Question

True or False? Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If the curve given by \(x=f(t), y=g(t),\) and \(z=h(t)\) is a line, then \(f, g,\) and \(h\) are first-degree polynomial functions of \(t\).

Short answer

The statement is false. While it's typical for the functions \(f(t)\), \(g(t)\), \(h(t)\) to be first-degree polynomials if the curve is a line, it's not a necessary condition. For example, \(f(t) = g(t) = h(t) = 2t^2\) for \(t \in [0, 1]\) are second-degree polynomials that describe a line in three-dimensional space.

Step-by-step solution

Understanding the statement

We start by understanding the problem. The problem provides a curve defined parametrically by three functions of t: \(x=f(t)\), \(y=g(t)\), and \(z=h(t)\). The statement we are attempting to validate or refute is whether these functions must always be first-degree if the curve they describe is a line.

Analyzing the condition

Next, we consider the nature of a line in three-dimensional space. A line can be defined uniquely by any two points on the line. The function describing such a line between two points would typically be a first-degree polynomial.

Contradicting the statement

However, it's not essential for the functions \(f(t)\), \(g(t)\), \(h(t)\) to be first-degree for the curve to be a line. For instance, consider \(f(t) = 2t^2\), \(g(t) = 2t^2\), and \(h(t) = 2t^2\), for \(t \in [0, 1]\). Both functions are second-degree polynomials, not first-degree. But they describe a line in three-dimensional space from point (0,0,0) to (2,2,2).

Key concepts

Parametric Representation of Lines

When we talk about the parametric representation of lines, we specifically refer to expressing the coordinates of points on a line as functions of a single parameter, often denoted as 't'. In a two-dimensional space, we would use two equations to represent a line: for the x-coordinate and y-coordinate respectively. However, when dealing with lines in three-dimensional space, we require three equations: one each for the x, y, and z coordinates.

Parametric equations allow us to capture all the points along a line by varying 't'. For instance, a simple parametric equation of a line might be given by \(x = t + 1\), \(y = 2t - 3\), and \(z = -t\), where 't' can take any real value. This approach to describing lines offers great flexibility and provides a powerful method for defining lines in geometry and calculus.

Degree of Polynomial Functions

The degree of a polynomial function is the highest power of the variable in the polynomial. For example, in the polynomial \(f(t) = a_nt^n + ... + a_1t + a_0\), the degree is 'n' if \(a_n \eq 0\). It's a fundamental concept in algebra that helps define the behavior and characteristics of the function.

First-degree polynomial functions, often called linear functions, are of the form \(at + b\) and graph as straight lines with a constant slope 'a'. Higher-degree polynomial functions, such as quadratics (\(t^2\)), cubics (\(t^3\)), and so on, have more complex curves and turning points. In the context of lines, it's commonly assumed that a straight line can be described by first-degree polynomials due to their constant rate of change.

Three-Dimensional Space Curves

In three-dimensional space, curves can take on much more complex shapes than in two dimensions. A space curve does not necessarily lie within a single plane and can extend in all three spatial dimensions. Parametric equations are especially useful for defining these curves because they can account for the independent movements along the x, y, and z axes.

For example, a helix can be represented by parametric equations like \(x = a\text{cos}(t)\), \(y = a\text{sin}(t)\), and \(z = bt\), where 't' is the parameter and 'a' and 'b' are constants that affect the helix's radius and pitch. Unlike lines, space curves are not restricted to straight paths, allowing for a variety of applications in physics, engineering, and math modeling.

Analyzing Mathematical Statements

A critical aspect of mathematical understanding is the ability to analyze statements for their validity. An analytical approach usually involves verifying examples that satisfy the conditions, seeking counterexamples, and comprehending the underlying principles that make the statement true or false.

In the case of the given exercise, analyzing whether parametric functions of a line must be first-degree involves recognizing that a straight line in space can be expressed in many different forms, not necessarily only as a first-degree polynomial. Counterexamples, as provided in the solution, demonstrate that even second-degree polynomial functions, under certain conditions, can represent a straight line. The capability to analyze such mathematical statements is invaluable for students as it cultivates critical thinking and a deeper understanding of mathematical concepts.