This course on Applications of Calculus aims to help learners understand the behavior of functions at distinct points and over intervals. By mastering calculus, students will gain powerful tools to visualize function characteristics and solve problems, especially in handling equations related to derivatives. The course covers topics such as maximum and minimum points, curve sketching, volumes of solids of revolution, implicit differentiation, logarithmic functions, and differential equations. The teaching method includes theoretical explanations, visual approaches, worked examples, and exam questions. This course is intended for individuals interested in advancing their knowledge of calculus and its practical applications.
Overview
Syllabus
Maximum & Minimum.
Turning Points.
Stationary Points.
Non-Stationary Turning Points (1 of 2).
Non-Stationary Turning Points (2 of 2).
Introduction to Points of Inflexion.
The Special Case of x^4.
Horizontal Points of Inflexion.
Overview of Critical Points (1 of 2).
Overview of Critical Points (2 of 2).
Overview of Critical Points: Examples (1 of 2).
Overview of Critical Points: Examples (2 of 2).
Finding and Confirming Turning Points.
Curve Sketching: Locating Stationary Points.
Curve Sketching: Determining Nature of SPs.
Curve Sketching: Drawing the Graph.
Sign of the First Derivative.
Second Derivative: A Physical Analogy.
Second Derivative: Concavity.
Second Derivative: Notation.
Second Derivative: Relationship w/ First Derivative.
Graphing w/ the First Derivative.
Graphing w/ the Second Derivative.
Choosing First or Second Derivative.
Graph Behaviour Chart.
Implicit Differentiation.
Implicit Differentiation - example question.
Visual Approach to Derivatives (1 of 2).
Visual Approach to Derivatives (2 of 2).
Y11 Mathematics Ext 1 Quiz (1 of 2: Curve sketching with calculus).
Introduction to Solids of Revolution.
Verifying Formulae for Cylinder, Cone & Sphere.
Compound Volumes (1 of 2).
Compound Volumes (2 of 2).
Volumes: Examples around x-axis & y-axis.
Subtraction of Volumes.
Subtraction of Volumes: Class Discussion.
Volume within a Cone (1 of 3: Separating variables and constants for differentiation).
Volume within a Cone (2 of 3: Finding Volume in terms of a single variable to differentiate).
Volume within a Cone (3 of 3: Finding the Stationary Points to determine the maximum volume).
Solids of Revolution (1 of 3: What happens when you rotate an area around an axis?).
Solid of Revolution (2 of 3: Finding Volume of the Solid of Revolution using Volume of a cylinder).
Solids of Revolution (3 of 3: Finding the Volume of an area rotated around the y axis).
Conical Volume (1 of 2: Derivation of the Volume of a Cone through Solids of Revolution).
Spherical Volume (2 of 2: Derivation of the Volume of a Sphere through Solids of Revolution).
Difference between Volumes (1 of 2: Method to finding the difference between volumes).
Difference Between Volumes (2 of 2: Investigating the relationship between Parabolas & Cylinders).
Intro to Solids of Revolution (1 of 3: Establishing the formula).
Intro to Solids of Revolution (2 of 3: Simple worked example).
Intro to Solids of Revolution (3 of 3: Other axes, volume of a sphere).
Solids of Revolution - Subtracting Volumes.
Non-Standard Integrals: "Differentiate, hence integrate" (1 of 2).
Non-Standard Integrals: "Differentiate, hence integrate" (2 of 2).
Areas Under Curves: Logarithmic Functions.
Integration & Logarithmic Functions: Log Integrands (1 of 2).
Integration & Logarithmic Functions: Log Integrands (2 of 2).
Integration & Logarithmic Functions: Non-Log Integrands (1 of 3).
Integration & Logarithmic Functions: Non-Log Integrands (2 of 3).
Integration & Logarithmic Functions: Non-Log Integrands (3 of 3).
Areas Under Curves: Logarithmic Functions (Alternative Approach).
Applications & Implications of d/dx(½v²): Concrete Example.
Motion Exam Question (1 of 2: Finding v(x) from a(x)).
Motion Exam Question (2 of 2: Finding x(t) from v(x)).
Differentiating x^x (3 of 3: Implicit Differentiation).
Implicit Differentiation (Differentiating a function without needing to rearrange for x or y).
Rates of Change (1 of 4: Finding the Volume of an unknown height of water with a diagram).
Maximum/Minimum with Quadratics (1 of 2: Axis of symmetry).
Maximum/Minimum with Quadratics (2 of 2: Completing the square).
Max/Min Question: Cutting a Wire in Two (1 of 2: Setting up the equations).
Max/Min Question: Cutting a Wire in Two (2 of 2: Finding the minimum).
Challenging Max/Min Exam Question.
Graphing Logarithmic Function with Calculus (1 of 2: Identifying features).
Graphing Logarithmic Function with Calculus (2 of 2: Constructing the sketch).
Differential Equation in terms of Dependent Variable (1 of 2: Partial Fractions).
Differential Equation in terms of Dependent Variable (2 of 2: Integration).
Taught by
Eddie Woo
