Answer:
Explanation:
1) If we obtain the derivative of a function and evaluate it for a value of x that corresponds to a local maximum value, the result of the evaluation will be zero. This is because at a local maximum, the slope of the function is horizontal, which means the derivative is equal to zero.
To understand why, let's consider a simple example. Suppose we have a function f(x) and we take its derivative f'(x). At a local maximum, the graph of the function is concave downward, and the slope of the tangent line at that point is horizontal.
When we evaluate the derivative f'(x) at this point, we are essentially finding the slope of the tangent line. Since the tangent line is horizontal, its slope is zero. Therefore, the derivative evaluated at a local maximum will be zero.
2) If we have a particle in motion and f(x) represents the function of the particle's acceleration, then the integral of f(x) represents the change in velocity of the particle over a specific time interval.
To understand this concept, let's break it down. The derivative of a function represents its rate of change. In this case, the derivative of the acceleration function f(x) gives us the rate at which the particle's velocity is changing at any given point in time.
When we integrate the acceleration function f(x), we are essentially summing up all the instantaneous changes in velocity over a given time interval. This integral gives us the total change in velocity of the particle during that time interval.
For example, if we integrate the acceleration function from time t1 to time t2, the result would be the change in velocity of the particle from t1 to t2. This change in velocity could be positive (if the particle is speeding up), negative (if the particle is slowing down), or zero (if the particle's velocity remains constant).
In summary, the integral of the acceleration function represents the change in velocity of the particle over a specific time interval.