Slope of Secant

From ProofWiki
Jump to: navigation, search

Theorem

Let $f: \R \to \R$ be a real function.

Let the graph of $f$ be depicted on a Cartesian plane.


SecantToCurve.png


Let $AB$ be a secant of $f$ where:

$A = \left({x, f \left({x}\right)}\right)$
$A = \left({x + h, f \left({x + h}\right)}\right)$

Then the slope of $AB$ is given by:

$\dfrac {f \left({x + h}\right) - f \left({x}\right)} h$


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 $\left({x + h}\right) - x = h$
the change in $y$ is $f \left({x + h}\right) - f \left({x}\right)$.

Hence the result.

$\blacksquare$


Sources