10D - Instantaneous rate of change

Learning intentions:

In this section we will examine:

    • Tangents to curves and their use in calculating the instantaneous rate of change.
    • Using CAS to determine the gradient of a tangent on a graph.

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 undertaken in section 11 and section 12.

Tangents

A tangent to a curve or graph at a point P, is a a line which touches the curve exactly once at P without intersecting (cutting the curve). 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 the point P.

Figure 1 - The instantaneous rates of change of f(x) at P is 2.

Approximating the gradient of a tangent

For this section we only require approximations of the gradient of a tangent. To approximate the gradient use the following steps:

  1. Using a ruler, draw a tangent onto the curve at the specified point, P.
  2. After extending the tangent a suitable length, select two points (x1, y1) and (x2, y2) that exist on the tangent line (select easy points to work with.
  3. Calculate the gradient, m, using the formula:

Note: Approximations will be most accurate if grid paper is used.

10D - VIDEO EXAMPLE 1:

For the graph on the grid below, approximate the instantaneous rate of change at:

  • x = 2
  • x = 6

10D - VIDEO EXAMPLE 2:

The temperature, T (degrees Celsius) inside an oven after t seconds is graphed below. Approximate the instantaneous rate of change at 50 seconds.

Determining the gradient of a tangent using CAS

A CAS calculator can be used to determine the gradient of a tangent to a graph.

Casio ClassPad II

To approximate the gradient of a tangent to a graph on the Casio ClassPad II, use the following steps:

    1. Type the equation of the curve into the Main screen.
    2. Drag the equation into the graph screen to sketch the graph.
    3. Click: Analysis → Sketch → Tangent.
    4. Using the hard keypad, enter the x-value you want that tangent at and hit OK.
    5. Hitting exe on the hard keypad will bring up the equation (y = mx + c).
    6. The value of m (the coefficient of x) is the approximate value of the gradient.

Figure 2 - Approximation of the gradient (3.2) of the tangent to the curve y = x2 at x = 1.6.

10D - VIDEO EXAMPLE 3:

Using a CAS calculator, determine the instantaneous rate of change of the function f(x) = ex

at x = -2. State the rate of change correct to 3 decimal places.

Success criteria:

You will be successful if you can:

    1. Draw a tangent to a curve at a point.
    2. Calculate the gradient of a tangent as a point.
    3. Interpret the real-world, contextual, meaning of the gradient of a tangent at a point.
    4. Use a CAS calculator to determine the gradient of a tangent at a point on a graph.