Differentiation can be used to identify increasing and decreasing functions. The intervals where a function is either increasing or decreasing can then be used to sketch the curve of a derivative.
Increasing & Decreasing Functions
Recall that upward sloping straight lines have a positive gradient whereas downward sloping straight lines have a negative gradient. The same applies to curves. Gradients on a curve are always changing but an upward sloping curve has a positive gradient and a downward sloping curve has a negative gradient.
Recall the graph of . You will notice that for positive x, the graph has a positive gradient; for negative x the graph has a negative gradient; and for x=0 the gradient is 0.
This can be seen from the gradient function . Find out more about differentiating. 2x is positive when x is positive, negative when x is negative and 0 when x is 0.
Now consider a slightly more complicated example of the cubic . Click here to see how to sketch cubics. Firstly, it can be seen from the graph that for x<1 the graph is increasing. At x=1, there is a stationary point where the gradient is 0. Next, for x between 1 and 3, the graph is decreasing. There is another stationary point at x=3. Then the curve increases again. We can say that the cubic is a strictly increasing function on x<1 or x>3 and a strictly decreasing function on x>3. Note that the inequalities are strict.
- The function f(x) is increasing on the interval [a,b] if for all x on [a,b]. Additionally, f(x) is strictly increasing if the inequality is strict.
- The function f(x) is decreasing on the interval [a,b] if for all x on [a,b]. Additionally, f(x) is strictly decreasing if the inequality is strict.
Now consider again the graph of the cubic above. We can attempt to sketch the graph of the derivative by first recognising that it should be quadratic. Then, recall that the stationary points are at x=1 and x=3 and so the derivative must cross the x-axis at these points. Finally, the derivative is negative for 1<x<3, i.e. below the x-axis, and positive otherwise. It follows that the curve of the derivative is as seen here. This has been achieved without finding an expression for the derivative.
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