Intermediate Value Theorem

How to use the applet

• Drag the point $a$ and $b$ along the $x$-axis to change the interval $[a,b]$.
• Drag the point/slider $c$ (the horizontal level) to choose the target value.
• The graph of $f$ on the chosen interval $[a,b]$.
• A horizontal line $y=c$.
• Intersection points between the graph and the line $y=c$ (up to a fixed maximum number), together with their projections onto the $x$-axis.
• If $f$ is continuous on $[a,b]$ and

then at least one intersection point appears with $x\in(a,b)$, meaning there exists $\xi\in(a,b)$ such that $f(\xi)=c$.

• By moving $c$, you can see that there may be more than one solution $\xi$.
• The applet also highlights the image $f([a,b])$:
  • green segment when the function is continuous on the chosen interval (no “gaps” in the values),
  • orange warning segment when the function is not continuous on the chosen interval (gaps can occur, and IVT can fail).

Description

Link to code.

Reference: Lecture Notes Calculus 1 ( May22,2022 ) Theorem4.7 page 62

This example illustrates how the Intermediate Value Theorem (IVT) holds for continuous functions on a closed interval $[a,b]$.

By moving points $a$ and $b$, one can adjust the domain.

If the functions is continuous and $\min(f(a), f(b))< c < \max(f(a), f(b))$, then there exists at least one $x \in (a,b)$ such that $f(x) = c$. By moving the point $c$ in between the $f(a)$ and $f(b)$ one can observe that there may be more than one such $x$.

To show these points i have created 3 (maximum roots of the function) intersection points with the lineC (parallel ot the x-axis) and graph.

board.create('intersection', [graph, lineC, 1]...

I then make the intersections and their projections to the x-axis visible, only when $c$ is in between $f(a)$ and $f(b)$ (isPointCinBetweenFaFb()) and the projections lie on the interval $[a,b]$ (isPointXCoordInIntervalAB())

A nice thing to notice is that, given continuity, the image of the function, over $[a,b]$, is again a closed interval (with no gaps).

The applets highligths the image of the function, with a green segment when the function is continuous on the given domain and with a “warning” orange segment when the function is not continuous at the given domain and its image contains gaps.