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Intermediate Value Theorem

Author: Sophia
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what's covered
In this lesson, you will analyze functions using the intermediate value theorem. Specifically, this lesson will cover:

Table of Contents

1. The Intermediate Value Theorem

Suppose at 7 AM, you walk outside and it is 40 degreeF. Then, at 11 AM, the temperature is 60 degreeF. We know at some point between 7 AM and 11 AM, the temperature had to be 50 degreeF. Why?

This is because temperature doesn’t “jump” from one level to the next, meaning that the temperature is a continuous function of time.

Another way to visualize this:

  1. Graph the points left parenthesis 7 comma space 40 right parenthesis and left parenthesis 11 comma space 60 right parenthesis.
  2. Connect the points with any continuous curve. Be creative.
  3. Does your curve have a point where y equals 50 between x equals 7 and x equals 11? The answer should be yes. Otherwise, your graph is not continuous.
This idea is generalized by the intermediate value theorem.

For the intermediate value theorem (IVT), suppose f open parentheses x close parentheses is a continuous function on the closed interval open square brackets a comma space b close square brackets. Let V be a value between f open parentheses a close parentheses and f open parentheses b close parentheses. Then, there is at least one value of c between a and b such that f open parentheses c close parentheses equals V.

A graph with an x-axis with the labels ‘a’, ‘c’, and ‘b’ such that c is a value between a and b and a y-axis with the labels ‘f(b)’, ‘V’, and ‘f(a)’ such that V lies between f(b) and f(a). A curve gently rises from a marked point at (a, f(a)) and then descends downward, passing through a marked point at (c, V) labeled ‘f(c) equals V’. The curve continues to descend toward another marked point at (b, (f(b)). The dashed lines y equals V and x equals c intersect at a marked point at (c, V).

EXAMPLE

Consider the continuous function f open parentheses x close parentheses equals x squared plus 1 on the closed interval left square bracket 1 comma space 4 right square bracket. Note that f open parentheses 1 close parentheses equals 1 squared plus 1 equals 2 and f open parentheses 4 close parentheses equals 4 squared plus 1 equals 17.

Choose a value between 2 and 17, say, the value 8. By the IVT, this means that there is at least one value of c between 1 and 4 such that f open parentheses c close parentheses equals 8. Let’s find this value.

Since we want f open parentheses c close parentheses equals 8, this means c squared plus 1 equals 8, which means c squared equals 7, or c equals plus-or-minus square root of 7. Since square root of 7 is between 1 and 4, this illustrates the existence of the value of c in the theorem.

Note that short dash square root of 7 is a solution to the equation but is not in the interval open square brackets 1 comma space 4 close square brackets. This value of c is not considered when applying the Intermediate Value Theorem in the interval open square brackets 1 comma space 4 close square brackets.

hint
When solving the equation f open parentheses c close parentheses equals V comma make sure to check that each value of c is between a and b (the endpoints of the interval). This idea is emphasized again in the next video.

watch
An example of the IVT for the function f open parentheses x close parentheses equals x squared minus 7 x on open square brackets short dash 3 comma space 1 close square brackets is presented in this video.

term to know
Intermediate Value Theorem (IVT)
Suppose f open parentheses x close parentheses is a continuous function on the closed interval open square brackets a comma space b close square brackets. Let V be a value between f open parentheses a close parentheses and f open parentheses b close parentheses. Then, there is at least one value of c between a and b such that f open parentheses c close parentheses equals V.

2. Real-World Applications

Here is an example of a real-world application in which the IVT can be useful.

EXAMPLE

Suppose a design requires a spherical shape with volume 200 in cubed, but the radius of the sphere is to be between 3 and 4 inches. Is it possible to meet these requirements?

First, identify the function, which is the volume of a sphere: V open parentheses r close parentheses equals 4 over 3 πr cubed. This problem translates to: Is V open parentheses r close parentheses equals 200 for some value in the interval open square brackets 3 comma space 4 close square brackets?

Since this is a polynomial function, we know V open parentheses r close parentheses is continuous. Now, evaluate V open parentheses r close parentheses at the endpoints:

  • V open parentheses 3 close parentheses equals 4 over 3 straight pi open parentheses 3 close parentheses cubed equals 36 straight pi almost equal to 113.1 space in cubed
  • V open parentheses 4 close parentheses equals 4 over 3 straight pi open parentheses 4 close parentheses cubed equals 256 over 3 straight pi almost equal to 268.1 space in cubed
By the IVT, there is a value of r between 3 and 4 inches that produces a volume of 200 in cubed.

