Applications of Differentiation to Functions and Graphs

Stationary Points

The points at which a function changes its direction are termed as stationary points, which are points where the derivative equals zero.
By finding the derivative of a function and setting it equal to zero, the values of x which correspond to stationary points can be located.

There are three types of stationary points: maxima, minima, and points of inflection. These are all points where the function’s gradient is zero.
A maximum stationary point is a high point on the graph and is determined by the change of the sign of the derivative from positive to negative.
A minimum stationary point is a low point on the graph is determined by the change in the derivative from negative to positive.
A point of inflection signifies a change in the nature of the curve, usually from a curve that is concave upwards to concave downwards, or vice versa.

The presence of a stationary point can be verified by taking the derivative of a function and setting it to zero, however, it does not disclose the nature of the stationary point whether it’s a maxima, minima or point of inflection.
The Second Derivative Test is useful to identify the nature of the stationary point. If f’‘(x) > 0, then f has a local minimum at x; if f’‘(x) < 0, then f has a local maximum at x; if f’‘(x) = 0, the test is inconclusive.

Differentiation can be used to determine how rapidly a quantity is changing at a particular instant, this is termed as the rate of change.
The derivative of a function provides the rate of change of the output value with respect to its input value.

Differentiation is fundamentally used to solve optimisation problems – problems where you have to make something as large or small as possible.
Many real-world problems in physics, engineering and economics can be modelled and solved using optimisation techniques, including maximising profit, minimising cost, or minimising waste.

It is crucial to understand how a graph of a function can be sketched using derivative information. The sketching involves determining critical points, intercepts, asymptotes, end behaviour, increasing / decreasing nature, and concavity.
The Point of inflection can be found by finding second derivative and setting it to zero.
By combining the information obtained from a function and its derivatives, a precise sketch can be created.