Extension Calculus – Application of Calculus

This lesson comprises six (6) master classes focusing on:

• Advanced differentiation techniques
• Advanced integration techniques
• Differential equations

Content:

ME-C2

• Find and evaluate indefinite and definite integrals using the method of integration by substitution, using a given substitution
  • change an integrand into an appropriate form using algebra
• Prove and use the identities \( \sin^2 nx=\frac{1}{2}(1− \cos 2nx) \) and \( \cos^2 nx=\frac{1}{2}(1+ \cos 2nx) \) to solve problems
• Solve problems involving \( \int \sin^2 nx\ dx \) and \( \int \cos^2 nx\ dx \)
• Find derivatives of inverse functions by using the relationship \( \frac{dy}{dx}=\frac{1}{\frac{dx}{dy}} \)
• Solve problems involving the derivatives of inverse trigonometric functions
• Integrate expressions of the form \( \frac{1}{\sqrt{a^2−x^2}} \) or \( \frac{a}{a^2+x^2 \)

ME-C3.1

• Calculate area of regions between curves determined by functions
• Sketch, with and without the use of technology, the graph of a solid of revolution whose boundary is formed by rotating an arc of a function about the x-axis or y-axis
• Calculate the volume of a solid of revolution formed by rotating a region in the plane about the x-axis or y-axis, with and without the use of technology
• Determine the volumes of solids of revolution that are formed by rotating the region between two curves about either the x-axis or y-axis in both real-life and abstract contexts 

ME-C3.2

• Recognise that an equation involving a derivative is called a differential equation
• Recognise that solutions to differential equations are functions and that these solutions may not be unique
• Sketch the graph of a particular solution given a direction field and initial conditions
  • form a direction field (slope field) from simple first-order differential equations
  • recognise the shape of a direction field from several alternatives given the form of a differential equation, and vice versa
  • sketch several possible solution curves on a given direction field
• Solve simple first-order differential equations
  • solve differential equations of the form \( \frac{dy}{dx}=f(x) \)
  • solve differential equations of the form \( \frac{dy}{dx}=g(y) \)
  • solve differential equations of the form \( frac{dy}{dx}=f(x)g(y) \) using separation of variables
• Recognise the features of a first-order linear differential equation and that exponential growth and decay models are first-order linear differential equations, with known solutions
• Model and solve differential equations including the logistic equation that will arise in situations where rates are involved, for example in chemistry, biology and economics