Informative line

### Graphs Of Derivatives

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?.

# Representation of Non Differentiability of a Continuous Function on the Graph

• 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 $$x\to a$$)

#### For the given graph of $$'f'$$, choose the correct option.

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  $$x$$.

×

$$'f'$$ is non differentiable at $$x=3$$ (sharp corner) and at $$x=5$$ as it is discontinuous there.

$$\therefore$$ Non differentiable at $$x=3$$ and $$x=5$$.

### For the given graph of $$'f'$$, choose the correct option.

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  $$x$$.

Option A is Correct

# Deriving Graph of f' from the Graph of f

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

#### Given below is the graph of a function $$f$$. Identify the correct graph of $$f'$$ from the following options.

A

B

C

D

×

$$'f'$$ is increasing in $$(-\infty,\,0)$$

$$\Rightarrow$$ $$f'$$ is positive in $$(-\infty,\,0)$$

$$f$$ has a horizontal tangent at $$x=0$$

$$\Rightarrow$$ $$f'=0$$ at $$x=0$$

$$f$$ is decreasing in $$(0,\,\infty)$$

$$\Rightarrow$$ $$f'$$ is negative in $$(0,\,\infty)$$

### Given below is the graph of a function $$f$$. Identify the correct graph of $$f'$$ from the following options.

A
B
C
D

Option D is Correct

# Reading Derivative Values from the Graph

• The value of derivative is 0 whenever the slope of tangent is horizontal or parallel to $$x$$ axis.

#### Given is the graph of a function $$f$$. The value of $$f'(1.5)$$ from the graph is

A 0

B 5

C –7

D 8

×

$$f$$ has a horizontal tangent at $$x=1.5$$

$$\Rightarrow$$ $$f'(1.5)=0$$

(Slope of tangent will be 0)

### Given is the graph of a function $$f$$. The value of $$f'(1.5)$$ from the graph is

A

0

.

B

5

C

–7

D

8

Option A is Correct

# Deriving Graph of f from the Graph of f'

• 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.

#### Given the following graph of $$f'$$ , choose an appropriate graph of $$f$$.

A

B

C

D

×

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 $$f$$.

Here $$f'<0$$ in  (0,3)

$$\therefore f$$ is decreasing (0,3)

$$f>0$$ in (3,4)

$$\therefore f$$ is increasing in (3,4)

$$f'=0$$ at $$x =3$$ therefore there is a  turning point at $$x =3$$

$$f'$$ has a turning point at $$x = 2$$

$$\therefore\,f$$ has an inflection point at $$x = 2$$ .

$$\therefore$$Correct option is 'b'.

### Given the following graph of $$f'$$ , choose an appropriate graph of $$f$$.

A
B
C
D

Option B is Correct