Understand the Intermediate Value Theorem.

The Intermediate Value Theorem (IVT) tells us that if a function is continuous, then to get from one point on the function to another point, we have to hit all [math]y[/math]-values in between at least once. For example, we know intuitively that the temperature of an object over time is a continuous function - it cannot change instantly, it cannot be infinite, and it must always exist (it must always have a temperature value). If an object had a temperature of [math]44^{\circ}[/math] an hour ago and now measures [math]73^{\circ}[/math], we know that it must have measured [math]50^{\circ}[/math] - and every other temperature between [math]44^{\circ}[/math] and [math]73^{\circ}[/math] - during that period.[br][br]Although the IVT might seem like an obvious statement not worthy of its own theorem, its power lies in telling us that we can indeed count on finding specific values between points of a continuous function. It is especially useful for finding zeroes of functions. If we know that [math]f(2)<0[/math] and [math]f(3)>0[/math], then we know that [math]f[/math] has at least one zero between [math]x=2[/math] and [math]x=3[/math].[br][br]To use the app, begin by selecting the function [math]f(x)=x[/math] from the drop-down box. Slide the [math]a[/math] and [math]b[/math] values (ends of the interval we are checking) to any location (as long as [math]a<b[/math]). You will see corresponding values of [math]f(a)[/math] and [math]f(b)[/math] identified on the graph as well as along the [math]y[/math]-axis. Now slide the [math]f(c)[/math] point along the [math]y[/math]-axis. Now try this with the next function, [math]x^2-2[/math]. As long as the selected function is continuous, and [math]f(c)[/math] is between [math]f(a)[/math] and [math]f(b)[/math], at least one [math]y[/math]-value equal to [math]f(c)[/math] will exist along the function, indicated by a point plotted on the curve. Also plotted for each intersection is the corresponding [math]x[/math]-value. This [math]x[/math]-value is the "[math]c[/math]" in the theorem: "...there exists at least one [math]c[/math] in [math](a, b)[/math]...". There should always be [i]at least one[/i] such [math]c[/math] when [math]f[/math] is continuous on [math](a, b)[/math] and [math]f(c)[/math] is between [math]f(a)[/math] and [math]f(b)[/math].[br][br]The next function ([math]sin(4x)+x[/math]) in the list are also continuous, but depending on how you set [math]a[/math] and b, you can get more than one "[math]c[/math]". This is OK - the IVT states that "[i]at least[/i] one [math]c[/math] exists...".[br][br]The next function, [math]floor(x)+0.1x[/math], illustrates the case where [math]f(x)[/math] is not continuous. Be sure to set [math]a[/math] and [math]b[/math] to land on different "steps" in order to ensure that a vertical gap will exist between them. Notice now that you can find values of [math]y[/math] between [math]a[/math] and [math]b[/math] where no [math]c[/math] exists.[br][br]The last function, [math]floor(x)+0.6sin(4x)[/math], is discontinuous, but will actually satisfy the conclusion of the IVT since no matter where you put [math]a[/math] and [math]b[/math], you will always get at least one [math]c[/math]. But the statement of the IVT is not bidirectional; in other words, a function satisfies the IVT if it is continuous, but satisfying the IVT does not necessarily mean the function is continuous.