Which is the proper way to calculate average speed.

Question

Let's say we have a graph, a distance-time graph, and there are couple points on the graph. The points are connected to each other with different slopes, so with different speeds between the points.

What is the correct way of calculationg speed?

Method 1

Subtract the initial distance from the final distance from the whole graph and divide by time.

Method 2

Subtract the initial distance from final distance of each segment of speed. Then divide each segment difference by time, abd average them all together.

I keep insisting on method 2 and my teacher on method 1. Which is more correct? By the way they produce similar but different results.

Answer 1

For a function $f(x)$, its average value on an interval $[a, b]$ is given as; $$f_{avg} =\frac{1}{b-a}\int_a^b f(x)dx$$

Suppose the given distance function respect to time ($t$) is given by $s(t)$. Then the velocity would be given as $v(t)=s'(t)$

Now if we were to calculate the average velocity, we would compute; $$v_{avg}=\frac{1}{b-a}\int_a^b v(t)dt$$

Using the fundamental theorem of calculus, $$v_{avg}=\frac{1}{b-a}\int_a^b v(t)dt=\frac{1}{b-a}[s(t)]_a^b$$

which is equal to $$\frac{s(b)-s(a)}{b-a}$$

Implication: Subtract initial position from the final position and divide it by whole time. So method 1 would be correct for calculating the average velocity.