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VCE Math Methods Unit 2 AOS 2
The Derivative: From Secant to Tangent
The derivative measures the instantaneous rate of change. We can visualize this by starting with a secant line connecting two points on a curve. As the distance between these points approaches zero, the secant line pivots to become the tangent line, and its gradient becomes the derivative at that point.
Interactive First Principles
Let’s find the gradient of at the point . Use the buttons to make the interval smaller and see how the secant gradient approaches the true gradient.
As approaches 0, the secant gradient approaches the tangent gradient, which is 4.
Differentiation Rules
While first principles are foundational, we use a set of efficient rules for differentiating polynomials. Master these to quickly find derivative functions.
The Power Rule
For any real number .
Sum/Difference Rule
Differentiate term by term.
Constant Multiple Rule
Constants are carried through.
Derivative of a Constant
The gradient of a horizontal line is zero.
Function Analysis with Derivatives
The derivative is a powerful tool for analyzing a function’s graph. It allows us to find the exact location and nature of stationary points, where the function momentarily stops increasing or decreasing.
Finding & Classifying Stationary Points
Let’s analyze the function . Follow the steps to find and classify its stationary points.
Step 1: Find the derivative.
Optimisation Problems
A key application of calculus is finding the maximum or minimum value of a quantity under certain constraints. This involves creating a function, finding its stationary points, and testing them to find the optimal solution.
Example: Maximising Garden Area
A rectangular garden uses an existing wall for one side and 40m of fencing for the other three sides. What dimensions for and maximise the area?
Wall
y
x
x
Current Dimensions:
Side = 10.0 m
Side = 20.0 m
Calculated Area:
200.0 m²
Kinematics: The Calculus of Motion
Calculus provides the language to describe motion. Velocity is the derivative of displacement, and acceleration is the derivative of velocity. This interactive demonstrates the relationship for a particle with displacement .
Particle Motion Simulation
Origin
Displacement
Velocity
Acceleration
Anti-differentiation: The Inverse Process
Anti-differentiation (or integration) is the reverse of differentiation. Given a derivative, we find the original function. However, since the derivative of a constant is zero, there are infinitely many possible original functions, all differing by a constant, .
The Family of Curves
If the derivative is , the general anti-derivative is . Use the slider to change the value of and see how it affects the final function and its y-intercept.
Resulting Function:
Y-Intercept:
