Everything Maths Grade 12

7.5 CHAPTER 7. DIFFERENTIAL CALCULUS

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4 3 − 2 − − 1 − (^1234) x
y

�(1; 1)

y = x^2

y = 2x− 1

Curve Sketching EMCBL

Differentiation can be used to sketch the graphsof functions, by helpingdetermine the turning points.

We know that if a graphis increasing on an interval and reaches a turning point, then the graphwill

start decreasing after theturning point. The turning point is also knownas a stationary point because

the gradient at a turningpoint is 0. We can then use thisinformation to calculateturning points, by

calculating the points atwhich the derivative ofa function is 0.

Tip

Ifx=a is a turning
point off(x), then:

f�(a)=0

This means that the
derivative is 0 at a
turning point.

Take the graph of y = x^2 as an example. We know that the graph of thisfunction has a turning point

at (0,0), but we can use the derivative of the function:

y�= 2x

and set it equal to 0 to find the x-value for which the graph has a turning point.

2 x = 0

x = 0

We then substitute thisinto the equation of thegraph (i.e. y = x^2 ) to determine the y-coordinate of

the turning point:

f (0) = (0)^2 = 0

This corresponds to thepoint that we have previously calculated.