critical point

Examples of critical point in the following topics:

• Maximum and Minimum Values

  • The second partial derivative test is a method used to determine whether a critical point is a local minimum, maximum, or saddle point.
  • The second partial derivative test is a method in multivariable calculus used to determine whether a critical point $(a,b, \cdots )$ of a function $f(x,y, \cdots )$ is a local minimum, maximum, or saddle point.
  • For example, if a bounded differentiable function $f$ defined on a closed interval in the real line has a single critical point, which is a local minimum, then it is also a global minimum (use the intermediate value theorem and Rolle's theorem).
  • Its only critical point is at $(0,0)$, which is a local minimum with $f(0,0) = 0$.
  • Apply the second partial derivative test to determine whether a critical point is a local minimum, maximum, or saddle point

• Concavity and the Second Derivative Test

  • The second derivative test is a criterion for determining whether a given critical point is a local maximum or a local minimum.
  • In calculus, the second derivative test is a criterion for determining whether a given critical point of a real function of one variable is a local maximum or a local minimum using the value of the second derivative at the point.
  • The test states: if the function $f$ is twice differentiable at a critical point $x$ (i.e.
  • Telling whether a critical point is a maximum or a minimum has to do with the second derivative.
  • Calculate whether a function has a local maximum or minimum at a critical point using the second derivative test

• Supercritical Fluids

  • A supercritical fluid is a substance at a temperature and pressure above its critical point, where distinct liquid and gas phases do not exist.
  • A supercritical fluid is any substance at a temperature and pressure above its critical point, where distinct liquid and gas phases do not exist.
  • However, close to the critical point, the density can drop sharply with a slight increase in temperature.
  • The critical point of a binary mixture can be estimated as the arithmetic mean of the critical temperatures and pressures of the two components,
  • At the critical point, (304.1 K and 7.38 MPa) there is no difference in density, and the two phases become one fluid phase.

• Optimization in Several Variables

  • To solve an optimization problem, formulate the function $f(x,y, \cdots )$ to be optimized and find all critical points first.
  • After finding out the function $f(x)$ to be optimized, local maxima or minima at critical points can be easily found.
  • (Of course, end points may have maximum/minimum values as well.)

• Maximum and Minimum Values

  • Maxima and minima are critical points on graphs and can be found by the first derivative and the second derivative.
  • The value of the function at this point is called maximum of the function.
  • Local extrema can be found by Fermat's theorem, which states that they must occur at critical points.
  • One can distinguish whether a critical point is a local maximum or local minimum by using the first derivative test or second derivative test.
  • Use the first and second derivative to find critical points (maxima and minima) on graphs of functions

• Interpreting Phase Diagrams

  • Phase diagrams can also be used to explain the behavior of a pure sample of matter at the critical point.
  • The critical point, which occurs at critical pressure (Pcr) and critical temperature (Tcr), is a feature that indicates the point in thermodynamic parameter space at which the liquid and gaseous states of the substance being evaluated are indistinguishable.
  • At this point and beyond it, the substance being evaluated exists as a "supercritical fluid".
  • At temperatures above the critical temperature, the kinetic energy of the molecules is high enough so that even at high pressures the sample cannot condense into the liquid phase.
  • A typical phase diagram illustrating the major components of a phase diagram as well as the critical point.

• An Overview of PowerPoint

  • PowerPoint is a presentation software program .
  • PowerPoint does have its critics, and the benefits of the software are continually debated.
  • Some critics of PowerPoint argue that condensing complex issues into simplified bullet points is detrimental, and compromises the quality of information provided to the audience.
  • Additionally, there are also some critics who say that rather than providing too little information, PowerPoint allows users to put too much information into presentations.
  • State the arguments for and against using PowerPoint as a visual aid

• Critical Thinking

  • Constructing your speech with an effective thesis or main point and evidence to support that thesis requires you as the speechwriter to use critical thinking to determine how you'll make those points.
  • As you pinpoint your thesis and main points, you'll begin to outline exactly how you plan to support your argument.
  • Consider your thesis from opposing points of view.
  • Using this critical thinking skill of discerning hidden values gives you a comprehensive way to approach your speech from all possible points of view.
  • By evaluating evidence with a critical eye, you'll strengthen your argument by selecting the most compelling evidence to make your point.

• Critical Thinking

  • In this case, "critically" does not mean that you are looking for what is wrong with a work (although in the course of your critical process, you may well do that).
  • Instead, thinking critically means approaching a work as if you were a critic or commentator.
  • If you disagree with a text, what is the point of contention?
  • Critical thinking has many uses.
  • Finding an error in someone else's argument can be the point of destabilization you need to make a worthy argument of your own.

• Critical Thinking

  • The essential skill of critical thinking will go a long way in helping one to develop statistical literacy.
  • Each day people are inundated with statistical information from advertisements ("4 out of 5 dentists recommend"), news reports ("opinion polls show the incumbent leading by four points"), and even general conversation ("half the time I don't know what you're talking about").
  • The essential skill of critical thinking will go a long way in helping one to develop statistical literacy.
  • Critical thinking is an inherent part of data analysis and statistical literacy.
  • Interpret the role that the process of critical thinking plays in statistical literacy.