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 \(x\to a\))

Illustration Questions

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. 

image
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

Illustration Questions

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.

image
A image
B image
C image
D image

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.

Illustration Questions

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

image
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.

Illustration Questions

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 \(f\).

image
A image
B image
C image
D image

Option B is Correct

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