Instantaneous Rate of Change at x=a
f'(x) = Negative
The instantaneous rate of change at x=a is the rate of change of a function at a specific point a, which is determined by finding the slope of the tangent line to the graph of the function at that point. Mathematically, we can express this as follows:
f'(a) = lim (x→a) [f(x) – f(a)] / (x – a)
where f'(a) denotes the derivative of the function f at the point a.
To find the instantaneous rate of change at x=a, we need to take the limit of the difference quotient as x approaches a. This formula gives us the slope of the tangent line to the curve at the point a. It tells us how fast the function changes at that specific point.
For example, suppose we have a function f(x) = x^2 + 3x – 5. We want to find the instantaneous rate of change at x=2. We can do this by using the derivative formula above:
f'(2) = lim (x→2) [(x^2 + 3x – 5) – (2^2 + 3(2) – 5)] / (x – 2)
= lim (x→2) [(x^2 + 3x – 5) – 5] / (x – 2)
= lim (x→2) [(x^2 + 3x – 10) / (x – 2)]
= 7
Therefore, the instantaneous rate of change of the function f(x) at x=2 is 7. This means that the function is changing at a rate of 7 units per unit of x at that specific point.
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