Pre-sessionals - Maths 2

Introduction to Derivatives

• Introduction to Derivatives

  • What is a Derivative?
    • Definition of a derivative as the instantaneous rate of change.
    • Notation: \(f'(x)\), \(\frac{dy}{dx}\).
    • Example: Calculate the derivative of \(f(x) = 3x^2 - 2x + 5\).
    • Exercise: Find the derivative of given functions.
  • The Concept of Instantaneous Rate of Change
    • Understanding how derivatives relate to slopes of tangent lines.
    • Example: Interpret the derivative of a position function as velocity.
    • Exercise: Interpret derivatives in real-world contexts.
  • Notation and Interpretation
    • Discuss the meaning of \(f'(x)\) and \(\frac{dy}{dx}\) in context.
    • Emphasize the connection between slope and rate of change.
    • Exercise: Match graphs of functions with their derivatives.

• Derivatives of Linear and Quadratic Functions

  • Finding the Derivative of Linear Functions
    • Deriving the derivative of linear functions.
    • Example: Find the derivative of \(L(x) = 2x + 4\).
    • Exercise: Differentiate linear functions with different coefficients.
  • Finding the Derivative of Quadratic Functions
    • Differentiating quadratic functions using the power rule.
    • Example: Differentiate \(q(x) = -3x^2 + 5x - 1\).
    • Exercise: Derive quadratic functions with various coefficients.
  • Tangent Lines and Rates of Change
    • Connecting derivatives to slopes of tangent lines.
    • Calculating slopes of tangent lines at specific points.
    • Example: Find the equation of the tangent line to \(f(x) = x^2\) at \(x = 2\).

• Basic Rules of Differentiation

  • Power Rule for Differentiation
    • Statement and derivation of the power rule.
    • Example: Differentiate \(g(x) = 4x^3 - 2x^2 + 7x\) using the power rule.
    • Exercise: Apply the power rule to various functions.
  • Constant Rule and Sum Rule
    • Using the constant rule and sum rule to differentiate functions.
    • Example: Find the derivative of \(h(x) = 3 + 2x^2 - 5x\).
    • Exercise: Differentiate expressions involving constants and sums.
  • Differentiating Polynomial Functions
    • Applying differentiation rules to polynomial functions.
    • Example: Differentiate \(p(x) = 6x^4 + 2x^3 - 9x^2 + 5\).
    • Exercise: Differentiate given polynomial functions.

• Applications of Derivatives

  • Finding Maxima and Minima
    • Using derivatives to locate critical points.
    • Identifying maxima, minima, and points of inflection.
    • Example: Find the critical points of \(g(x) = 2x^3 - 9x^2 + 12x\).
  • Tangent Lines as Approximations
    • Using tangent lines for linear approximations.
    • Example: Estimate \(\sqrt{9.2}\) using the tangent line at \(x = 9\) for \(f(x) = \sqrt{x}\).
  • Simple Optimization Problems
    • Applying derivatives to solve basic optimization problems.
    • Example: Find the dimensions of a rectangle with maximum area given a fixed perimeter.

• Conclusion and Recap

  • Summarize the key concepts covered in the lecture.

• Q&A Session

• References (with exercises)