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Integrals involving non-rational functions are often difficult to evaluate using standard techniques, especially when radicals appear in the integrand. Rationalizing substitution provides a systematic method for simplifying such integrals by converting them into rational forms that are easier to handle.Consider a rod whose linear mass density depends on a constant linear density, a characteristic length, and the distance from the left end of the rod. Determining the total mass requires...
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Rational expressions are algebraic fractions in which both the numerator and the denominator are polynomials. These expressions follow the arithmetic rules of numerical fractions but require extra care due to the presence of variables. A fundamental part of working with rational expressions is identifying values that make the expression undefined, typically those that result in division by zero or undefined radicals.Determining the DomainThe domain of a rational expression includes all real...
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A rational function is defined as the quotient of two polynomials:  where Q(x)≠0, These functions often exhibit asymptotes, which are the lines that the graph approaches but never touches. These asymptotes are classified based on how the function behaves near specific values of the input.Vertical asymptotes occur where the denominator is zero, and the numerator is not, causing the function to be undefined. These are found by solving Q(x)=0. For example:  has a vertical...
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Cognitive-behavioral therapies (CBTs) are grounded in the belief that our thoughts profoundly influence our emotions and actions. Advocates of CBT emphasize three core assumptions: first, that cognitions are identifiable and measurable; second, that they are central to psychological functioning; and third, that irrational or maladaptive beliefs can be replaced with rational and adaptive ones. This transformative approach to therapy has paved the way for specific models such as Albert...
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Rational functions are expressions written as the ratio of two polynomials, and their integrals are evaluated by simplifying the integrand into manageable parts. These functions are classified as proper or improper based on the degrees of the numerator and denominator.A rational function is proper when the degree of the numerator is less than the degree of the denominator. In this case, partial fraction decomposition is used to rewrite the function as a sum of simpler rational terms. The...
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Related Experiment Video

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Assessing Working Memory in Children: The Comprehensive Assessment Battery for Children – Working Memory (CABC-WM)
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Rational Higher-Order Belief Revision in Young Children.

Katherine Kimura1, Alison Gopnik1

  • 1University of California, Berkeley.

Child Development
|September 4, 2018
PubMed
Summary

Preschoolers revise higher-order beliefs when conflicting evidence is strong and prior beliefs are weak. Young children

Area of Science:

  • Cognitive Development
  • Developmental Psychology
  • Belief Revision

Background:

  • Belief revision occurs at multiple levels, including higher-order beliefs.
  • The interaction between conflicting evidence and prior beliefs in higher-order belief revision is not well understood.

Purpose of the Study:

  • To investigate how preschoolers respond to evidence that conflicts with their causal higher-order beliefs.
  • To determine the factors influencing higher-order belief revision in young children.

Main Methods:

  • Study involved 96 preschoolers aged 4 and 5 years.
  • Children were presented with evidence that directly conflicted with their prior causal higher-order beliefs.
  • The strength of prior beliefs and counterevidence was varied.

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Related Experiment Videos

Last Updated: Feb 5, 2026

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Main Results:

  • Preschoolers largely maintained higher-order beliefs when a single event violated a strongly supported prior belief.
  • Children revised their higher-order beliefs when prior beliefs were weakly supported and counterevidence was stronger.

Conclusions:

  • Young children are capable of revising their higher-order beliefs.
  • Higher-order belief revision in preschoolers is dependent on the relative strengths of prior beliefs and incoming evidence.