Can instantaneous rate of change be zero?

Answer 1

It is true that a zero instantaneous rate of change is possible.

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.

In terms of physics, if I remain motionless for five minutes, my rate of change of position would be zero. Similarly, if I walked for five minutes at a steady pace of 2.5 miles per hour, my rate of change of velocity would also be zero.

To have the instantaneous rate of change at a given point, however, a function need not be constant throughout. For instance, let's say that I walk forward for five seconds, and then I turn around and walk back the way I came for another five seconds, returning to my starting position. Assuming that I was initially walking in the 'positive' direction, I started walking in the negative direction when I started walking back.

Since velocity is the product of speed and direction, it follows that while my initial velocity was positive, it was negative for the second half of the walk. If my velocity function is continuous, then at some point, between the forward and return walks, I must have had a velocity of 0—that is, I must have stopped and turned around. At that point, my instantaneous rate of change of position was zero.

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Answer 2

Yes, the instantaneous rate of change can be zero. This occurs at points where the function has a local maximum, local minimum, or point of inflection.

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Answer 3

Yes, the instantaneous rate of change can be zero. This occurs when the function is momentarily flat or stationary at a specific point, meaning its slope (rate of change) at that point is zero.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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