07Aratesofchange

toc = Rates of Change =


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


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


 * 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.

Constant Rates

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


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

Average Rate of Change



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


 * This 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 **.


 * 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// **.

math . \qquad \text{average rate of change}=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{f(x_2)-f(x_1)}{x_2-x_1} \qquad. math

Instantaneous Rate of Change



 * 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** //.


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


 * We will study three ways to find the instantaneous rate of change at a point,
 * Find the derivative at **//P//** from first principles
 * See 7B Differentiation by First Principles ... (or)
 * Differentiate using the appropriate formula and substitute the x value at **//P,//**
 * see Summary of Differentiation Techniques ... (or)
 * Use the CAS calculator
 * see Differentiating on the Classpad

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