nonlinear

(noun)

A polynomial expression of degree 2 or higher.

Related Terms

  • conic section
  • inequality
  • system of equations
  • nonlinear function

(noun)

An algebraic term that is raised to the power of two or higher; equivalently, a function with a curved graph.

Related Terms

  • conic section
  • inequality
  • system of equations
  • nonlinear function

Examples of nonlinear in the following topics:

  • Nonlinear Systems of Inequalities

    • Systems of nonlinear inequalities can be solved by graphing boundary lines.
    • A nonlinear inequality is an inequality that involves a nonlinear expression—a polynomial function of degree 2 or higher.
    • All points below the line $y=x+2$ satisfy the linear equality, and all points above the parabola $y=x^2$ satisfy the parabolic nonlinear inequality.
    • This need not be the case with all nonlinear inequalities, but reversing the direction of both inequalities in the previous example would lead to an infinite solution area.
    • This graph of a cubic function is an example of a nonlinear equation.
  • Models Involving Nonlinear Systems of Equations

    • Nonlinear systems of equations can be used to solve complex problems involving multiple known relationships.
    • The conservation of mechanical energy can produce a system of nonlinear equations when there is an elastic (perfectly bouncy) collision.
    • Rendering and visualizing these objects, and formulating a plan for constructing them, requires the software to solve nonlinear systems.
    • In addition to practical scenarios like the above, nonlinear systems can be used in abstract problems.
    • Extend the ideas behind nonlinear systems of equations to real world applications
  • Nonlinear Systems of Equations and Problem-Solving

    • As with linear systems, a nonlinear system of equations (and conics) can be solved graphically and algebraically for all of its variables.
    • Nonlinear systems of equations, such as conic sections, include at least one equation that is nonlinear.
    • A nonlinear equation is defined as an equation possessing at least one term that is raised to a power of 2 or more.
    • Since at least one function has curvature, it is possible for nonlinear systems of equations to contain multiple solutions.
    • Solving nonlinear systems of equations algebraically is similar to doing the same for linear systems of equations.
  • Dependence of Resistance on Temperature

    • Resistivity and resistance depend on temperature with the dependence being linear for small temperature changes and nonlinear for large.
    • For larger temperature changes, α may vary, or a nonlinear equation may be needed to find ρ.
  • Describing linear relationships with correlation

    • Nonlinear trends, even when strong, sometimes produce correlations that do not reflect the strength of the relationship; see three such examples in Figure 7.11.
    • Try drawing nonlinear curves on each plot.
  • Polynomial Regression

    • Polynomial regression fits a nonlinear relationship between the value of $x$ and the corresponding conditional mean of $y$, denoted $E(y\ | \ x)$, and has been used to describe nonlinear phenomena such as the growth rate of tissues, the distribution of carbon isotopes in lake sediments, and the progression of disease epidemics.
    • Although polynomial regression fits a nonlinear model to the data, as a statistical estimation problem it is linear, in the sense that the regression function $E(y\ | \ x)$ is linear in the unknown parameters that are estimated from the data.
    • Explain how the linear and nonlinear aspects of polynomial regression make it a special case of multiple linear regression.
  • Beginning with straight lines

    • If data show a nonlinear trend, like that in the right panel of Figure 7.6, more advanced techniques should be used.
  • Conditions for the least squares line

    • If there is a nonlinear trend (e.g. left panel of Figure 7.13), an advanced regression method from another book or later course should be applied.
  • Introduction to line fitting, residuals, and correlation

    • We will discuss nonlinear trends in this chapter and the next, but the details of fitting nonlinear models are saved for a later course.
    • A linear model is not useful in this nonlinear case.
  • Linear Equations

    • Since terms of linear equations cannot contain products of distinct or equal variables, nor any power (other than $1$) or other function of a variable, equations involving terms such as $xy$, $x^2$, $y^{\frac{1}{3}}$, and $\sin x$ are nonlinear.
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