Question

Use the given graph to estimate the value of each derivative. Then sketch the graph of .f'

Answer 1

Step 1 : 

(a)

Approximate a tangent line to the given curve at the point .

Slope of the tangent line is the derivative of the function.

Draw the tangent line at .

 

Observe the graph.

The tangent line passes through the points and .

Slope of the tangent line at is .

Step 2 :

(b)

Approximate a tangent line to the given curve at the point .

Slope of the tangent line is the derivative of the function.

Draw the tangent line at .

 

Observe the graph.

Since the tangent line is horizontal, .

Slope of the tangent line at is .

Step 3 : 

(c)

Approximate a tangent line to the given curve at the point .

Slope of the tangent line is the derivative of the function.

Draw the tangent line at .

 

Observe the graph.

The tangent line passes through the points and .

Slope of the tangent line at is .

Answer 2

Contd..........

Step 4 : 

(d)

Approximate a tangent line to the given curve at the point .

Slope of the tangent line is the derivative of the function.

Draw the tangent line at .

 

Observe the graph.

The tangent line passes through the points and .

Slope of the tangent line at is .

Step 5 :

(e)

Approximate a tangent line to the given curve at the point .

Slope of the tangent line is the derivative of the function.

Draw the tangent line at .

 

Observe the graph.

The tangent line passes through the points and .

Slope of the tangent line at is .

Step 6 :

(f)

Approximate a tangent line to the given curve at the point .

Slope of the tangent line is the derivative of the function.

Draw the tangent line at .

 

Observe the graph.

Since the tangent line is horizontal, .

Slope of the tangent line at is .

Answer 3

Contd.......

Step 7 : 

(g)

Approximate a tangent line to the given curve at the point .

Slope of the tangent line is the derivative of the function.

Draw the tangent line at .

 

Observe the graph.

The tangent line passes through the points and .

Slope of the tangent line at is .

Step 8 :

Make table for the slopes of these lines for given values of x :

 Graph : 

1) Draw the co-ordinate plane.

2) Plot the points.

3) Connect the points with a smooth curve.

Solution :

(a)

(b)

(c)

(d)

(e)

(f)

(g)

Graph of the derivative f '.