Find the value of derivative whenever the slope of tangent is horizontal or parallel to x axis. Practice for graphing using first and second derivatives, deriving graph of f given the graph of f?.

- If a function is continuous at \(x=a\), then it is non differentiable at \(x=a\) if the graph of \('f'\) has a sharp corner or a kink at \(x=a\).

- No tangent can be drawn at sharp corner, so there is no tangent and hence \(f'\) can not be found.
- Three ways a function can be non differentiable at \(x=a\).

Vertical tangent at \(x=a\)

(Tangent line becomes steeper & steeper as \(x\to a\))

A \('f'\) is non differentiable at \(x=3 ,5\)

B \('f'\) is non differentiable at \(x=-3\)

C \('f'\) is differentiable at \(x=5\)

D \('f'\) is non differentiable at three values of \(x\).

Consider the graph of a function \(f\).

To sketch the graph of \(f'\) from \(f\), follow these points.

(1) We find (or estimate) the value of derivative at some points by examining the slope value of tangent at that point.

e.g. Slope at \((-3,\,2)\) is say \((1.2)\) then \((-3,\,1.2)\) will be a point on \(f'\). Do this for several points.

(2) Tangent at A and B are horizontal, meaning the value of derivative is 0 at these points, so for \(A(-2.5,\,3)\) there will be a point \((-2.5,\,0)\) on \(f'\) and for \(B\left(-\dfrac{1}{2},\,-2\right)\) there will be a point \(B\left(-\dfrac{1}{2},\,0\right)\) on \(f'\).

(3) When tangents have positive slope \(\to\) \(f'\) is positive.

When tangents have negative slope \(\to\) \(f'\) is negative.

\(\therefore\) Graph of \(f'\) , as shown in figure below,

At \(\left(-\dfrac{1}{2},\,-2\right)\) and \((-2.5,\,3)\) there is a turning point of \(f\)

\(\;\therefore\;f'\) will be 0

- The value of derivative is 0 whenever the slope of tangent is horizontal or parallel to \(x\) axis.

- If we are given the graph of the derivative of function \(f\) i.e. \(f'\) and we desire to sketch the graph of \(f\) , then the following points should be noted.

1. When \(f'\) is positive (i.e. graph is above \(x\) axis) the graph of \(f\) is increasing, while when \(f'\) is negative (i.e graph is below \(x\) axis ) the graph of \(f\) is decreasing.

2. When \(f' =0\) i.e the point where graph crosses \(x\) axis, the graph of \(f\) has a turning point.

3. The points where \(f'\) has a turning point , will be points of inflection for \(f\) .

- Note that when \(f'\) is given , the graph of \(f\) is not unique.
- e.g. If \(f'(x) = cos \,x \Rightarrow f(x) = sin \,x+c\)

\(\therefore\) there are many choices of \(f\) possible.