quotients

Examples of quotients in the following topics:

• Reaction Quotients

  • The reaction quotient is a measure of the relative amounts of reactants and products during a chemical reaction at a given point in time.
  • The reaction quotient, Q, is a measure of the relative amounts of reactants and products during a chemical reaction at a given point in time.
  • Just as for the equilibrium constant, the reaction quotient can be a function of activities or concentrations.
  • Three properties can be derived from this definition of the reaction quotient:
  • Calculate the reaction quotient, Q, and use it to predict whether a reaction will proceed in the forward or reverse direction

• Difference Quotients

  • The difference quotient is used in algebra to calculate the average slope between two points but has broader effects in calculus.
  • It is also known as Newton's quotient:
  • The difference quotient is the average slope of a function between two points.
  • In this case, the difference quotient is know as a derivative, a useful tool in calculus.
  • Relate the difference quotient in algebra to the derivative in calculus

• Dividing Polynomials

  • For example, find the quotient and the remainder of the division of $x^3 - 12x^2 -42$, the dividend, by $x-3$, the divisor.
  • Multiply the divisor by the result just obtained (the first term of the eventual quotient): $x^2 \cdot (x − 3) = x^3 − 3x^2$.
  • For example, find the quotient and the remainder of the division of $x^3 - 12x^2 -42$, the dividend, by $x-3$, the divisor.
  • Multiply the divisor by the result just obtained (the first term of the eventual quotient): $x^2 \cdot (x − 3) = x^3 − 3x^2$.
  • The calculated polynomial is the quotient, and the number left over (−123) is the remainder:

• Logarithms of Quotients

  • By applying the product, power, and quotient rules, you could write this expression as:
  • Relate the quotient rule for logarithms to the rules for operating with exponents, and use this rule to rewrite logarithms of quotients

• Predicting the Direction of a Reaction

  • Equilibrium constants and reaction quotients can be used to predict whether a reaction will favor the products or the reactants.
  • If a reaction is not at equilibrium, you can use the reaction quotient, Q, to see where the reaction is in the pathway:

• Strategy for General Problem Solving

  • If the units are ignored, the quotients do not numerically equal 1, but 1/12 or 12.
  • Since the two quotients are equal to 1, multiplying or dividing by the quotients is the same as multiplying or dividing by 1.
  • You can also use these quotients to convert from inches to feet or from feet to inches.
  • If there is confusion regarding which quotient to use in the conversion, just make sure the units cancel out correctly.
  • The units behave just like numbers in products and quotients—they can be multiplied and divided.

• The Derivative and Tangent Line Problem

  • The slope of the secant line passing through $p$ and $q$ is equal to the difference quotient
  • As the point $q$ approaches $p$, which corresponds to making $h$ smaller and smaller, the difference quotient should approach a certain limiting value $k$, which is the slope of the tangent line at the point $p$.
  • Then there is a unique value of $k$ such that, as $h$approaches $0$, the difference quotient gets closer and closer to $k$, and the distance between them becomes negligible compared with the size of $h$, if $h$ is small enough.
  • This leads to the definition of the slope of the tangent line to the graph as the limit of the difference quotients for the function $f$.

• The Remainder Theorem and Synthetic Division

  • Synthetic division is a technique for dividing a polynomial and finding the quotient and remainder.
  • This gives the quotient $x^2-9x-27$ and the remainder $-123$.
  • So the quotient must be the second degree polynomial $x^2 - 9x - 27$.

• History of Intelligence Testing

  • The abbreviation "IQ" comes from the term intelligence quotient, first coined by the German psychologist William Stern in the early 1900s (from the German Intelligenz-Quotient).
  • He proposed that an individual's intelligence level be measured as a quotient (hence the term "intelligence quotient") of their estimated mental age divided by their chronological age.
  • The original formula for the quotient was Mental Age/Chronological Age x 100.

• Division and Factors

  • So given two polynomials $D(x)$ (the dividend) and $d(x)$ (the divisor), we are looking for two polynomials $q(x)$ (the quotient) and $r(x)$ (the $remainder)$ such that $D(x) = d(x)q(x) + r(x)$ and the degree of $r(x)$ is strictly smaller than the degree of $d(x).$
  • Conceptually, we want to see how many copies of $d(x)$ are contained in $D(x)$ (this is the quotient) and then how far $D(x)$ is away from being a multiple of $d(x)$ (this is the remainder).
  • Again looking at the highest degree terms, we see that $4x^2 = 2x\cdot2x$, so we write down $2x$ as the second term in the quotient and proceed as before:
  • We see that the quotient $q(x)$ $3x^2+2x+6$ and the remainder $r(x)$ is $22$, so
  • (Of course, the quotient will also be a factor.)