Slope of Secant
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Theorem
Let $f: \R \to \R$ be a real function.
Let the graph of $f$ be depicted on a Cartesian plane.
Let $AB$ be a secant of $f$ where:
- $A = \tuple {x, \map f x}$
- $A = \tuple {x + h, \map f {x + h} }$
Then the slope of $AB$ is given by:
- $\dfrac {\map f {x + h} - \map f x} h$
Proof
The slope of $AB$ is defined as the change in $y$ divided by the change in $x$.
Between $A$ and $B$:
- the change in $x$ is $\paren {x + h} - x = h$
- the change in $y$ is $\map f {x + h} - \map f x$.
Hence the result.
$\blacksquare$
Sources
- 2008: Ian Stewart: Taming the Infinite ... (previous) ... (next): Chapter $8$: The System of the World: Calculus