Using the Curve Sketching Tool

Curve sketching is a methodical process used in calculus to draw the graph of a function by analyzing its properties without plotting thousands of points manually. This Curve Sketching Tool automates the calculus steps to reveal the “interesting” parts of the graph: the peaks, valleys, and twists.

Critical Points (f'(x) = 0)

The first derivative, f'(x), represents the slope of the tangent line.

• Local Maximum: Occurs where the slope changes from positive to negative (top of a hill).
• Local Minimum: Occurs where the slope changes from negative to positive (bottom of a valley).

Inflection Points (f”(x) = 0)

The second derivative, f”(x), measures concavity (how the curve bends).

• Concave Up (Smile): f”(x) > 0. The curve holds water.
• Concave Down (Frown): f”(x) < 0. The curve spills water.
• Point of Inflection: A point where the concavity changes (e.g., from up to down). This happens where f”(x) = 0 (and changes sign).

Why use a Curve Sketching Tool?

Manual calculations involving the second derivative test or solving cubic equations for roots can be lengthy and error-prone. This tool provides instant verification, allowing you to focus on understanding the relationship between the algebraic derivatives and the geometric shape of the curve.