Principles of Calculus

Curriculum details

Strand

Calculus (Strand 3)

Sub-strand

Principles of Calculus (3.1)

Content standard

2.3.1.CS.1 - Determine the appropriate rule to use in finding the derivative of a function and relations. 2.3.1.LO.1 Determine the appropriate rule and use it to find the derivative of a function. 2.3.1.LO.2 Estimate the area under a curve using the trapezoid rule.

Indicator

2.3.1.LI.5 - Generalise the behaviour with respect to the slope of a moving object along a curve.

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.388: 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. 388

Exemplars (from the NaCCA curriculum)

Talk for Learning, Think-pair-share, Experiential Learning, and Group Work/Collaborative Learning.
Activity 1: Classify the behaviour of a curve In pairs or groups task, learners to discuss how to determine the slope of a curve at a point and use it to classify its behaviour.
Activity 2: Derivative of Exponential and Logarithm function Learners work in convenient groups (ability, mixed-ability and mixed gender) to find and classify slopes of a curve at a point. Hint: Given a function 𝑦 = 𝑓(𝑥) if, <A
- | > 0 then the slope of 𝑦 = 𝑓(𝑥) is positive at 𝑥 = 𝑐 <0 04; <A
- | <0 then the slope of 𝑦 = 𝑓(𝑥) is negative at 𝑥 = 𝑏 <0 04'
- <A | =0 then the slope of 𝑦 = 𝑓(𝑥) is zero and (d, f(d)) is a turning point or a <0 04< saddle point
Example 1 Find the slope and classify the behaviour of the curve at the point. State whether the slope is positive or negative or a turning point. !
- 𝑓(𝑥) = at 𝑥 = 4 (0*")
- 𝑓(𝑥) = 4𝑥 − 7𝑥 ) at 𝑥 = −3
- 𝑓(𝑥) = (2𝑥 + 5𝑥 . )(𝑥 − 2) at 𝑥 = 1 ()0$.0 ( )(0*))
- 𝑓(𝑥) = at 𝑥 = −2 (0*")
- 𝑓(𝑥) = (4𝑥 − 7𝑥 )(3𝑥 − 9) !( at 𝑥 = 0 )
- 𝑓(𝑥) = (3𝑥 − 4√𝑥) !( at 𝑥 = 2
- 𝑓(𝑥) = (3𝑥 − 9) !( at 𝑥 = 1.5
Example 2 Given that the slope of the curve 𝑓(𝑥) = cx ) + 𝑑 + 4 at 𝑥 = 2 is 5 and turns at x=1, find the zeros and the turning point of 𝑓
Teaching and Learning Resources:
- Reading resource
- Colour pens
- Notebook
- Graph sheets
- Mathematical sets
- Technological tools
Assessment (2.3.1.AS.5). The document marks these depth-of-knowledge levels for this indicator: Level 2 Skills of conceptual understanding; Level 4 Extended critical thinking and reasoning.