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.

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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. .

• 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. .

• 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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