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NSW Syllabuses

Mathematics Extension 1 Stage 6 - Year 12 Extension 1 - Calculus ME-C3 Applications of Calculus

Outcomes

A student:

  • ME12-1

    applies techniques involving proof or calculus to model and solve problems

  • ME12-4

    uses calculus in the solution of applied problems, including differential equations and volumes of solids of revolution

  • ME12-6

    chooses and uses appropriate technology to solve problems in a range of contexts

  • ME12-7

    evaluates and justifies conclusions, communicating a position clearly in appropriate mathematical forms

Subtopic Focus

The principal focus of this subtopic is to develop an appreciation for certain applications of calculus in a practical context, including the use of differential equations and volumes of solids of revolution, to solve problems.

Students develop awareness of the use of calculus to solve practical problems.

Content

  • C3.1: Differential equations
  • Students:
  • recognise that an equation involving a derivative is called a differential equation, and that solving a differential equation involves finding a function which satisfies the differential equation
  • recognise that solutions to differential equations (if they exist) are functions that may not be unique
  • sketch the graph of a particular solution given a direction field and initial conditions (ACMSM131)
  • form a direction field (slope field) from simple 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
  • recognise the features of a first-order linear differential equation and that exponential growth and decay population models are first-order linear differential equations, with known solutions
  • solve simple first-order differential equations (ACMSM130)
  • 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  
  • model and solve differential equations including but not limited to the logistic equation that will arise in situations where rates are involved, for example in chemistry, biology and economics (ACMSM132) AAM
  • C3.2: Volumes of solids of revolution
  • Students:
  • sketch and calculate the volume of a solid of revolution formed by rotating a region in the plane about the \(x\)-axis or \(y\)-axis, using digital technology or otherwise (ACMSM125) AAM cctict
  • derive and use the formula \(V = \pi\int_a^b \left[f(x)\right]^2 \, dx\) CCT
  • determine the volumes of solids of revolution about either the \(x\) or \(y\) axis that are formed by rotating the area between two curves in both real-life and abstract contexts CCT