slope =MTAN

5

Est. Equation of tangent line

Do not give equation to students until all estimated slopes are found

Then find the exact slopes using the formal definition of the derivative of the function

whose graph is above and when expanded will yield

5 15y x

1( ) ( 3)( 3)( 1)5

f x x x x

5

This exercise will help the students estimate slopes of tangent lines readily and help them with graphing derivative functions from the graph of given functions, with understanding when functions are increasing and decreasing and how to figure out when functions are concave up or concave down.

10. 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.

We Introduce the concept of a limit of a function defined on an interval containing a, but not necessarily at and follow that experience with numerical calculation and graphical inspection and algebraic calculation of the instantaneous rate of change

Cauchy used in his definition of the limit of a function at a point. He used for difference, the French word for difference between x and . He used for erreur which is the

French word error to indicate how far is the function from the limit L.

When studying the formal definition of the limit, students need to understand two important points: (1) the dynamic of the given epsilon challenge and the delta response and (2) the investigation phase and the proof phase of the proof. An example is shown below.

The definition of , given a (challenge) there exists a( response) >0 such

that if 0<|x-a|< then

Example: For linear functions prove that

Investigation phase: Consider

Proof Phase:

3 21( ) ( 9 9)5

f x x x x

( , )

a

( , ) a

lim ( )x a

f x L

0

| ( ) |f x L

lim ( ) lim( )x a x a

f x mx b ma b

| ( ) | exp ( )| ( ) | si| || || | | |

| || |

f x L Write the ression for f xmx b ma b Distribute the negative gn and simplifymx ma Factor out mm x a Divide by m

x am

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. If ,

then which equals

, which equals

which equals

as required to prove

Also create a table for students to see the challenge and response dynamic.

where and

11. Use Students Prior Knowledge to create new knowledge: Two Examples. 1) Power Rule by Synthetic Division (as an alternative to the Binomial Theorem)

Example 1: Power Rule by Synthetic Division (as an alternative to the Binomial Theorem)

Using Student’s Prior knowledge of factoring or synthetic Division

Using the alternative definition of the derivative of a function differentiable at a point a

, one can prove the power rule for where n is a positive

number, namely, using factoring or Synthetic division. As a motivation first compute the three derivatives

I) , 2) 3)

for , , respectively. Then compute

| |Let

m 0 | |

| |x a

m

| || |m x a

| |mx ma

| ( ) |mx b ma b

| ( ) |f x L

0 | |m

12

12 | |m

110

110 | |m

( )ma b f x ma b | | | |a x a

m m

( )f x

( ) ( )'( ) limx a

f x f af ax a

( ) nf x x1'( ) nf x nx

3 3

'( ) limx a

x ag ax a

4 4

'( ) limx a

x ah ax a

5 5

'( ) limx a

x at ax a

3( )g x x 4( )h x x 5( )t x x

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Q.E.D.

Example 2: Product and Quotient rules for Derivatives Using Logarithmic Differentiation Most calculus books use the formal definition of the derivative to prove the product and quotient rules. For a restricted class of functions the product and quotient rules can be easily obtained using logarithmic differentiation

The Product Rule

such that for all x in their domains, and

are differentiable functions of x, then

. Taking the derivative of both sides yields

The Quotient Rule

are defined as above.

. Differentiating both sides,

. Multiply both sides by to get

'( ) 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

( ) ( ) ( )If f x u x v x ( ) 0 ( ) 0u x and v x

u and v

ln ( ) ln[ ( ) ( )] ln ( ) ln ( )f x u x v x u x v x

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

( ) '( ) ( ) '( ) ( ) '( ) ( ) '( )'( ) ( ) ( ) ( )( ) ( ) ( ) ( )' ( ) '( ) ( ) '( ).

f x u x v xf x u x v x

v x u x u x v x v x u x u x v xf x f x u x v xu x v x u x v xf x v x u x u x v x

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

( )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

( )f 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

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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.

12. The Four Important Theorems of Differential Calculus

When teaching the four important theorems of differential calculus. IVT, EVT, RT and MVT. Ask about the similarities and differences of the requirement of each theorem. I find that it is easier to learn them together by comparing and contrasting them than by learning each one alone. See attached Handout

13. Areas under a curve

When teaching areas under curves start by finding the area under a linear function, with a non-zero slope, which yields the area of a trapezoid and then the area under a step function. Handling the partition of the interval and finding the area under the curve becomes much easier.When finding the area using the limit of a sum, attempt that in three steps.See attached handout.

