# Applications of Differentiation to Functions and Graphs

# 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.

## Types of Stationary Points

- 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.

## Determining the Nature of Stationary Points

- 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.

## Rate of Change

- 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.

## Optimisation Problems

- 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.

## Sketching Curves

- 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.