instantaneous rate of change
(f(x + h) – f(x))/h
The instantaneous rate of change is a concept in calculus and refers to the rate at which a given function changes at a specific point. In other words, the instantaneous rate of change gives us the exact rate of change in a function at a certain single point, as opposed to an average rate of change over a certain interval.
To calculate the instantaneous rate of change of a function at a point, we take the limit of a difference quotient as the interval approaches zero. More specifically, we take the limit of the difference quotient:
(f(x + h) – f(x))/h
as h approaches 0. This limit gives us the slope of the tangent line to the graph of the function at the point x.
The formula for the instantaneous rate of change of a function f(x) at a point x=a is:
lim(h → 0) [(f(a+h) – f(a))/h]
Alternatively, we can also use the derivative of the function to calculate the instantaneous rate of change at a point. The derivative gives us a formula that describes the rate of change of the function at any point.
Overall, the instantaneous rate of change is a very important concept in calculus as it allows us to study the behavior of functions at specific points and in turn, use this information to solve a variety of complex problems.
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