difference quotient

(noun)

the function difference $\Delta F$ divided by the point difference $\Delta x$: $\Delta F(x) / \Delta x$

Related Terms

Examples of difference quotient in the following topics:

  • Difference Quotients

    • The difference quotient is used in algebra to calculate the average slope between two points but has broader effects in calculus.
    • The function difference divided by the point difference is known as the difference quotient, attributed to Isaac Newton.
    • 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

  • 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 $f$.

  • Differentiation Rules

    • In many cases, complicated limit calculations by direct application of Newton's difference quotient can be avoided by using differentiation rules.

  • Sums, Differences, Products, and Quotients

  • Logarithms of Quotients

    • The logarithm of the ratio of two quantities is the difference of the logarithms of the quantities.
    • Similarly, the logarithm of the ratio of two quantities is the difference of the logarithms:
    • 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

  • 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 difference is that Q applies when the reaction is at non-equilibrium conditions, and therefore its value can vary.
    • 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

  • Strategy for General Problem Solving

    • To convert a measured quantity to a different unit of measure without changing the relative amount, use a conversion factor.
    • Chemistry, along with other sciences and engineering, makes use of many different units.
    • Since there are so many different units that can be used, it is necessary to be able to convert between the various units.
    • 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.

  • 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.
    • He further argued that g should be free of cultural bias such as differences in language and education type.

  • Concentration of Cells

    • When reactant concentrations differ from standard conditions, the cell potential will deviate from the standard potential.
    • Here, ΔG is the change in Gibbs free energy, T is absolute temperature, R is the gas constant, and Q is the reaction quotient.
    • In chemistry, a reaction quotient is a function of the activities or concentrations of the chemical species involved in a chemical reaction.

  • Homogeneous versus Heterogeneous Solution Equilibria

    • The equilibrium constants for homogeneous and heterogeneous solutions need to be calculated differently.
    • The equilibrium constants for reactions that contain substances that are all in the same phase, and reactions that contain substances in different phases, need to be calculated differently.
    • The reaction quotient measured at equilibrium is the equilibrium constant K.