A secant line to a function \(f\) on a closed interval \([a,b]\), is the line passing through the two points \(\left(a,f(a)\right)\) and \(\left(b,f(b)\right)\). See the graph below:
If the slope of this secant line is calculated, then \(m=\large{\frac{f(b)-f(a)}{b-a}}\). In general, for any \(x\) in \([a,b]\), we can repeat this same calculation to get: $$m_{sec}=\frac{f(x)-f(a)}{x-a}$$ This is known as the average rate of change of \(f\) over \([a,x]\).
Find the average rate of change of \(f(x)=x^2+4x\) on \([a,x]\).
| $$\frac{f(x)-f(a)}{x-a}$$ | $$\implies$$ | $$\frac{x^2+4x-\left(a^2+4a\right)}{x-a}$$ |
| $$\implies$$ | $$\frac{x^2+4x-a^2-4a}{x-a}$$ | Distribute the subtraction |
| $$\implies$$ | $$\frac{x^2-a^2+4x-4a}{x-a}$$ | Regroup terms |
| $$\implies$$ | $$\frac{(x-a)(x+a)+4(x-a)}{x-a}$$ | Factoring |
| $$\implies$$ | $$\frac{(x-a)[(x+a)+4]}{x-a}$$ | |
| $$\implies$$ | $$\frac{x+a+4}{1}$$ | Cancel out \(x-a\) |
| $$\implies$$ | $$x+a+4$$ | Simplified answer\(\;\color{green}{\checkmark}\) |
The above average rate of change is also an example of a Difference Quotient. In general, for any function \(f\) and any value \(x\), if \(h\) is a small distance to the right of \(x\) then the Difference Quotient is: $$\frac{f(x+h)-f(x)}{x+h-x}\implies\frac{f(x+h)-f(x)}{h}$$ The goal is to simplify this Difference Quotient by algebraic methods.
Simplify the Difference Quotient for \(f(x)=\large{\frac{1}{x}}\).
| $$\frac{f(x+h)-f(x)}{h}\implies\frac{\frac{1}{x+h}-\frac{1}{x}}{h}\implies\frac{\frac{1}{x+h}\cdot\color{red}{\frac{x}{x}}-\frac{1}{x}\cdot\color{red}{\frac{x+h}{x+h}}}{h}\implies\frac{\frac{x}{x(x+h)}-\frac{x+h}{x(x+h)}}{h}$$ |
| $$\implies\frac{\frac{x-(x+h)}{x(x+h)}}{h}\implies\frac{\frac{x-x-h}{x(x+h)}}{h}\implies\frac{\frac{-h}{x(x+h)}}{h}\implies\frac{-h}{x(x+h)}\cdot\frac{1}{h}\implies\frac{-1}{x(x+h)}\;\color{green}{\checkmark}$$ |