As we have discussed in previous lessons, the slope of a straight line is constant, whereas the slope of a curve is always changing!
There are a couple of different ways for calculating the slope of a curve. One of these ways is by finding the secant.
This can be done by finding the Average Rate of Change between 2 points. This can be expressed algebraically as:
\(m_{sec} = \cfrac{\Delta y}{\Delta x} = \cfrac{\textcolor{red}{f(b)} - \textcolor{blue}{f(a)}}{\textcolor{red}{b} - \textcolor{blue}{a}}\)
When calculating the secant, \(a\le x\le b\) and \(x∈[a,b]\).
Example
Determine the slope of the secant for the function \(f(x) = 3x^2\ - 2\) where \(a = 2\) and \(b = 4\).
First, we can evaluate \(f(b)\) by substituting \(4\) for \(x\) in the original function:
\(f(4) = 3(4)^2 - 2\)
\(f(4) = 3(16) - 2\)
\(f(4) = 48 - 2\)
\(\textcolor{red}{f(4) = 46}\)
Next, we can evaluate \f(a)\) by substituting \(2\) for \(x\) in the original function:
\(f(2) = 3(2)^2 - 2\)
\(f(2) = 3(4) - 2\)
\(f(2) = 12 - 2\)
\(\textcolor{blue}{f(2) = 10}\)
Finally, we can substitute all pertinent values into the secant formula to determine its slope at these points:
\(\cfrac{\Delta y}{\Delta x} = \cfrac{\textcolor{red}{f(b)} - \textcolor{blue}{f(a)}}{\textcolor{red}{b} - \textcolor{blue}{a}}\)
\(= \cfrac{\textcolor{red}{46} - \textcolor{blue}{10}}{\textcolor{red}{4} - \textcolor{blue}{2}}\)
\(= \cfrac{36}{2}\)
\(= 18\)
Therefore, we can determine that the slope of the secant is \(18\).
Determine the slope of the secant for the function \(p(x) = \cfrac{4}{2x}\) where \(a = 5\) and \(b = 8\).
Show Answer
First, we can evaluate \(p(b)\) by substituting \(8\) for \(x\) in the original function:
\(p(8) = \cfrac{4}{2(8)}\)
\(p(8) = \cfrac{4}{16}\)
\(\textcolor{red}{p(8) = \cfrac{1}{4}}\)
Next, we can evaluate \(p(a)\) by substituting \(5\) for \(x\) in the original function:
\(p(5) = \cfrac{4}{2(5)}\)
\(p(5) = \cfrac{4}{10}\)
\(\textcolor{blue}{p(5) = \cfrac{2}{5}}\)
Finally, we can substitute all pertinent values into the secant formula to determine its slope at these points:
\(\cfrac{\Delta y}{\Delta x} = \cfrac{\textcolor{red}{f(b)} - \textcolor{blue}{f(a)}}{\textcolor{red}{b} - \textcolor{blue}{a}}\)
\(= \cfrac{\textcolor{red}{\cfrac{1}{4}} - \textcolor{blue}{\cfrac{2}{5}}}{\textcolor{red}{8} - \textcolor{blue}{5}}\)
\(= \cfrac{\cfrac{5}{20} - \cfrac{8}{20}}{3}\)
\(= \cfrac{-\cfrac{3}{20}}{3}\)
\(= -\cfrac{3}{60} = -\cfrac{1}{20}\)
Therefore, we can determine that the slope of the secant is \(-\cfrac{1}{20}\).