One particularly useful application of the IVT is locating x-intercepts. Here is the important point:

big idea
If f open parentheses a close parentheses and f open parentheses b close parentheses have different signs (one is positive and one is negative), then there is a value of c in the interval open parentheses a comma space b close parentheses such that f open parentheses c close parentheses equals 0.

EXAMPLE

Let f open parentheses x close parentheses equals x minus cos x. Show that there is an x-intercept on the interval open square brackets 0 comma space 1 close square brackets.

First, note that f open parentheses x close parentheses is continuous. Next, evaluate the function at the endpoints:

  • f open parentheses 0 close parentheses equals 0 minus cos 0 equals short dash 1
  • f open parentheses 1 close parentheses equals 1 minus cos 1 almost equal to 0.46
Since f open parentheses 0 close parentheses and f open parentheses 1 close parentheses have opposite signs, it follows from the IVT that there is a value of x in the interval open square brackets 0 comma space 1 close square brackets such that f open parentheses x close parentheses equals 0.

Here is a graph to help illustrate. As you can see, the x-intercept occurs when x almost equal to 0.739, which is inside the interval open square brackets 0 comma space 1 close square brackets.

A graph with an x-axis and a y-axis ranging from –3 to 3. A parabolic portion starts from a marked point at (0, –1) on the y-axis, extends to another marked point at (0.739, 0) on the x-axis, and ends at a marked point at (1, 0.46) in the first quadrant.

try it
Let f open parentheses x close parentheses equals x minus 5 square root of x.
Use the IVT to determine if there is a guaranteed value of x for which f (x) = 20 on the interval [36, 100].
Since f open parentheses x close parentheses is continuous on open square brackets 36 comma space 100 close square brackets with f open parentheses 36 close parentheses equals 6 and f open parentheses 100 close parentheses equals 50, there must be a value of x for which f open parentheses x close parentheses equals 20 on the interval open square brackets 36 comma space 100 close square brackets.

try it
Let f open parentheses x close parentheses equals x minus e to the power of short dash 2 x end exponent.
Use the IVT to determine if this function is guaranteed an x-intercept on the closed interval [0, 2].
Since f open parentheses x close parentheses is continuous on open square brackets 0 comma space 2 close square brackets with f open parentheses 0 close parentheses equals short dash 1 and f open parentheses 2 close parentheses almost equal to 1.98, there must be a value of x for which f open parentheses x close parentheses equals 0 on the interval open square brackets 0 comma space 2 close square brackets.

summary
In this lesson, you learned about the intermediate value theorem (IVT), which is very useful in determining if an input is guaranteed in an interval open parentheses a comma space b close parentheses for which the output is V when you have a continuous function on a closed interval open square brackets a comma space b close square brackets. Specifically, the IVT states that if you have a continuous function on a closed interval open square brackets a comma space b close square brackets comma and if V is between f open parentheses a close parentheses and f open parentheses b close parentheses comma you are guaranteed at least one input, c, in the interval open square brackets a comma space b close square brackets for which f open parentheses c close parentheses equals V.

You also learned about several useful real-world applications of the IVT, such as determining if x-intercepts exist on a closed interval. It is important to remember that if f open parentheses a close parentheses and f open parentheses b close parentheses have different signs (one is positive and one is negative), then there is a value of c in the interval open parentheses a comma space b close parentheses such that f open parentheses c close parentheses equals 0.

Source: THIS TUTORIAL HAS BEEN ADAPTED FROM CHAPTER 1 OF "CONTEMPORARY CALCULUS" BY DALE HOFFMAN. ACCESS FOR FREE AT WWW.CONTEMPORARYCALCULUS.COM. LICENSE: CREATIVE COMMONS ATTRIBUTION 3.0 UNITED STATES.

Terms to Know
Intermediate Value Theorem (IVT)

Suppose f open parentheses x close parentheses is a continuous function on the closed interval open square brackets a comma space b close square brackets. Let V be a value between f open parentheses a close parentheses and f open parentheses b close parentheses. Then, there is at least one value of c between a and b such that f open parentheses c close parentheses equals V.

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