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- Mar 10, 2012

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To define what it means to say that a car is going at the speed of 2mph would again require the use of limits. If $\lim_{h\to 0}\frac{x(t_0+h)-x(t_0)}{h}=2$ then the speed at $t=t_0$ is $2$, where $x(t)$ is the displacement of the car from any given reference at time $t$. But I guess you already knew that. I don't what more can I say.I'm not really sure how this concept makes sense. We give it meaning by way of a limit but, to my mind, at any instant the car is motionless so how can it have a velocity? What, in essence, does it mean to say that at some point in time a car is going 2 mph?

Can you elaborate on "the car is motionless at every instant"??

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- Mar 10, 2012

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Can you say something more about the following?

At time t, the car is stationary...

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In fact, the 'issue' generalises to differentiation of non-linear functions. Take f(x)=x^2.

when x=2, the derivative is 2. Now if one is asked to explain what a derivative tells you, then you would probably say along the lines 'the rate of change of y with x'. But what does the rate of change mean at a single point? I hope this helps you to understand.

- Mar 10, 2012

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Would you agree that the 'average rate of change' between $2$ and $4$ of the function $y=x^2$ is $(16-4)/2=6$?In fact, the 'issue' generalises to differentiation of non-linear functions. Take f(x)=x^2.

when x=2, the derivative is 2. Now if one is asked to explain what a derivative tells you, then you would probably say along the lines 'the rate of change of y with x'. But what does the rate of change mean at a single point? I hope this helps you to understand.

If yes, then what is the average rate between $2$ and $2+h$ for h=1, 0.9, 0.8,...,0.1 ?

Does some pattern emerge? As $h$ gets smaller and smaller, you don't have average rate

Of course this is just an intuitive way of looking at it. The real thing is limits. But one should know that the concept of limits is just a formalization of our intuition, just as most of mathematics is.

In fact, the 'issue' generalises to differentiation of non-linear functions. Take f(x)=x^2.

when x=2, the derivative is 2. Now if one is asked to explain what a derivative tells you, then you would probably say along the lines 'the rate of change of y with x'. But what does the rate of change mean at a single point? I hope this helps you to understand.

It means exactly what the definition says. The rate of change at a point is the limit of the average rate of change over intervals containing the point as the length of the intervals decrease to zero (with due allowances for end effects which I can't be bothered with here).

If you want to quibble about average rate of change over an interval the answer is just the same, it is defined to be the change in the function value divided by the corresponding change in x (the interval length).

There is no metaphysical mystery here, just a definition.

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