Instantenous Rate of Change
Slope of tangent line at a point, value of derivative at a point
The instantaneous rate of change is the rate at which a function is changing at a specific point. In calculus, the instantaneous rate of change is determined using the derivative of the function.
The derivative of a function f(x) at a point x=a gives the instantaneous rate of change of the function at that point. The formula for the derivative is:
f'(a) = lim(h->0) [f(a+h) – f(a)]/h
where f'(a) is the derivative of the function f(x) at the point x=a, and h is a small value that approaches zero.
To find the instantaneous rate of change of a function at a specific point, we can use the following steps:
1. Determine the function f(x) and the point x=a.
2. Find the derivative of the function f(x) using the above formula.
3. Substitute the value of a in the derivative formula to get the instantaneous rate of change at that point.
For example, if we have a function f(x) = x^2, and we want to find the instantaneous rate of change at x=2, we can follow these steps:
1. f(x) = x^2 and a=2
2. f'(a) = lim(h->0) [(2+h)^2 – 2^2]/h
3. f'(2) = lim(h->0) [4h + h^2]/h = lim(h->0) [4 + h] = 4
Therefore, the instantaneous rate of change of the function f(x) = x^2 at x=2 is 4. This means that the slope of the tangent line to the function at that point is 4.
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