Principles of Calculus · SHS1 Additional…

Lesson notes

Learning Objectives

Indicator: 1.3.1.LI.3 - Distinguish between continuous and discontinuous functions near an input value on its domain and investigate them with the use of technology or any other means appropriate.

By the end of the lesson, learners can:

State the three conditions required for a function f to be continuous at a point a.
Determine, both graphically and algebraically, whether a function is continuous or discontinuous at a given point on its domain.
Identify the point(s) or interval(s) of discontinuity of a function from its graph and from its algebraic expression.
Classify a range of functions (polynomial, rational, absolute value, piecewise) as continuous or discontinuous at specified input values, justifying each answer using the three conditions.
Use GeoGebra (or graph paper as an alternative) to investigate and verify points of continuity and discontinuity of given functions.

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Curriculum details

Strand: Calculus (Strand 3)
Sub-strand: Principles of Calculus (3.1)
Content standard: 1.3.1.CS.1 - Demonstrate understanding of the limit of a function, investigate the behaviour of a function near a value in its domain and establish the derivative of a function. 1.3.1.LO.1 Describe graphically and algebraically the behaviour of the function about an input value and determine its derivative.
Indicator: 1.3.1.LI.3 - Distinguish between continuous and discontinuous functions near an input value on its domain and investigate them with the use of technology or any other means appropriate. Collaborative learning, Discussions, Experiential Learning, and initiate Talk for Learning.
Suggested placement: Semester 2, Week 9 (Week 29 of the year)
Our suggestion, laid out in curriculum order across three terms of twelve weeks. NaCCA does not fix the week, so follow your school's scheme of learning.
Source document note: The official curriculum document has a fault in this entry, so the curriculum text above is transcribed exactly as printed:

exemplars - p.192: adjacent duplicated CambriaMath characters were collapsed, but this PDF also maps some equation glyphs to the wrong letter or operator; verify every mathematical expression in this row against the rendered page
Curriculum reference: NaCCA curriculum document, p. 192

Exemplars (from the NaCCA curriculum)

Learning Experience: Learners working in mixed-ability groups investigate and classify various functions into continuous and discontinuous functions and relate these to things around them.
Activity 1 (Continuous and Discontinuous functions) Learners are to work in groups to identify the point(s)/interval(s) of continuity or discontinuity of a function on a graph and algebraically.
Definition (Continuous at a Point) A function f is continuous at a point a if 𝑓(𝑥) = 𝑓(𝑎) Theorem: If f(x) is continuous at a, then the following three conditions hold;
- f(a) is defined,
- 𝑓(𝑥) must be defined (thus, the left and right limits must be the same and
- 𝑓(𝑥) = 𝑓(𝑎)
Example:

A worked mathematical visual containing displayed two-dimensional notation and its visible labels.

Example 1: Find the interval(s)/point(s) 𝑔(𝑥) (Refer to Fig. 1) is(are) continuous or discontinuous Solution: The function 𝑔 is discontinuous at x= -7 or x=-4 because there are jumps at these points.
Example 2: Determine whether the following functions are continuous at x = 2. Why? a. f(x)= 3x + 1 0*, b. 𝑓(𝑥) = ! 0 *,
- ℎ(𝑥) = |𝑥|, 𝑥 ∈ 𝑅 Solution: h is continuous for all, 𝑥 ∈ 𝑅
Activity 2: Learners are to work in groups and discuss how to use GeoGebra/technological tools or any appropriate means to investigate and identify the points of discontinuity: ! - 𝑓(𝑥) = ! 𝑓 5 ≤ 𝑥 ≤ 6 0 *"0*,
NB:
- Learners should be made to understand that from the concepts of limit, the value f(c) plays no role in the definition of the limit of 𝑓(𝑥) as 𝑥 tends to 𝑐. We do not even assume that 𝑓 is defined at 𝑐.
- A definition of "limit" that requires f to be defined at c would not apply to many important applications. Even when 𝑓(𝑐) is defined, the value 𝑓(𝑐) does not have to be related to 𝑓(𝑥) in any way. In a sense, 𝑓(𝑥) is what we anticipate that 𝑓(𝑥) will equal at 𝑥 = 𝑐, not necessarily what 𝑓(𝑥) actually equals when 𝑥 = 𝑐.
- When we drive along a high-speed road, we frequently cannot see very far ahead. We learn to drive as though the portion of the road immediately beyond our vision is the obvious extension of the portion of the road that we can see. What we are doing is computing the limit of what we see. Usually, the actual state of the road coincides with what we have anticipated it will be (that is, the road is continuous). If the road has been damaged, or if a bridge is out, then the actual state of the road does not coincide with what we have anticipated (the road is discontinuous). In this section, we develop mathematical analogues of these ideas.
Teaching and Learning Resources:
- GeoGebra
- PhET
- Technology tools
- Mathematical sets
- Calculators
- Learners textbooks
- Graph sheets
Assessment (1.3.1.AS.3). The document marks these depth-of-knowledge levels for this indicator: Level 3 Strategic reasoning; Level 4 Extended critical thinking and reasoning.