Exploring MeanValue Theorem Activities

The graph represents the orange line as the secant and the black line as the tangent at a point C.

Complete the sentence:

1) The slope of the secant is not equal to the slope to the tangent line when:

the slopes are not parallel the slopes are parallel

2) The slope of the secant is defined as:

average rate of change between two points neither equation of a line passing through two points

3) The slope of the tangent of the line represents:

slope of the curve. instantaneous rate of change at that point derivative of the function.

Activity : Exploring Mean Value Theorem

In the following activity you are going to explore the relation between the slopes of the tangent and the secant line that that lies within specific interval [a,b].[br]Use the graph to answer the questions below.[br]Express your own conclusion and then compare it with your classmate.

Drag the slider to construct the tangent and answer the following questions

2) Drag the slider until the tangent line is parallel to the secant line. [br]Let's call the interval on the x-axis between the points of secancy a,b : [a,b].[br]Compare the results when the function f(x) is in the following cases: [br]a) Differentiable[br]b) Non-differentiable[br]c) discontinous at x=b[br] In which state You found a tangent line with point of tangency in the open interval (a,b)[br]that is parallel to the corresponding secant line?[br][br]

discontinuous Differentiable Non-differntiable

2) Deduce the the formula of the slopes of: the tangent at the point of tangency c and the secant line passing through the points A(a, f(a)) and B(b, f(b))

When the lines are parallel:

3) Will this always be possible for any function and any interval? Why or why not?

[br]Tick [b]ALL[/b] the conditions for the tangent line with point of tangency between the interval (a,b) is parallel to the corresponding secant line AB.

Continuous Differentiable Only in polynomials Discontinuous Krips curve

Your Findings:

Write with your own words a hypothesis indicating relation between the slopes of a tangent line at the point of tangency c and the slope of the secant line of interval [a,b].[br]Hint: use the parts 2 and 3 above