Significance of the derivative
Before we look in detail at curves and sketching lets have a look back at what the gradient is; the gradient (slope) of a straight line measures the rate of change of y with respect to the change in x (video here if you need it). There are 3 cases that we must consider when looking at the derivative.
f'(x) > 0
f'(x) < 0
f'(x) = 0
See below:
f'(x) > 0
f'(x) < 0
f'(x) = 0
See below:
From the above examples what do you notice about the curve when:
f'(x) > 0
f'(x) < 0
f'(x) = 0
A curve is considered to be monotonic increasing/decreasing, if it is always increasing or always decreasing.
f'(x) > 0
f'(x) < 0
f'(x) = 0
A curve is considered to be monotonic increasing/decreasing, if it is always increasing or always decreasing.
Now its time for you to have a go, try some of the questions below and if they're too easy try the extension questions below that!