 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

Lessons

#### Area between Two Functions involving Signed Areas 3 Topics 