A Student S First Journey Into Calculus Siham Alfred

A student’s first journey into Calculus
Siham Alfred

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.
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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.
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 laterregretted his decision. As he said, he lacked the understanding of the world in motion. His view of it remained static.

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)

CALCULUS
The Study of Change, rates of Change and Accumulated Change
DIFFERENTIATION / INTEGRATION
Definitions, Techniques, methods, / Definitions, Techniques, methods
Theorems and Applications / Theorems and Applications

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 reviewswithin the lesson, homework & labs
Attached 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

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.

A simple Activity to Engage the Students from Day 1

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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 =

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

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
.
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.
- Theirinstantaneous 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
isthe 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

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

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.

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 fordifference, the French word for difference between x and. He usedforerreurwhich 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:
. 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

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

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.

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

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.

Included List of Handouts

I.Class activity: On the definition of a function

II.Class activity: On Inverse Trigonometric Functions

III.Class activity: Creating a Concept Map

IV.Lab: Four Existence Theorems in Differential Calculus

V.Lab Estimating and Computing Areas Under the Curve

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).

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.