VCE Mathematical Methods Units 3 and 4 Calculus
Course

VCE Mathematical Methods Units 3 and 4 – Calculus

3.1 Rates of Change
3.2 Differentiation
3.3 Application of Differentiation
3.4 Integration
3.5 Application of Integration

44 Lessons

In this area of study students cover graphical treatment of limits, continuity and differentiability of functions of a single real variable, and differentiation, anti-differentiation and integration of these functions. This material is to be linked to applications in practical situations.

This area of study includes:

  • review of average and instantaneous rates of change, tangents to the graph of a given function and the derivative function
  • deducing the graph of the derivative function from the graph of a given function and deducing the graph of an anti-derivative function from the graph of a given function derivatives of \( x^n \) , for \( n ∈ Q, e^x , \log_e(x), \sin(x), \cos(x) \) and \( \tan(x) \)
  • derivatives of \( f (x) \pm g(x), f (x) \times g(x), \displaystyle \frac{f(x)}{g(x)} \) and \( f (g (x)) \) where \(f\) and \(g\) are polynomial functions, exponential, circular, logarithmic or power functions and transformations or simple combinations of these functions
  • application of differentiation to graph sketching and identification of key features of graphs, identification of intervals over which a function is constant, stationary, strictly increasing or strictly decreasing, identification of the maximum rate of increase or decrease in a given application context (consideration of the second derivative is not required), identification of local maximum/minimum values over an interval and application to solving problems, and identification of interval endpoint maximum and minimum values
  • anti-derivatives of polynomial functions and functions of the form \( f(ax + b) \) where \(f\) is \(x^n\) , for \( n ∈ Q, e^x , \sin(x), \cos(x) \) and linear combinations of these
  • informal consideration of the definite integral as a limiting value of a sum involving quantities such as area under a curve, including examples such as distance travelled in a straight line and cumulative effects of growth such as inflation
  • anti-differentiation by recognition that \( F^{\prime} (x) = f(x) \) implies \( \displaystyle \int f(x) dx = F(x) +c \)
  • informal treatment of the fundamental theorem of calculus, \( \displaystyle \int_{a}^{b} f(x)dx = F(b)-F(a) \)
  • properties of anti-derivatives and definite integrals
  • application of integration to problems involving finding a function from a known rate of change given a boundary condition, calculation of the area of a region under a curve and simple cases of areas between curves, distance travelled in a straight line, average value of a function and other situations.

source – VCE Mathematics Study Design

RATES OF CHANGE

Constant Rates

Average Rates of Change

Gradient using Rates of Change

Limits at Constants

Limits at Infinity

DIFFERENTIATION

Basic Differentiation Rules

Differentiation of Surds

Chain Rule

Product Rule

Quotient Rule

Higher Derivatives

Derivative of Sine Functions

Derivatives of Cosine Functions

Derivatives of Tangent Functions

Derivatives of Exponential Functions

Derivatives of Logarithmic Functions

APPLICATION OF DIFFERENTIATION

Tangents

Normals

Increasing and Decreasing Functions

Turning Points

Points of Inflection and Concavity

Curves of Rational Functions in Quadratics by Linear Form

Sketching Graphs Containing Stationary Points

Gradient Functions

Sketching Curves from Derivatives

Curves of Rational Functions in Quadratics by Quadratic Form

INTEGRATION

Basic Integration Rules

Basic Integration Rules involving Algebra

Integration of Power Functions

Particular Values

Graphs of Antiderivatives of Functions

Upper and Lower Rectangles

Definite Integrals

Definite Integration of Power Functions

APPLICATION OF INTEGRATION

Area Under a Curve

Signed Areas

Area between Two Functions

Area between Two Functions involving Signed Areas

Kinematics using Integration