Good question . . . but the answer is No.
1) If is the position function of a particle moving in a straight line,
would you be able to find its total distance traveled in, say 3 seconds, by finding ,
and calculating the absolute value between each of them and then summing those values,
as opposed to differentiating the function first, setting the derivative to 0, and solving for ?
Would you get the same answer? . . . . No
It worked for this problem because the turning points occured at integral values of
Suppose we have: .
And we want the distance traveled in the first two seconds.
By your method, we would have:
Our interpretartion would be:
. . In the first second, the particle moved 31 units to the right.
. . In the next second, it moved 1 unit to the right.
Hence, the total distance moved is 32 units . . . But this is wrong!
We have: .
. . .Then: .
To find turning points, solve
Hence, turning points occur at: .
Our table would look like this:
In the first second, the particle moved 31 units to the right.;
In the next half-second, it moved 2.75 units to the right.
In the last half-second, it moved 1.75 units to the left.
Therefore, the total distance is: .