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)
• 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)
• derive and use the formula $$V = \pi\int_a^b \left[f(x)\right]^2 \, dx$$
• 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