14. Included List of Handouts

I. Class activity: On the definition of a functionII. Class activity: On Inverse Trigonometric FunctionsIII. Class activity: Creating a Concept MapIV. Lab : Four Existence Theorems in Differential CalculusV. Lab Estimating and Computing Areas Under the Curve

15. Resources

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

Single Variable Calculus: Early Transcendentals, James Stewart, 6 edition, 2007, Thompson Brooks /Cole

Jorge Sarmiento, MATYCNJ Presentation, April 2012, County College of Morris(Product and Quotient Rule using logarithms).

2( ) '( ) ( ) '( )'( ) [ ( )]

v x u x u x v xf x v x

9

Class Activity I.On the Definition of a Function

Done individually or in groups of two

Write the definition of a function

Given the 10 functions below, state the domain, the rule, the range of the function then check the required uniqueness property: for each element in the domain, there is a unique element in the range.

1.

The Domain The Rule The Range Check Uniqueness

2.


10

3.


4. The following table depicts the number of students who passed with a B or better in Calculus I class in the years from 2004 to 2010

x 2004 2005 2006 2007 2008 2009 201012 9 9 8 7 13 7


5.


6. The function is given by the graph below

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7. The set of ordered pairs where each ordered pair consists of a student’s Registration number, and his or her height.


8. A set of 52 ordered pairs where each ordered pair depicts the number of the week in the year and the total food expenditure for that week, example of an ordered pair would be (week 6, $152.87)


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9.

Use the graph to answer question 9


10.


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11. Given the two functions: , where x is a real number and where x is an integer. Graph both functions on the planes provided. Then determine if the functions are the same or different and provide a valid reason for your answer.

Graph of Graph of

Same or different? ____________ Because:

When are two functions equal?

12. Write a summary statement of what you and/or your group learned from this activity. Be sure to include a word about function representation and why uniqueness is required in its definition. Attach additional paper if needs be.

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Activity II. On Inverse Trigonometric Functions

1. Graph the restricted sine function and its inverse on the two separate planes given below. Then write the domain and range for each function below its graph where indicated

Graph of restricted sine Graph of inverse sine

Write its domain and range Write its domain and range

Domain__________________ Domain___________________

Range___________________ Range____________________

2 Show how you obtain the derivative of

3. Use additional papers with the same format as above to answer questions 1 and 2 for

, . and

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4. Alternative way to find the derivative of . We proved in class the

trigonometric identity: for . Use the identity to find the derivative of the cosine inverse of x

5. Similarly, prove and use the identity: to find the derivative of

6. Prove and use the identity: to find the derivative of

Activity III: Concept Map

Create a Concept Map for the terms below then explain to your group partner the logical connections of your concept map, namely why you ordered it the way you did.

Functions Differentiable functionsRelationsContinuous FunctionsIntegrable Functions

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Activity IV

Lab on the Four Existence Theorems in Differential Calculus

Applications on Continuity

I. The Intermediate Value Theorem (IVT)

1. State the Intermediate Value Theorem (IVT) and give a graphical illustration of the theorem showing all pertinent information.

2. Prove that the function f(x) = has a root in the interval [2, 3]

3. Let f(x) = . Although f (2) = - 4 and f(4) = 8 there is no value c such that f(c) = 0. Does that contradict the IVT? Explain.

4. Suppose $8000 is invested in a savings account with an annual interest rate r compounded monthly for 10 years (120 months). The amount of money in the account after 10 years is

. Using the Intermediate Value Theorem to show that there is a value r in (0, .04) that allows you to reach your savings goal of $10,000.

Graph and approximate on the graph the interest rate r that is required to reach your goal of $10,000

The Extreme Value Theorem

5. State the Extreme Value Theorem (EVT) and give a graphical illustration of the givens in the theorem.

6. Let g(x) = be a continuous function defined on the closed interval

[ , 4]. Show that g(x) satisfies the Extreme Value Theorem. What are the maximum and minimum values of g(x)?

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7. Given g(x) = is continuous and defined on [3, ). g(x) has no minimum value on the interval [3, ). Does this function contradict the EVT? Explain.

Application on Differentiability

III Rolle’s Theorem (RT)

8. State Rolle’s Theorem and draw a graph to illustrate all the givens in the theorem.

9. Verify that h(x) = defined on [0,2] satisfies Rolle’s Theorem and find all values c that satisfy the conclusion of Rolle’s Theorem.

