05 1rates

# Rates of Change

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A **rate of change** describes how one quantity changes with respect to another.

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**Speed**is a rate of change.- Speed measures the distance travelled (change of position) with respect to the time taken.

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- The
**gradient**(or slope) of a graph is a rate of change.- Gradient measures how the vertical distance changes with respect to the horizontal distance.

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## Constant Rates

- The
**gradient**of a straight line is the same no matter which two points we use to find it.- The rate of change of
**y**with respect to**x**is constant.

- The rate of change of

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- Water flowing out of a tap (usually) has a constant rate.
- For example, water efficient shower heads allow a constant flow of 9 litres of water per minute.

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## Average Rate of Change

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- An
**average rate of change**describes how one quantity changes with respect to another across an interval (usually time).

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- The
**average rate**is found by calculating the gradient of the straight line joining the endpoints of the interval.- The straight line joining two points is called a secant.

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- The average rate of change of a function $y = f(x)$ between two points
**P**and**Q** - is equal to the gradient of the straight line passing through
**P**and**Q**.

- average rate of change =$\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{f(x_2)-f(x_1)}{x_2-x_1}$

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## Instantaneous Rate of Change

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- If the rate is variable, it is often useful to know the rate of change at a specific time or point.

- This is referred to as the
**instantaneous rate of change**.

- The
**instantaneous rate of change**of a function**y = f(x)**at a point**P**is equal to the gradient of the tangent of the graph at**P**.

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- The process of finding the gradient of a graph of the function at a given point P is called
**differentiation**.

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We will study three ways to find the instantaneous rate of change at a point,

- Find the derivative at P from first principles … (or)
- Differentiate using the appropriate formula and substitute the x value at P, … (or)
- Use the CAS calculator

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