Assignment 2a- N

Nermin Fialkowski Dr. Monica Kelly AIL 622 6 September, 2018 Assignment 2A- Brainstorm Unit of Study: Limits Standard(s): Mathematics Content Standards for California Public Schools Calculus 1.0 Students demonstrate knowledge of both the formal definition and the graphical interpretation of limit of values of functions. This knowledge includes one-sided limits, infinite limits, and limits at infinity. Students know the definition of convergence and divergence of a function as the domain variable approaches either a number or infinity: 1.1 Students prove and use theorems evaluating the limits of sums, products, quotients, and composition of functions. 1.2 Students use graphical calculators to verify and estimate limits. 2.0 Students demonstrate knowledge of both the formal definition and the graphical interpretation of continuity of a function.

Inquiry Question(s): 1. Before I can reach my desk from the classroom door, I must first walk halfway between the door and my desk. Then I’ll walk another halfway distance, then another halfway, and another halfway. When will I be able to reach my desk? ! " % & !)) 2. How can I find the value of " + $ + & + ' + ⋯ + "** + ⋯ without adding up 200+ numbers? ! 3. What is the value of !,***,***,***,***,***,***? Essential Question(s): 1. How can I predict the behavior of a graph? 2. What conclusions can be made about a graph that never stops growing? 3. What additional conclusions can still be made about a graph that has no value?

Big Ideas: Intuitive approach using Benny & Bertha the Bug As Benny & Bertha “get closer, and closer” to said x-value, how high are they getting (yvalue)? Will Benny & Bertha meet at the same place? Use for both One-Sided Limit and Definition of a Limit The existence of a point is irrelevant for a limit to be possible. What matters is where Benny & Bertha go and if they go to the same place 𝑓(𝑎) does not have to equal lim 𝑓(𝑥) 3→5

Algebracially solving a limit Direct Substition Numberical Value (convergent limit) ! →Vertical Asysmptote (divergent limit) * *

→Indeterminate→Do “algebra” Factor Hole Rationalize Formal definition of continutity 𝑓(𝑎) exists lim 𝑓(𝑥) exists *

3→5

lim 𝑓(𝑥) = 𝑓(𝑎)

3→5

Resources: 1. College Preparatory Mathematics (CPM) Textbook: Pre-Calculus with Trigonometry 2. Calculators TI-84 Plus Desmos Online Graphing Calculator 3. Student Laptops Websites: 1. Pre-Calculus with Trigonometry (https://cpm.org/pct/) 2. Desmos Online Graphing Calculator (https://www.desmos.com/calculator) 3. Desmos Activities (https://teacher.desmos.com/activitybuilder/custom/574de5cdab71b5085a2aad42) 4. Limits Student Lab (http://www.mesacc.edu/~davvu41111/LimitStudent.html#MapleAutoBookmark5)

Print: Homework: 5.2.1 Examine each graph, then evaluate the limit. 1.)

2.)

3.)

Evaluate each limit. You may use the provided graph to sketch the function. 4.) 5.)

Evaluate each limit. 7.)

8.)

Classwork: Limits Sketch each function on the graph. Evaluate each limit. 1.) lim 2𝑥 + 3 2.) lim −√𝑥 + 3

3.) lim 𝑥 " + 3

4.) Given 𝑓(𝑥) = 23 − 3

6.) lim> −𝑥 + 2

3→8"

3→$

3→"

5.) lim −2𝑥 + 5 3→8"

3→8

find lim 𝑓(𝑥) & lim 𝑓(𝑥) 3→8@

3→@

6.) lim 𝑥 $ + 𝑥 " − 𝑥 3→@

8.) lim

3?

3→@ %3 B A$

?

7.) lim

83A"

3→@ 3 B A3A!

9.) lim

3A!

3→@ "3 B A"3A!

10.) Give an example of a limit that goes to 4 as x goes to ∞. Hint: Draw a graph first.

