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The 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 degree straight F. At 11 AM, the temperature is 60 degree straight F. Then, we know at some point between 7 AM and 11 AM, the temperature had to be 50 degree straight F. Why?

This is because temperature doesn’t “jump” from one level to the next, meaning that the temperature is a continuous function of time. A continuous function contains no breaks.

Another way to visualize this:

  1. Graph the points open parentheses 7 comma space 40 close parentheses and open parentheses 11 comma space 60 close parentheses.
  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.
Here are three examples of curves that meet the requirements:

A graph contains a horizontal line and and a rising curve that opens downward. The horizontal line is 50 units above the x-axis. The curve begins at the point (7, 40), contains the point (8, 50), and ends at the point (11, 60). A graph contains a horizontal line and and a curve. The horizontal line is 50 units above the x-axis. The curve begins at the point (7, 40), descends to the point (7.5, 30), then ascends to the point (10, 72), then descends to the point (11, 60). A graph contains a horizontal line and and a curve. The horizontal line is 50 units above the x-axis. The curve begins at the point (7, 40), ascends to the point (8.25, 55), then descends to the point (9.5, 45), then ascends to the point (11, 60).

This idea is generalized by 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.

Here is a picture to help visualize the IVT.

A graph defined between x equals a on the left and x equals b on the right. The graph rises from the point (a, f of a), then descends through a point marked (c, V), then continues to descend until reaching the point (b, f of b). The value of V is between the values of f of a and f of b. 


















big idea





Polynomial functions have smooth curves with no breaks. If f open parentheses x close parentheses is a polynomial function, then f open parentheses x close parentheses is continuous, and therefore the IVT can be applied to polynomial functions.











EXAMPLE

Consider the function f open parentheses x close parentheses equals x squared plus 1 on the closed interval open square brackets 1 comma space 4 close square brackets.



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. To illustrate, let’s look at the graph of f open parentheses x close parentheses on the interval open square brackets 1 comma space 4 close square brackets.



 The graph of the function f of x equals x squared plus 1 starts from the marked point (1, 2) and ends at the point (4, 17). A horizontal dashed line labeled ‘y equals 8’ extends from the second to the first quadrant by passing through the point (0, 8) and intersects the curve at the point (2.65, 8). 



To find the value of c, set f open parentheses c close parentheses equals 8 and solve. Then, there is at least one value of c between 1 and 4 that is guaranteed by the intermediate value theorem. 








c squared plus 1 equals 8


Set f open parentheses c close parentheses equals 8.






c squared equals 7


Subtract 1 from both sides.






c equals plus-or-minus square root of 7


Use the square root principle.







Since c equals short dash square root of 7 is not between 1 and 4, this value of c is not guaranteed by the theorem. However, since c equals square root of 7 is between 1 and 4, this is the value of c guaranteed by the theorem.




















watch





Here is a video in which you can see how the intermediate value theorem is applied to the quadratic 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.














 















try it





Consider the function f open parentheses x close parentheses equals 2 x cubed plus 4 x plus 1.




Is there a value of c on the interval [1, 2] such that f   (c  ) = 20?

Since f is continuous with f open parentheses 1 close parentheses equals 7 and f open parentheses 2 close parentheses equals 25 comma and 20 is between 7 and 25, there is a value of c between 1 and 2 such that f open parentheses c close parentheses equals 20.


























terms to know






Continuous (Function)


A function whose graph contains no breaks. 


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. Applications of the Intermediate Value Theorem






Here is an example of a real-world application in which the IVT can be useful. In this example, you will use the formula for the volume of a sphere. If you wish to review that and other geometry formulas, please consult the formula sheet that is referenced at the end of this tutorial. 







EXAMPLE

Suppose a design requires a spherical shape with volume 200 space in cubed comma 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 straight pi r 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 almost equal to 113.097 space in cubed




V open parentheses 4 close parentheses equals 4 over 3 straight pi open parentheses 4 close parentheses cubed almost equal to 268.083 space in cubed



By the IVT, there is a value of r between 3 and 4 inches that produces a volume of 200 space 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 at least one value of c in the interval open parentheses a comma space b close parentheses such that f open parentheses c close parentheses equals 0.







This idea can be used to determine if there is a guarantee that the graph of f open parentheses x close parentheses has at least one x-intercept on the interval open square brackets a comma space b close square brackets.







EXAMPLE

Let f open parentheses x close parentheses equals x cubed minus 4 x squared plus 10 x minus 3. 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 a polynomial and therefore is continuous. 



Next, evaluate the function at the endpoints: 




f open parentheses 0 close parentheses equals 0 cubed minus 4 open parentheses 0 close parentheses squared plus 10 open parentheses 0 close parentheses minus 3 equals short dash 3




f open parentheses 1 close parentheses equals 1 cubed minus 4 open parentheses 1 close parentheses squared plus 10 open parentheses 1 close parentheses minus 3 equals 4



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 at least one 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 is approximately 0.343, which is inside the interval open square brackets 0 comma space 1 close square brackets.



A curve starts from the fourth quadrant below the point (0, −2) and extends upward into the first quadrant by passing through the marked point labeled ‘(0.343, 0)’.




















try it





Consider the function f open parentheses x close parentheses equals short dash x to the power of 4 plus 2 x squared plus 3.




Does the IVT guarantee at least one x-intercept on the graph of f on the interval [1, 2]?

Since f is continuous with f open parentheses 1 close parentheses equals 4 and f open parentheses 2 close parentheses equals short dash 5 comma the IVT guarantees that there is at least one x-intercept on the interval open square brackets 1 comma space 2 close square brackets.


























try it





Consider the function f open parentheses x close parentheses equals x cubed minus 6 x squared plus 9 x minus 1.




Does the IVT guarantee at least one x-intercept on the graph of f on the interval [2, 4]?

While f is continuous, f open parentheses 2 close parentheses equals 1 and f open parentheses 4 close parentheses equals 3 comma which are not opposite in sign. Therefore, the IVT does not guarantee that there is an x-intercept on the interval open square brackets 2 comma space 4 close square brackets.













If you were to graph f open parentheses x close parentheses equals x cubed minus 6 x squared plus 9 x minus 1 comma you will notice that there are two x-intercepts on open square brackets 2 comma space 4 close square brackets. So, what happened? 
















big idea





It turns out, this can happen. Assuming f open parentheses a close parentheses and f open parentheses b close parentheses have opposite signs, the IVT guarantees that there is at least one x-intercept on the interval open square brackets a comma space b close square brackets. If f open parentheses a close parentheses and f open parentheses b close parentheses have the same sign, the IVT cannot be used to tell us that there is not an x-intercept on open square brackets a comma space b close square brackets.

















summary






In this lesson, you learned that since polynomial functions are continuous functions (i.e., containing no breaks), the intermediate value theorem is very useful for determining a value of c in the closed interval open square brackets a comma space b close square brackets comma, such that f open parentheses c close parentheses equals V, given that V is between f open parentheses a close parentheses and f open parentheses b close parentheses. Realizing that locating x-intercepts is also important, you explored some applications of the IVT, which illustrate how the IVT is helpful in determining if intercepts are guaranteed on a closed interval.








SOURCE: THIS TUTORIAL HAS BEEN ADAPTED FROM OPENSTAX "PRECALCULUS” BY JAY ABRAMSON. ACCESS FOR FREE AT OPENSTAX.ORG/DETAILS/BOOKS/PRECALCULUS-2E. LICENSE: CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL.











































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Terms to Know









Continuous (Function)





A function whose graph contains no breaks.





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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