Notes:

1. The main purpose of differentiation is to find the slope of a tangent to a curve.

Solution: f ' ( x) = 3x² + 4x + 3 and when x = 1, f ' ( x) = 3(1)² + 4(1) + 3 = 10.

i.e. at (1 , 8) the slope of the tangent to the curve f(x) = x³ + 2x² + 3x + 2 is 10.

2. The slope of the x - axis is 0
1. The slope of a line is defined as the tan of the angle made between the line and the positive direction of the x - axis. (Note the angle is always measured in the anti-clockwise direction)

Solve the equation = 0, to give the values of x at which the maximum and minimum points occur.

Let us now consider the graph f(x) = x³ - 6x² + 9x - 10. In order to draw this graph we do out a table:

To find the slopes of the tangents to the graph at each of the above points we make out another table using f ' ( x) = 3x² -12x + 9 and graph this as well

To find the slopes of the tangents to this second graph at each of the above points we repeat the procedure i.e. we make out another table using f ' ' ( x) = 6x - 12.

These three graphs should show us the relationship between the turning points and the differentiation process. We draw these three graphs underneath each other as follows:

The following points should be clear from the graph:

1. At the point a(1,-6) there is a local maximum, f ' (1) = 0 and f ' ' (1) is negative.

In General: For a maximum point f ' (x) = 0 and f ' ' (x) < 0

2. At the point c(3,-10) there is a local minimum, f ' (3) = 0 and f ' ' (3) is positive.

In general: For a minimum point f ' (x) = 0 and f ' ' (x) > 0

3. At the point (2,-8) there is a point of inflection (i.e. the point where the curve goes from being concave downwards to concave upwards or vice versa), f For a maximum point f ' ' (2) = 0

In general: For a point of inflection, f ' ' (x) = 0

4. A curve is increasing where f ' (x) > 0 (Here for x < 1 and x > 3)

 

1. A curve is decreasing where f ' (x) < 0 (Here for 1 < x < 3)

Notes 4 and 5 above lead to a definition for the maximum and minimum points:

Local Maximum Point: A point on a curve at which the curve changes from an increasing function to a decreasing function.

Local Minimum Point: A point on a curve at which the curve changes from a decreasing function to an increasing function.

Point of inflection: A point on a curve at which the curve changes from being concave downwards to concave upwards or vice versa.

A quadratic function has 1 maximum OR minimum

A cubic equation has 1 maximum, 1 minimum and 1 point of inflection. It can happen that the three of these may coincide (e.g. in the function y = x³). When this occurs the resulting point is called a Saddle point.