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CAPE PURE MATHEMATICS. UNIT 1 WORKSHOP. SEPTEMBER 20 – 22, 2017. OUTLINE. LIMITS & CONTINUITY DIFERENTIATION FROM FIRST PRINCIPLES LOCI. Some Factors affecting Lessons. Time Syllabus is long, use Summarized Handouts/Videos etc. to speed up the class.
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CAPE PURE MATHEMATICS UNIT 1 WORKSHOP SEPTEMBER 20 – 22, 2017
OUTLINE • LIMITS & CONTINUITY • DIFERENTIATION FROM FIRST PRINCIPLES • LOCI
Some Factors affecting Lessons • Time • Syllabus is long, use Summarized Handouts/Videos etc. to speed up the class. • Do not reinvent the wheel, PowerPoint presentations/activities are available online for easy downloads as well as editing for your purposes. • Basic Mathematics • Fractions, Manipulating the Algebra impact how the students perform with the advanced topics. ….More Practice?
UNIT 1 MODULE 3 LIMITS & CONTINUITY
KEY AREAS • Understanding what a limit it and when it exists • Ways to Evaluate Limits • Substitution • Factorization • Conjugate • Limits to Infinity • Special Limit (Limit of Sine) • Continuity
Misconception Students often have difficulties over whether the limit can actually be reached. Or that the limit as x approaches a certain value is affected by the value of the function there. Students put the value in the function, find that it is undefined at that point and conclude limit does not exist.
INVESTIGATION: CONCEPT OF A LIMIT Fill in the following table: What do you think should be here?
LIMITS – Definition WHAT IS A LIMIT? • A limit is the intended height of a function at a given value of x,whether or not the function actually reaches that height at the given x. • As x gets closer to a , f(x) approaches b. b a
Examining What is happening at x = -2? Why? -5 Reinforce previous knowledge: What is the domain of g(x)? Remember, the function doesn’t actually have to exist at a certain point for a limit to exist—the function only has to have a clear height it intends to reach.
Misconception How to determine the let hand limit and the right hand limit and therefore its impact on determining when a limit exist.
Right- Hand Limit Approach a from the right hand side, the function gets closer to L L a
Left - Hand Limit Approaching a from the left hand side, the functions gets closer to M M a
When does a limit exist? • The value obtained approaching the limit from the left must be equal to the value obtained from approaching the limit from the right. That is,
INVESTIGATION 2 THE ACTIVITY: (TO FURTHER CLARIFY MISCONCEPTIONS – EMPHASIZING WHAT SHOULD BE THERE) Make a four stations around the classroom with each one having 3 or 4 different functions represented either graphically or algebraically. The goal for the students to determine the left, right and overall limits of each function. For the tabular limits cover the approached x-value of the function in the tables with coloured paper. Ask them to first determine the left, right and overall limits before lifting up the coloured piece of paper. Then they are to lift it up and check out what the behaviour of the function is like at the approached point, and see how their answers change.
INVESTIGATION 2 CONTINUED After everyone passed through all the stations, have a discussion. Discussion Questions: • What changed when you lifted up the paper and what didn’t • What was the point of the activity?
LIMIT SENTENCES – 5 mins Group Activity In groups of four: • Using the limit sentence words, create true statements or sentences that correspond to the graphs provided. Each of the cards will be used once and only one. • When you are finished, check your work with another group."
Problem - Evaluating Limits Poor algebra skills......
where k is a constant Rules of Limits Certain rules allow you to manipulate and solve limit problems. Rule 1 Rule 2 Rule 3 Rule 4 Rule 5 Rule 6 as long as
Evaluating Limits • Direct Substitution • Factorising Method • Conjugate Method
Direct Substitution Direct substitution is the first method to attempt when evaluating a limit. Examples: 1) 2)
Factorization & Conjugate Methods Indeterminate forms occur when substitution in the limit results in 0/0. In such cases either factor or rationalize the expressions. Notice form Ex. Factor and cancel common factors
Limits at Infinity For all n > 0, provided that is defined. Divide by Ex.
Limits to Infinity • When the degree of the function is greater than 0, the limit is infinity or negative infinity, • When the degree of the functions is less than 0, the limit is 0. Eg • For rational functions , if • the degree of P is less than the degree of Q the limit is 0. • The degree of P and Q are the same then divide the coefficients of the terms with the largest exponent. • The degree of P is greater than the degree of Q, then the limit is positive or negative infinity. This is determined by looking at the signs of the terms with the largest exponent
FUN ACTIVITIES WHILE INCORPORATING THE OBJECTIVES Limits Activity Joke Review Sheet
REVIEW OF LIMITS See activity
Theorem 1: Limit of Sine Note, there is nothing special about the variable x, but the angle that appears in the numerator should be the same value in the denominator.
Evaluate The problem is that the ‘angle’ or ‘argument’ is not the same in the numerator and denominator.
Evaluate In order to use the theorem, We need to divide both the numerator and denominator by x
Continuity A function f is continuous at the point x = a if the following are true: f(a) a
Continuity A function f is continuous at the point x = a if the following are true: f(a) a Graphs can be drawn without lifting the pencil from the paper
