Pre-sessionals - Maths 2
Contact
- Name: Domenico Mergoni
- Email: d.mergoni -at- lse.ac.uk
- Work: London School of Economics
Tip
Internet is a great resource. Use it. Some resources I like:
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.
- What is a Derivative?
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 \(x = 2\).
- Finding the Derivative of Linear Functions
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.
- Power Rule for Differentiation
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.
- Finding Maxima and Minima
Conclusion and Recap
- Summarize the key concepts covered in the lecture.
Q&A Session