10. Two runners start a race at the same time and finish in a tie. Prove that at sometime during the race they have the same velocity. (Hint: Consider f(t) = g(t) – h(t) where g and h are the position functions of the two runners.)

IV The Mean Value Theorem (MVT)

11. State the Mean Value Theorem.

11. Use a graphing utility to Graph the function T(x) = x + on the interval [1,8]. Draw the secant line from (1, T(1)) and ( 8, T(8)).

Find c on the x axis in the interval (1,8) such that c satisfies the conclusion of the Mean Value Theorem.

Draw the tangent line to T at the point (c, T(c)) where x = c that you found.

What is the relationship between the secant line you drew and the tangent line at (c, T(c))?

12. Given S(x) = 1 - defined on [-1, 8]. Show that there is no value c in [-1,8] that satisfies the Mean Value Theorem. Does this function contradict the Mean Value Theorem? Explain.

13. If D(x) = x + 2cosx represents the distance traveled by a particle from x = 0 to x = 2 . Find an x value in (0, 2 ) for which the instantaneous velocity equals the average velocity.

14. Two stationary patrol cars equipped with radar are 6 miles apart on a highway. A truck passes a patrol car and its speed is clocked at 55 mph. 5 minutes later when the truck passes the second patrol its speed is clocked at 50 mph. Prove that the truck driver must

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have exceeded the speed limit of 55mph sometimes during the 5 minutes.

15. Which of the four theorems you learned require the function to be continuous on a closed interval [a,b]?

16. Which of the four theorems you learned require the function to be differentiable on an open interval (a,b)?

17. Which of the four theorems you learned require f(a)= f(b)?

18. Which of the four theorems you learned may help you find intervals where the function has a root?

Activity V. Lab: Estimating and Computing Areas under Curves

Activity 1 Approximating Areas under Curves

1. Find the area under the curve bounded by the x axis and on the interval [3, 7]. 2. Find the area under the curve of the greatest integer function [x] on the interval [0,6].

3. Estimate to four decimal places the area under the curve from [2, 5] using , the Left Hand Sum for n = 6. Show all your work. Draw the region and identify the rectangles

If the exact area is , find the percent error in the Left Hand Sum estimate

4. Estimate to four decimal places the area under the curve from [2,5] using , the Right Hand Sum for n = 6

If the exact area is , find the percent error in the Right Hand Sum estimate. Which method gave a better estimate of the area under the curve?

5. Find the average of the left hand sum and the right hand sum. This average is equivalent to the Trapezoid Method. Find its percent error.

6. Estimate to four decimal places the area under the curve from [2, 5] using, the Sum by the Midpoint Rule for n = 6. Show all your work. Draw the region and identify the rectangles

If the exact area is , find the percent error in the Sum by the Midpoint Rule7. Which of the four methods used gave you the least error.

Method Value obtained Percent ErrorLeft Hand RuleRight Hand Rule

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Trapezoid RuleMidpoint Rule

Activity 2: Finding Areas under Curve as a Limit of a sum

8. Find

9. Suppose that is defined on a closed interval which is divided into n equal subintervals

of length . If is a point in the ith subinterval for i= 1,2,3, …,n

Then the Riemann Sum of on is . Show that:

For Left Riemann Sum ,

For Right Riemann Sum,

For Midpoint Riemann Sum,

Method 1 2 3 … iLeft = +(1-1) +(2-1) +(3-1) +(i-1)Right Skip . + +2 +3 +iMidpoint

10. The sum gives the area under the curve defined on the interval {a, b}, in terms of n, where n is a fixed but arbitrary integer.

What is =? What is =? Find

11. Use the limit process to find the area of the region between the graph of from [1, 4] (No credit will be given using any other method) Hint: let n be fixed but arbitrary. Begin by finding the important building blocks

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Then write the sum and simplify it. Finally take the limit as n approaches infinity of

the Riemann Sum, namely

12. Use the limit process to find the area of the region between the graph of from [1, 3] (No credit will be given using any other method)

Hints: let n be fixed but arbitrary. What is , what is , and what is ?=

13. Powers of x by Riemann Sums: Consider the integral where p is a positive integer.a. Write the left Riemann sum for the integral with n subintervals.b. Use part a to prove the fact(proved by the 17th Century mathematicians Fermat and Pascal)

that

(Source for problem 11: Calculus Early Transcendental by Briggs &

Cochran, Addison Wesley, 2010 p. 334)

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