Definition:Difference Quotient
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Definition
Let $V$ be a vector space over the real numbers $\R$.
Let $f: \R \to V$ be a function.
A difference quotient is an expression of the form:
- $\dfrac {\map f {x + h} - \map f x} h$
where $h \ne 0$ is a real number.
Left Difference Quotient
Let $V$ be a vector space over the real numbers $\R$.
Let $f: \R \to V$ be a function.
A left difference quotient is an expression of the form:
- $\dfrac {\map f {x + h} - \map f x} h$
where $h < 0$ is a strictly negative real number.
Right Difference Quotient
Let $V$ be a vector space over the real numbers $\R$.
Let $f: \R \to V$ be a function.
A right difference quotient is an expression of the form:
- $\dfrac {\map f {x + h} - \map f x} h$
where $h > 0$ is a strictly positive real number.
Geometric Interpretation
The difference quotient is the slope of the secant line of the graph of $f$ connecting points $P_1 = \tuple {x, \map f x}$ and $P_2 = \tuple {x + h, \map f {x + h} }$.
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Also known as
A difference quotient is also known as a Newton quotient, for Isaac Newton
Also see
- Results about difference quotients can be found here.
Sources
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): difference quotient
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Newton quotient