• To understand the nature of curves. To know when a curve is increasing and decreasing. • To fi nd local maxima and minima. • To sketch different types of curves. • To work out rate of change problems. • To solve modelling and optimisation problems.
ACTION ACTIVITY
ACTION
Understanding the nature of slopes (1)
OBJECTIVE To draw tangents to curves and explore the geometrical meaning of differentiating curves
21.1 Curves Y 3
When you draw the curve that describes a function y = f (x), you always read the curve from left to right.
This means that the x co-ordinate is always increasing: x2 − x1 = 3 − 1 = +2.
Imagine that you go for a walk in the hills, starting at A, in the graph below.
As you go from A to B, the value of the y co-ordinate is increasing (I ) as you are getting higher. The curve then fl attens out at B (stationary point). As you go downhill from B to C, the value of the y co-ordinate decreases (D). Now that you are getting lower, the curve fl attens out again at C (stationary point). Climbing from the bottom of the valley from C to E, the value of y increases (I ) again.
