Introduction to Derivatives: Slope of the Tangent Line

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 $(3,0)$ .