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Size: 109 KB

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Categories:

1. Home Study
2. Visualization

Subjects:

1. Plane Curves
2. Equation Solving
3. Polynomial Algebra
4. Lines
5. Differentiation

Title: Implicit Functions and Tangent Pencils

Book Description: An equation of the form f(x,y) = 0 (where f is smooth) often defines a curve in the plane. When this is the case, we say that the curve is defined implicitly (as opposed to explicitly) because we may not be able to solve for y in terms of x. Nevertheless, assuming that a local solution exists, the method of implicit differentiation often allows us to solve for yprime = dy/dx as a function of x and y at any point (x,y) along this curve. It fails when a local solution is not guaranteed existence (the solution curves "turns back").
In this WorkBook, we can visualize this phenomenon by drawing tangent pencils to an implicit curve. When the reader supplies an equation of the form f(x,y) = 0, the system draws the curve and it attempts to solve (symbolically) for yprime = dy/dx as a function of x and y. If it succeeds, it prints the expression for this derivative. Next, if the reader asks for a pencil of tangent lines, it uses this expression to generate such a pencil at points along the curve. The lines all have a fixed-length projection to the x-axis that is determined also by the reader. As these lines become more and more vertical, they therefore become longer and longer. Thus, they display the critical behavior at points along the curve where it "turns back". At such points of course the yprime function is undefined.

This is based on an earlier idea of Jan Ray.

Author: James E. White

Suggested Use: Visualize tangent pencils and singularities

Topics: implicit functions, implicit curves, pencils of tangent lines, implicit equations, implicit differentiation

Number of Pages: 2

Animation: No

Grade Level: