[1.2] Rates of Change

Homework notes

Here's a simpler version of problem 23, which might help you with that one:

Find the average rate of change of $f(x)=x^2$ between the point $(x,f(x))$ and $(x+h,f(x+h))$.

The average rate of change between two point $(x_1,y_1)$ and $(x_2,y_2)$ is $$\text{average rate of change}=\frac{y_2-y_1}{x_2-x_1}=\frac{\Delta y}{\Delta x}. \label{eq:rate}$$

In this problem we'll use exactly this approach. Let's make the connection between this formula and the problem to be solved like this: $$x_1=x;\ \ x_2=x+h;\ \ y_1=f(x);\ \ y_2=f(x+h)$$

The change in the input values is: $$\Delta x = x_2-x_1=(x+h)-x=h.$$

To find the change in the output values we need $y_1$ and $y_2$. $y_1$ is $f(x)$ which is... $$y_1=f(x)=x^2.$$

To evaluate $y_2=f(x+h)$ we need to substitute in $(x+h)$ for each occurrence of $x$ in the formula for $f$: $$y_2=f(x+h)=(x+h)^2=x^2+2xh+h^2.$$

Now, take the difference: $$\Delta y=y_2-y_1=x^2+2xh+h^2-x^2=2xh+h^2.$$

Putting $\Delta x$ and $\Delta y$ back into equation \eqref{eq:rate} $$\text{average rate of change}=\frac{\Delta y}{\Delta x}=\frac{2xh+h^2} {h}=\frac{h(2x+h)}{h}=\color{red}{2x+h}.$$