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Learning Guide Unit 2 -MATH 1211
Calculus (University of the People)
Studocu is not sponsored or endorsed by any college or university
Learning Guide Unit 2 -MATH 1211
Calculus (University of the People)
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MATH 1211-01 Calculus - AY2023-T3
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MATH 1211-01 - AY2023-T3
2 February - 8 February
Learning Guide Unit 2
Learning Guide Unit 2
Learning Guide Unit 2
Overview
Unit 2: Limits and Continuity - Limit of a function, Limit laws, Continuity/Discontinuity of
a function.
Topics:
A Preview of Calculus
The Limit of a Function
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The Limit Laws
Continuity and Discontinuity of a Function
Learning Objectives:
By the end of this Unit, you will be able to:
1.
Examine how the slope of a tangent line is connected to the average rate of a function.
2.
Recognize the slope of the tangent line is the limit of the slopes of the secant lines 3.
Examine the area problem and how it connects to the concept of integral. 4.
Determine the limit of a function.
5.
Evaluate the limit of a function by implementing different algebraic tools. 6.
Identify the three conditions for continuity of a function at a point.
7.
Evaluate intervals in which the function is continuous.
Tasks:
Peer assess Unit 1 Written Assignment
Read the Learning Guide and Reading Assignments
Participate in the Discussion Assignment (post, comment, and rate in the Discussion Forum)
Make entries to the Learning Journal
Take the Self-Quiz
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Introduction
Image credit: NASA
We begin this unit by examining why limits are so important. The limit is central to Calculus because all the theories of how to find Related Rates, Shape of a Graph, Area of a Region, Volume of an Object, Approximations. Optimizations, and all (I can keep going!) depends on the concept of limit. My 9
th
grade teacher in India told us, “if you do not understand the concept of Limits then your learning of mathematics will be limited.”
Next, we will describe how to find the limit of a function at a given point. Many functions will not have limits at points where the function is discontinuous. If the function is not continuous Downloaded by Whitney turner (ct8682126@gmail.com)
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then the concept of limit fails. But there are ways to help it if the discontinuity is a removable discontinuity.
This unit has been created in an informal, intuitive fashion, but this is not always enough if we need to prove a mathematical statement involving limits. The proofs of limit theorems are tedious and need a deeper understanding of the concepts. For example, why and how the limit of the slope of the secant line becomes the slope of the tangent line at a point. Intuitively we are making h becomes smaller and smaller as we write h→0.
Reading Assignment
1. Herman, E. & Strang, G. (2020). Calculus volume 1
. OpenStacks. Rice University.
https://openstax.org/books/calculus-volume-1/pages/2-introduction
Read Chapter 2, pages 125-209 o
sections 2.1, 2.2, 2.3, 2.4, and 2.5
Review the completed examples to understand the concepts and then complete some problems on your own. Review and practice will contribute to your success in this course.
As you read through the textbook, you should attempt the problems indicated below within each section. Use the book and other resources to understand how to answer the problems. You are also welcomed to ask about these in the discussion forum.
o
2.1 1, 3
o
2.2 30, 31, 35, 59-64, 77
o
2.3 83-85,90,102, 107, 117
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o
2.4 135, 136, 139, 140, 145, 153, 154, 157
o
2.5 177, 188, 192
2. Limits: An introduction
[Video]. (n.d.). Khan Academy. https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ab-1-
2/v/introduction-to-limits-hd
3. Limits to define continuity
[Video]. (n.d.). Khan Academy. https://www.khanacademy.org/math/in-in-grade-12-
ncert/xd340c21e718214c5:continuity-differentiability/xd340c21e718214c5:continuity-at-a-
point/v/limits-to-define-continuity
Discussion Assignment
In the discussion forum, you are expected to participate often and engage in deep levels of discourse. Please post your initial response by Sunday evening and continue to participate throughout the unit. You are required to post an initial response to the question/issue presented in the Forum and then respond to at least 3 of your classmates’ initial posts. You should also respond to anyone who has responded to you. Downloaded by Whitney turner (ct8682126@gmail.com)
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Discussion prompt:
If an n-sided regular polygon is inscribed in a circle of radius r, as shown in the figure below, then n-isosce
Based on the statement and figure above answer the following:
1. Express h and the base b of the isosceles triangle shown in terms of θ and r
.
2. Express the area of the isosceles triangle in terms of θ and r
. Use trig identities as needed.
3. Describe what happens as n goes to infinity, (notice the polygon fills the circle, the angle θ goes to zero)
4. Use special limit rules to discuss your response, you may use any graphing tool to support your response.
Your Discussion should be a minimum of 250 words in length and not more than 450 words. Please include a wo
APA standard, use references and in-text citations for the textbook and any other sources. Learning Journal
Assignment instructions:
Read the reading assignment Section 2.3 and section 2.4 on limit laws, continuity, and the
three types of discontinuity, Downloaded by Whitney turner (ct8682126@gmail.com)
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Herman, E. & Strang, G. (2020). Calculus volume 1. OpenStacks. Rice University. Calculus_Volume_1_-_WEB_68M1Z5W.compressed.pdf (uopeople.edu)
Provide your answers to the following problems. Write each step clearly when you answer each question.
Read the rubric on how you are going to be assessed for grading. 1. Use the limit laws to solve the problem below.
2. Explain the continuity of a function at any point. Explain the procedure to check continuity using a simple example.
3. A rock is dropped from a height of 16 ft. It is determined that its height (in feet) above ground t seconds later (for 0≤t≤3) is given by s(t)
=-2
t
2 + 16. Find the average velocity of the rock
over [0.2,0.21] time interval. 4. Evaluate each of the following limits. Identify any vertical asymptotes of the function Downloaded by Whitney turner (ct8682126@gmail.com)
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(i) (ii) (iii) 5. Evaluate the trigonometric limit 6. Find the value of k that makes the following function is continuous over the given interval.
7. Determine at the point 5, if the following function is discontinuous. Classify any discontinuity as jump, removable, infinite, or other.
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You will be assessed by your instructor using the available on the assignment in the course homepage.
Self-Quiz
The Self-Quiz gives you an opportunity to self-assess your knowledge of what you have learned so far.
The results of the Self-Quiz do not count towards your final grade, but the quiz is an important part of the University’s learning process and it is expected that you will take it to ensure understanding of the materials presented. Reviewing and analyzing your results will help you perform better on future Graded Quizzes and the Final Exam.
Please access the Self-Quiz on the main course homepage; it will be listed inside the Unit
.
Checklist
Peer assess Unit 1 Written Assignment
Read the Learning Guide and Reading Assignments
Participate in the Discussion Assignment (post, comment, and rate in the Discussion Forum)
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Make entries to the Learning Journal
Take the Self-Quiz
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