13D - Instantaneous rate of change
Previously we have examined finding the average rate of change by determining the gradient of a line segment between two points on a curve/graph. However, this method only gave us an average rate of change over an interval. In many situations we are interested in the rate of change at any given instant; that is, the instantaneous rate of change. This understanding is the precursor to our study of calculus which is undertaken in Section 14 and Section 15.
Approximating the instantaneous rate of change using tangents:
The gradient of the tangent is identical to the gradient of an infinitely small interval on the curve. Therefore, by calculating the gradient of the tangent, at a point \(P\), you determine the instantaneous rate of change at \(P\).
For this section we only require approximations of the gradient of a tangent. To approximate the gradient use the following steps:
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■ A tangent to a curve or graph at a point \(P\), is a line which touches the curve exactly once at \(P\) without intersecting (cutting) the curve at that point. |
13D - Example 1: Approximating the instantaneous rate of change using tangents
13D - Example 2: Approximating the instantaneous rate of change using tangents
Determining the gradient of a tangent using CAS:
To approximate the gradient of a tangent to a graph on the Casio ClassPad II, use the following steps:
To approximate the gradient of a tangent to a graph on the Casio ClassPad II, use the following steps:
- Type the equation of the curve into the Main screen.
- Drag the equation into the graph screen to sketch the graph.
- Click: Analysis → Sketch → Tangent.
- Using the hard keypad, enter the \(x\)-value you want that tangent at and hit OK.
- Hitting exe on the hard keypad will bring up the equation (\(y=mx+c\)).
- The value of \(m\) (the coefficient of \(x\)) is the approximate value of the gradient.
13D - Example 3: Approximating the instantaneous rate of change (CAS)
Using a CAS calculator, determine the instantaneous rate of change of the function \(f(x)=e^x\) at \(x=-2\). State the rate of change correct to 3 decimal places.
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13D - Example 3: Video solution
13D - Example 3: Practice
Question 1: ABC Question 2: ABC 13D - Example 3: Solutions
Question 1: ABC Question 2: ABC |