SHS1 Additional Mathematics · Semester 2 Week 11: Applications of Calculus

Lesson notes

Learning Objectives

Indicator: 1.3.2.LI.1 - Use knowledge of differentiation to determine the equation of tangents and normal to curves at a given point.

By the end of the lesson, learners can:

Calculate the gradient of a tangent to a curve at a given point by substituting the x-coordinate into the derived function.
Use the point-slope form of a straight line to write the equation of a tangent when the gradient and point are known.
Determine the gradient of a normal using the perpendicular relationship m₁ × m₂ = -1.
Derive the equation of a normal to a curve at a given point.
Verify their tangent and normal equations by checking that the point lies on both the curve and the line.

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Curriculum details

Strand: Calculus (Strand 3)
Sub-strand: Applications of Calculus (3.2)
Content standard: 1.3.2.CS.1 - Demonstrate the ability to apply differentiation to find equations of tangents, normal to a curve and rate of change. 1.3.2.LO.1 Determine the equation of tangents and normal to a curve at a given point
Indicator: 1.3.2.LI.1 - Use knowledge of differentiation to determine the equation of tangents and normal to curves at a given point.
Suggested placement: Semester 2, Week 11 (Week 31 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.207: 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. 207

Exemplars (from the NaCCA curriculum)

Collaborative learning, Experiential Learning, and initiate Talk for Learning.
Learning Experience: Learners working in mixed-ability groups explore ways of finding the equation of the tangent to a curve.
Activity 1: Think-pair share: Learners in their mixed-ability groups use their previous knowledge of finding the equation of a line to investigate and discuss how to determine the equation of a line tangent to a curve.
Example 1: Find the equation of the tangent line to the curve y=6x 2 at the point (1,6)
Solution: Step 1: Find the slope of the tangent: <A = 12𝑥, so at (1,6), the slope of the tangent will be 𝑚 = 12(1) = 12 <0 Step 2: Using the knowledge in finding the equation of a line, we have, 𝑦 = 12𝑥 − 6
Activity 2: Group work/Collaborative learning: Task learners in their mixed-ability groups to use their previous knowledge of finding the equation of a line to find the equation of a line normal to a curve y=6x 2 at the point (1,6)
Note: Learners must understand that the tangent line to a circle is always perpendicular to the radius of a circle at a point. If the tangent line passes through a point, then the circle lies on one side of the circle. The tangent line passes through a point and intersects at only that point.
Teaching and Learning Resources:
- GeoGebra and PhET
- Technology tools, mathematical sets and calculators
- Learners' textbooks and graph sheets curriculum
Assessment (1.3.2.AS.1). The document marks these depth-of-knowledge levels for this indicator: Level 1 Recall; Level 4 Extended critical thinking and reasoning.