Limits
and Their Properties

Erica Reese, Math Instructor

Emma Willard School

Troy, NY

Erica Reese, Math Instructor

Emma Willard School

Troy, NY

Limits and Their Properties
NAME:
___________________________

Learning Choices Folders will be collected Wednesday 9/24

C LAYER: (Choose 2 from each Topic)

Day One Topics:

A Preview of Calculus

o The Tangent Line Problem: Exploration pg. 44

o The Area Problem: Exploration pg. 45

o Exercises on pg. 46: 10, 11

Finding Limits Graphically and Numerically

o An Introduction to Limits: Exploration pg. 47

o Exercises on pg. 53: 1, 2, choose 1 from 3-10, 11-20, 22, 24, choose 1 from 25-36

o Exercises on pg. 53: 2, choose 1 from 3-10, 11-20, 22, 24, choose 3 from 25-36

o Exercises on pg. 55: 41, 42

o Listen to the 12 minute lecture and take notes. (9/17)

o The informal description of a limit; limit exists only if answer is a real number

o Limit fails to exist in Examples 3, 4, and 5

o The formal definition of a limit; why are epsilon and delta needed?

o pg. 54: 21

Evaluating Limits Analytically

o Summarize in three paragraphs three techniques to find limits

o Exercises on pg. 64: Choose 10 problems (2 from each section)

o Exercises on pg. 65: Choose 6 problems (1 from each section) and, 85 or 86

o Listen to the 12 minute lecture and take notes. (9/17)

o Direct Substitution and when we can use it

o Three Strategies for Finding Limits

Day Two Topics:

Continuity and One-Sided Limits

o Exercises on pg. 75: 1, 8-10, 11-20 (x3), 28, 29

o Exercises on pg. 75: 2, 5-7, 12-21 (x3), 27, 30

o Writing / Graphing pg. 77: 72, 74, 76, 86

o Listen to the 10 minute lecture and take notes. (9/17)

o If a Function is Continuous, then what?

o 2 Different Kinds of Continuous

o Testing for Continuity

Infinite Limits

o Exploration on pg. 80

o Exercises on pg. 84: 1-4, 8, 14, 26, 48 or 50

o Listen to the 10 minute lecture and take notes. (9/17)

o Infinite Limits

o When Vertical Asymptotes Occur

B LAYER: (Choose 2)

o Planning to take the AP Calculus Examination? Exercises (see handout)

o Lab Activity 8 (see handout)

o Write your Mathematical Autobiography: (Rough Draft due Monday)

I. Your earliest memories of mathematics or numbers

II. Your elementary and junior high experiences

III. Your high school experiences

IV. Calculator / Computer Experiences

V. Expectations for this year

A LAYER: (Choose 1)

o Absolute Zero is the topic of the example on page 71. This temperature has never been obtained. (Rough Draft due Monday)

I. Summarize how Jacques Charles determined the absolute zero of the Celsius scale.

II. Find the absolute value of the Fahrenheit scale.

III. If no one has ever obtained the lowest temperature ever, find the three closest attempts. (Who? When? Where? typed in newspaper format)

o Take It to the Limit (Rough Draft due Monday)

While limits are a fundamental concept in calculus, the idea of a limit can be found elsewhere. Music, visual arts, advertising, and other areas of popular culture often use the concept. Find an example of a song, poem, picture, or other item and explain how it uses or demonstrates the concept of a limit.

o Archimedes is often given credit for discovering Pi (3.14). Research how he used limits to find the value of Pi. (Rough Draft due Monday)

I. Who is Archimedes? (Who? When? Where?)

II. How did he find the value of Pi? Include several diagrams with your explanation.

III. How is this method related to a limit?

Learning Choices Folders will be collected Wednesday 9/24

C LAYER: (Choose 2 from each Topic)

Day One Topics:

A Preview of Calculus

o The Tangent Line Problem: Exploration pg. 44

o The Area Problem: Exploration pg. 45

o Exercises on pg. 46: 10, 11

Finding Limits Graphically and Numerically

o An Introduction to Limits: Exploration pg. 47

o Exercises on pg. 53: 1, 2, choose 1 from 3-10, 11-20, 22, 24, choose 1 from 25-36

o Exercises on pg. 53: 2, choose 1 from 3-10, 11-20, 22, 24, choose 3 from 25-36

o Exercises on pg. 55: 41, 42

o Listen to the 12 minute lecture and take notes. (9/17)

o The informal description of a limit; limit exists only if answer is a real number

o Limit fails to exist in Examples 3, 4, and 5

o The formal definition of a limit; why are epsilon and delta needed?

o pg. 54: 21

Evaluating Limits Analytically

o Summarize in three paragraphs three techniques to find limits

o Exercises on pg. 64: Choose 10 problems (2 from each section)

o Exercises on pg. 65: Choose 6 problems (1 from each section) and, 85 or 86

o Listen to the 12 minute lecture and take notes. (9/17)

o Direct Substitution and when we can use it

o Three Strategies for Finding Limits

Day Two Topics:

Continuity and One-Sided Limits

o Exercises on pg. 75: 1, 8-10, 11-20 (x3), 28, 29

o Exercises on pg. 75: 2, 5-7, 12-21 (x3), 27, 30

o Writing / Graphing pg. 77: 72, 74, 76, 86

o Listen to the 10 minute lecture and take notes. (9/17)

o If a Function is Continuous, then what?

o 2 Different Kinds of Continuous

o Testing for Continuity

Infinite Limits

o Exploration on pg. 80

o Exercises on pg. 84: 1-4, 8, 14, 26, 48 or 50

o Listen to the 10 minute lecture and take notes. (9/17)

o Infinite Limits

o When Vertical Asymptotes Occur

B LAYER: (Choose 2)

o Planning to take the AP Calculus Examination? Exercises (see handout)

o Lab Activity 8 (see handout)

o Write your Mathematical Autobiography: (Rough Draft due Monday)

I. Your earliest memories of mathematics or numbers

II. Your elementary and junior high experiences

III. Your high school experiences

IV. Calculator / Computer Experiences

V. Expectations for this year

A LAYER: (Choose 1)

o Absolute Zero is the topic of the example on page 71. This temperature has never been obtained. (Rough Draft due Monday)

I. Summarize how Jacques Charles determined the absolute zero of the Celsius scale.

II. Find the absolute value of the Fahrenheit scale.

III. If no one has ever obtained the lowest temperature ever, find the three closest attempts. (Who? When? Where? typed in newspaper format)

o Take It to the Limit (Rough Draft due Monday)

While limits are a fundamental concept in calculus, the idea of a limit can be found elsewhere. Music, visual arts, advertising, and other areas of popular culture often use the concept. Find an example of a song, poem, picture, or other item and explain how it uses or demonstrates the concept of a limit.

o Archimedes is often given credit for discovering Pi (3.14). Research how he used limits to find the value of Pi. (Rough Draft due Monday)

I. Who is Archimedes? (Who? When? Where?)

II. How did he find the value of Pi? Include several diagrams with your explanation.

III. How is this method related to a limit?