Instantaneous Rate of Change at x=a
f'(x) = Negative
The instantaneous rate of change at x=a is the slope of the curve at the point x=a. It represents the rate at which the value of the function is changing at that precise moment.
To find the instantaneous rate of change at x=a, we can calculate the derivative of the function at that value using the limit definition of the derivative.
We can use the following formula to find the instantaneous rate of change at x=a:
$f'(a) = \lim\limits_{h \to 0} \frac{f(a+h)-f(a)}{h}$
Here, h represents the change in x, and as we take the limit as h approaches 0, we get the slope of the tangent line to the curve at x=a.
Alternatively, if we already have the equation of the tangent line at x=a, we can simply take the slope of that line to find the instantaneous rate of change at x=a.
Overall, the instantaneous rate of change provides valuable information about the behavior of the function at a specific point, and can help us make predictions about its future behavior.
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