This section gives an informal introduction to average rate of change, which is important in Calculus.
You must understand rate of change to fully appreciate the special property of the natural exponential function.
In an earlier section we talked about ‘walking along
a curve’ and paying attention to how the ‘elevation’ changes. In taking each ‘step’ along the curve (at right), and the vertical movements (shown in green) UP is positive, DOWN is negative for horizontal movements: TO THE RIGHT is positive, TO THE LEFT is negative 

Let's do some computations. It doesn't matter if you move from left to right, or right to left. The average rate of change computation is the same. Why? Average rate of change between two points is just the slope of the line between the two points! 



Given two points $\,A(x_1,y_1)\,$ and $\,B(x_2,y_2)\,$, you use exactly the same formula for:
On the curve at right, the average rate of change between points $\,A\,$ and $\,B\,$ is $\displaystyle\,\frac{3 \text{ up}}{4 \text{ right}} = \frac{3}{4}\,$.
Imagine walking along the thick black curve at right from point $\,A\,$ to point $\,B\,$. Alternately, imagine that you're in a pitch black room. Look at the blue dashed segments on the picture. On average, in moving from $\,A\,$ to $\,B\,$, you move $\,\frac 34\,$ units up for each unit to the right! 
On average, in moving from $\,A\,$ to $\,B\,$, you move $\,\frac 34\,$ units up for each unit to the right! 
The average rate of change of a function $\,f\,$ on an interval $\,[a,b]\,$ 
The black curve is the graph of $\,f\,$. The average rate of change of $\,f\,$ on $\,[a,b]\,$ is the slope of the red line. 
You should be able to compute average rate of change in all the following situations:
GIVEN:  AVERAGE RATE OF CHANGE: 
a change in ‘outputs’: $\,\Delta y\,$ a change in ‘inputs’: $\,\Delta x\,$ 
$\displaystyle\,\frac{\Delta y}{\Delta x}$ 
two points: $\,A(x_1,y_1)\,$ and $\,B(x_2,y_2)\,$  $\displaystyle\,\frac{y_2  y_1}{x_2  x_1}$ 
a function $\,f\,$ and an interval $\,[a,b]\,$  $\displaystyle\,\frac{f(b)  f(a)}{b  a}$ 
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
