Introduction to Derivatives: Slope of the Tangent Line

Introduction to Derivatives

Derivative Definition: The derivative of a function represents the instantaneous rate of change of the function at a specific point. It is fundamentally the slope of the tangent line to the curve at that point.
Calculus Terminology: The word 'derivative' is a cornerstone of calculus, frequently used to describe rates of change for non-linear functions.

Function: The function being analyzed is f(x) = 3x - x^2. This can be factored as f(x) = x(3-x) to easily find intercepts.
Intercepts: Setting f(x)=0 reveals x-intercepts at x=0 and x=3. The point (0,0) is an intercept, and (3,0) is another.
Plotting Points: Additional points include:
- At x=1, f(1) = 3(1) - (1)^2 = 2, so the point is (1,2).
- Line of Symmetry: The graph is a parabola. The line of symmetry is at x = 1.5.
- Vertex: At x=1.5, f(1.5) = 3(1.5) - (1.5)^2 = 4.5 - 2.25 = 2.25, so the vertex is (1.5, 2.25).
Visual Representation: The function forms a downward-opening parabola starting at (0,0), peaking at (1.5, 2.25), and returning to (3,0).