Understanding Rates of Change

The term rate of change refers to how one quantity changes in relation to another. In calculus, it often signifies the slope of a function.
The instantaneous rate of change at a certain point is the slope of the line which is tangent to the function at that point, this can often be found using derivatives.
The average rate of change over a given interval is calculated as the change in output values divided by the change in input values. It is the slope of the secant line between two points on the graph of the function.

Calculating Rates of Change

To find the average rate of change of a function over a given interval, subtract the function value at the starting point from the function value at the ending point, then divide by the change in input value.
The instantaneous rate of change is more complex as it involves finding the limit as the interval over which the change is measured becomes infinitesimally small. This is essentially calculating the derivative of the function at that point.

Calculating rates of change is crucial in many scientific, engineering, and mathematical scenarios. For instance, in physics, velocity is the rate of change of distance with respect to time, and acceleration is the rate of change of velocity with respect to time.
Similarly, in economics, rates of change are used to examine trends over time, such as the change in the value of a currency or changes in population size.

A Point of Inflection is a point on the curve where the curve changes its direction of curvature. Mathematically, it is classified as a point where the second derivative changes sign.
Increasing and Decreasing Functions: If a function’s derivative is positive over an interval, the function is increasing over that interval. If the derivative is negative, the function is decreasing.

Average Rate of Change: (f(b) - f(a)) / (b - a)
Instantaneous Rate of Change: df/dx evaluated at the point of interest, often written as f’(x).
Inflection Point: A point where the second derivative of a function f changes sign, often written as f’‘(x) = 0.