Can instantaneous rate of change be zero?

1 Answer
Aug 12, 2014

Yes, it is possible for the instantaneous rate of change to be 0.

For a specific example, imagine the function #f(x) = 3#. This is a horizontal line parallel to the x-axis at the value y=3. This function is unchanging for any value of x, therefore its rate of change is zero.

For a physics example, if I stand still in a spot for 5 minutes, then my rate of change of position after those 5 minutes is 0. Another physics example, if I walked for 5 minutes at a constant velocity of 2.5 miles per hour, then my rate of change of velocity would be zero.

However, a function need not be constant throughout to have the instantaneous rate of change at a given point be 0. For example, suppose that for five seconds I walk forward, and after those five seconds I turn and walk back the way I came for five seconds, ending up at my original position. When I started walking back, assuming that I was walking in the 'positive' direction initially, I began walking in the negative direction.

Thus, my initial velocity was positive (recall that velocity is not just speed, but speed and direction combined), but for the second half, my velocity was negative. If my velocity function is continuous, then at some point between when I was walking forward and when I walked back i must have had a velocity of 0; in other words, I must have stopped and turned around. At that point, my instantaneous rate of change of position was zero.