C1 Curve Through 7 Points and Their 7 Tangents

This example demonstrates how to construct a C1-continuous smooth curve that passes through 7 specified points while matching the provided 7 tangents at those points, ensuring the curve is tangent to each target line and has consistent first-derivative behavior. The key mathematical ideas are C1 continuity (function value and first derivative agree at joins), tangents and derivative slopes, and the interpolation/spline-style principle of satisfying both point constraints and derivative (tangent) constraints. In GeoGebra, the build uses commands such as Textfield for parameter input/labels, Sequence to generate parameter/index lists, and Zip to bundle the objects associated with A–G and their tangent counterparts tA–tG. Derivative-related objects (e.g., lderiv) are used to obtain and visualize derivative information, while helper objects like lpts, δ (step/parameter increment), and intermediate expressions (e.g., m3, xyy′) support constructing the curve segments under tangent conditions. Suitable for advanced high-school to pre-university and early calculus/graphing teaching, especially for teacher-led demonstrations on tangency constraints and smooth curve modeling.