Further Mathematics - Senior Secondary 2 - Differentiation

Differentiation

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TERM: 2ND TERM

WEEK: 5
Class: Senior Secondary School 2
Age: 16 years
Duration: 40 minutes of 4 periods
Subject: Further Mathematics
Topic: Differentiation
Focus: Higher derivatives and differentiation of implicit functions

SPECIFIC OBJECTIVES:
By the end of the lesson, students should be able to:

  1. Identify and understand the concept of higher derivatives.
  2. Differentiate higher order functions.
  3. Differentiate implicit functions.
  4. Apply differentiation to solve real-life problems involving higher derivatives and implicit functions.

INSTRUCTIONAL TECHNIQUES:
• Question and answer
• Guided demonstration
• Discussion
• Practice exercises
• Real-life applications
• Visual aids (charts and diagrams)

INSTRUCTIONAL MATERIALS:
• Whiteboard and markers
• Chart showing areas of application for differentiation (e.g., rates of change in physics, economics, etc.)
• Flashcards with examples of higher derivatives and implicit functions
• Worksheets for practice

PERIOD 1 & 2: Introduction to Higher Derivatives
PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Introduces the concept of higher derivatives. Explains that a higher derivative is the derivative of a derivative. Starts with a function and demonstrates how to find the first and second derivatives.

Students listen and ask questions to clarify their understanding of first and second derivatives.

Step 2 - Higher Derivatives

Uses examples like f(x) = x³ to show how to find the first, second, and third derivatives. For example, if f(x) = x³, f'(x) = 3x², f''(x) = 6x, and f'''(x) = 6.

Students follow the steps to differentiate the function and take notes.

Step 3 - Applications of Higher Derivatives

Discusses the significance of higher derivatives in real-world problems, such as acceleration (second derivative of displacement) and jerk (third derivative of displacement).

Students engage in a discussion about real-life applications of higher derivatives.

Step 4 - Guided Practice

Provides various functions for students to practice finding higher derivatives, such as f(x) = 4x⁴ and f(x) = 5x³ + 2x. Works through examples on the board.

Students work on examples individually or in pairs. Teacher offers guidance as needed.

NOTE ON BOARD

Higher Derivatives: f'(x) = first derivative, f''(x) = second derivative, f'''(x) = third derivative, etc.

Students copy the note on higher derivatives from the board.

EVALUATION (5 exercises):

  1. Find the first and second derivatives of f(x) = x⁴.
  2. Find the first, second, and third derivatives of f(x) = 2x³ - 5x² + 4.
  3. What is the significance of the second derivative in physics?
  4. Find the third derivative of f(x) = 6x⁵.
  5. In what real-life scenario might the third derivative of a function be useful?

CLASSWORK (5 questions):

  1. Differentiate f(x) = x² + 3x.
  2. Find the second derivative of f(x) = x³ + 2x² - x.
  3. Differentiate f(x) = 7x⁴ - 4x².
  4. Find the third derivative of f(x) = 2x⁵ + 3x³.
  5. What is the first derivative of f(x) = 3x² + 2x?

ASSIGNMENT (5 tasks):

  1. Differentiate f(x) = 5x⁴ - 6x² + x.
  2. Find the first and second derivatives of f(x) = x⁵ - x³ + 2x.
  3. Find the third derivative of f(x) = 6x⁶ - 2x⁴.
  4. Discuss an application of the second derivative in economics.
  5. Calculate the second derivative of f(x) = 2x⁴ + x² - 3x.

PERIOD 3 & 4: Differentiation of Implicit Functions
PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction to Implicit Differentiation

Introduces implicit differentiation, explaining that it is used when the function is not explicitly written in terms of x or y (e.g., x² + y² = 25). Guides students on how to differentiate implicitly by treating y as a function of x.

Students listen carefully and take notes.

Step 2 - Implicit Differentiation Process

Works through an example: Given x² + y² = 25, differentiate both sides with respect to x, leading to 2x + 2y(dy/dx) = 0. Solves for dy/dx.

Students follow the example and understand how to differentiate implicitly.

Step 3 - Solving Implicit Functions

Provides more complex examples like x³ + y³ = 6xy and demonstrates how to differentiate both sides and solve for dy/dx.

Students work through similar problems in pairs or individually with teacher guidance.

Step 4 - Guided Practice

Gives various problems for students to practice implicit differentiation, such as x² + y² = 10 and x³ + y³ = 6xy. Teacher assists where necessary.

Students work through problems, asking questions when needed.

NOTE ON BOARD

Implicit Differentiation: 1. Differentiate both sides with respect to x. 2. Apply the chain rule when differentiating y. 3. Solve for dy/dx.

Students copy the steps of implicit differentiation from the board.

EVALUATION (5 exercises):

  1. Differentiate x² + y² = 10.
  2. Differentiate x³ + y³ = 6xy.
  3. Differentiate x² + 2y = 5.
  4. Find dy/dx if x² + y² = 20.
  5. What rule do you apply when differentiating y with respect to x in implicit differentiation?

CLASSWORK (5 questions):

  1. Differentiate x² + y² = 16.
  2. Differentiate x³ + y³ = 3xy.
  3. Find dy/dx for x² + y² = 25.
  4. Differentiate x³ + y³ = 6xy.
  5. Solve for dy/dx in 2x + 3y = 7.

ASSIGNMENT (5 tasks):

  1. Differentiate x² + y² = 50.
  2. Differentiate x³ + y³ = 3x²y.
  3. Differentiate x² + 2y = 5.
  4. Solve for dy/dx if x²y + y² = 10.
  5. Find the derivative of x² + y² = 100.

Teacher’s Note:

  • Use real-life examples (e.g., velocity and acceleration) to show the importance of higher derivatives and implicit differentiation in problem-solving.

Provide extra practice for students who may struggle with the concepts, using simpler examples before moving on to more complex ones.