nonlinear

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.