Instantaneous Rate of Change

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Jeterion85

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What is Instantaneous rate of change means?
For example if i have the function x2 the derivative 2x, and for x=2 is 4.
How does that 4 means?
Thank you!
 
Jeterion85 said:
What is Instantaneous rate of change means?
For example if i have the function x2 the derivative 2x, and for x=2 is 4.
How does that 4 means?
Thank you!
Click to expand...

Instantaneous rate of change means that rate of change of something at a given moment. In this case it is for the parabola y = x^2. To find the rate of change at x=2, we can use the derivative function y=2x and evaluate for x=2, which is 4. So yes, the slope of the line tangent at x=2 is 4.
 
[MATH]\dfrac{2.1^2 - 2^2}{2.1 - 2} = \dfrac{4.41 - 4}{0.1} = \dfrac{0.41}{0.1} = 4.1.[/MATH]
[MATH]\dfrac{1.9^2 - 2^2}{1.9 - 2} = \dfrac{3.61 - 4}{-0.1} = \dfrac{-0.39}{-0.1} = 3.9.[/MATH]
[MATH]\dfrac{2.01^2 - 2^2}{2.01 - 2} = \dfrac{4.0401 - 4}{0.01} = \dfrac{0.0401}{0.01} = 4.01.[/MATH]
[MATH]\dfrac{1.99^2 - 2^2}{1.99 - 2} = \dfrac{3.9601 - 4}{-0.01} = \dfrac{-0.0399}{-0.1} = 3.99.[/MATH]
[MATH]\dfrac{2.001^2 - 2^2}{2.01 - 2} = \dfrac{4.004001 - 4}{0.001} = \dfrac{0.004001}{0.001} = 4.001.[/MATH]
[MATH]\dfrac{1.999^2 - 2^2}{1.999 - 2} = \dfrac{3.996001 - 4}{-0.001} = \dfrac{-0.003999}{-0.1} = 3.999.[/MATH]
Is it not clear to you intuitively that when x gets close enough to 2, the change in x^2 is so close to 4 times the change in x that the difference is completely insignificant?
 
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