Asymptotes of Rational Functions Based on the asymptotes, match each equation with its graph 1. _____

2. _____

3. _____

4. _____

5. _____

6. _____

7. _____

8. _____

9. _____

10. ____ 11. ____

Homework: 5.2.5 1.) Let 𝑓(𝑥) = E

𝑥 " − 3 𝑓𝑜𝑟 𝑥 ≤ 2 5 − 𝑥 𝑓𝑜𝑟 𝑥 > 2

a. Find limK 𝑓(𝑥) b. Find limL 𝑓(𝑥) c. Find lim 𝑓(𝑥) or explain why it does not exists 3→"

3→"

3→"

2.) Sketch a function g(x) defined for all x such that • lim 𝑔(𝑥) = 3 • •

3→*

lim 𝑔(𝑥) = 4

3→"K

lim 𝑔(𝑥) = 2

3→"L

23 𝑓𝑜𝑟 𝑥 < 2 3.) Let 𝑓(𝑥) = E 4 𝑓𝑜𝑟 𝑥 ≥ 2 a. Find lim 𝑓(𝑥) 3→"

b. lim 𝑓(𝑥) 3→&

Movies/Video Clips/Animations/Visuals: 1. Achilles & The Tortoise: https://youtu.be/skM37PcZmWE 2. Area of a Circle: https://youtu.be/YokKp3pwVFc 3. The Limit Does Not Exist

4. Definition of A Limit lim 𝑓(𝑥) = 𝐿 3→5

5. Limits Student Lab (http://www.mesacc.edu/~davvu41111/LimitStudent.html#MapleAutoBookmark5)

Activities: Limits & Continuity: Desmos Activity https://teacher.desmos.com/activitybuilder/custom/574de5cdab71b5085a2aad42 Differentiated Learning Students will create their own piecewise graphs (like below) and pose various questions about the graph. Students will then quiz their classmates on the answers.

𝑓(−1)

𝑓(1)

lim 𝑓(𝑥)

3→!

lim 𝑓(𝑥)

3→@

lim 𝑓(𝑥)

3→8@

lim 𝑓(𝑥)

3→8%

𝑓(−4)

Strategies: Think-Pair-Share During instructional time Student Differentiation Students create their own piecewise functions and questions associated with their graph Quiz-Quiz-Trade For differentiated Learning activity Station Rotations For review day, one question per station, and students rotate around the classroom in groups Group Work For station rotations Exit Slips At the end of each section Technology in the classroom Laptops for Desmos activity Graphing calculator (Desmos and TI-84) for graphing functions and creating tables to evaluate limits

Assessments: Chapter 5 Test 1.) Evaluate the following limits. If the limit does not exist, explain why. a. lim

3 B A"

b. lim

3→@ 3 ? 8!

3→8@

"3 ? 8%3 B A38&

c. lim

3 B A$3A"

$3 ? A"

3→8@ 8)3 ? AS

𝟏

2.) Given 𝒇(𝒙) = 𝒙8𝟑 + 𝟐 sketch the graph, then evaluate the accompanying limits. b. lim 𝑓(𝑥)

a. Graph the function of 𝑓(𝑥)

3→@

c. limL 𝑓(𝑥) 3→$

d. lim 𝑓(𝑥) 3→$

e. lim 𝑓(𝑥) 3→&

𝒙𝟑 + 𝟐𝒙 − 𝟏 𝒇𝒐𝒓 𝒙 ≤ 𝟏 3.) Given 𝒇(𝒙) = Y 8𝟒𝒙A𝟕 evaluate the following limits. 𝒇𝒐𝒓 𝒙 > 𝟏 𝒙A𝟏 a. lim 𝑓(𝑥) 3→8"

b. lim 𝑓(𝑥)

c. lim 𝑓(𝑥)

3→!

3→$

7.) Given the piecewise function f (x) shown below, evaluate the following expressions. a. lim 𝑓(𝑥)

b. lim 𝑓(𝑥)

c. limL 𝑓(𝑥)

d. limK 𝑓(𝑥)

e. 𝑓(−3)

f. lim 𝑓(𝑥)

3→8@

3→)

3→@

3→)

3→8$