Hello, BrownianMan!

Good question . . . but the answer isNo.

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 atvalues ofintegral

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 unitsto the left.

Therefore, the total distance is: .

Get it?