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Average gradient

We notice that the gradient of a curve changes at every point on the curve, therefore we need to work with the average gradient. The average gradient between any two points on a curve is the gradient of the straight line passing through the two points.

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For the diagram above, the gradient of the line is
This is the average gradient of the curve between the points and .

What happens to the gradient if we fix the position of one point and move the second point closer to the fixed point?

Activity 1: Investigation: Gradient at a single point on a curve

The curve shown here is defined by . Point is fixed at and the position of point varies.

Complete the table below by calculating the -coordinates of point for the given -coordinates and then calculating the average gradient between points and .

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Table 1

Average gradient

−2    
−1,5    
−1    
−0,5    
0    
0,5    
1    
1,5    
2    
  1. What happens to the average gradient as moves towards ?
  2. What happens to the average gradient as moves away from ?
  3. What is the average gradient when overlaps with ?

In the example above, the gradient of the straight line that passes through points and changes as moves closer to . At the point where and overlap, the straight line only passes through one point on the curve. This line is known as a tangent to the curve.

Table 2
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We therefore introduce the idea of the gradient at a single point on a curve. The gradient at a point on a curve is the gradient of the tangent to the curve at the given point.

Example 1: Average gradient

Question

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  1. Find the average gradient between two points and on a curve .
  2. Determine the average gradient between and .
  3. Explain what happens to the average gradient if moves closer to .

Answer

Assign labels to the -values for the given points

Determine the corresponding -coordinates

Using the function , we can determine:

Calculate the average gradient


The average gradient between and on the curve is .

Calculate the average gradient between and

The -coordinate of is and the -coordinate of is therefore if we know that and , then .

The average gradient is therefore

When moves closer to

When point moves closer to point , gets smaller.

When the point overlaps with the point , and the gradient is given by .

We can write the equation for average gradient in another form. Given a curve with two points and with and . The average gradient between and is:
This result is important for calculating the gradient at a point on a curve and will be explored in greater detail in Grade 12.

Example 2: Average gradient

Question

Given .

  1. Draw a sketch of the function and determine the average gradient between the points , where , and , where .

  2. Determine the gradient of the curve at point .

Answer

Examine the form of the equation

From the equation we see that , therefore the graph is a “frown” and has a maximum turning point. We also see that when , , therefore the graph passes through the origin.

Draw a rough sketch

Image

Calculate the average gradient between and

Calculate the average gradient for


At point , and .

Therefore

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