Calculus Theorem That if F is Continuous Between a and B

Essentially, IVT says that if a function has no discontinuities, there is a point between the endpoints whose y-value is between the y-values of the endpoints. The IVT holds that a continuous function takes on all values between and .

Since the function is continuous, IVT says that there is at least one point between a and b that has a y-value between the y-values of a and b - StudySmarter Original

Uses and Applications of the Intermediate Value Theorem in Calculus

The Intermediate Value Theorem is an excellent method for solving equations. Suppose we have an equation and its respective graph (pictured below). Let's say we are looking for a solution to . The Intermediate Value Theorem says that if the function is continuous on the interval and if the target value that we're searching for is between and , we can find using .

The Intermediate Value Theorem guarantees the existence of a solution c - StudySmarter Original

The Intermediate Value Theorem is also foundational in the field of Calculus. It is used to prove many other Calculus theorems, namely the Extreme Value Theorem and the Mean Value Theorem.

Examples of the Intermediate Value Theorem

Example 1

Prove that has at least one solution. Then find the solution.

Step 1: Define f(x) and graph

We'll let

Step 2: Define a y-value for c

From the graph and the equation, we can see that the function value at is 0.

Step 3: Ensure f(x) meets the requirements of the IVT

From the graph and with a knowledge of the nature of polynomial functions, we can confidently say that is continuous on any interval we choose.

We can see that the root of lies between 1 and 1.5. So, we'll let our interval be [1, 1.5]. The Intermediate Value Theorem says that must lie between and . So, we plug in and evaluate and .

Step 4: Apply the IVT

Now that all of the IVT requirements are met, we can conclude that there is a value in such that .

So, is solvable.

Example 2

Does the function take on the value on the interval ?

Step 1: Ensure f(x) is continuous

Next, we check to make sure the function fits the requirements of the Intermediate Value Theorem.

We know that is continuous over the entire interval because it is a polynomial function.

Step 2: Find the function value at the endpoints of the interval

Plugging in and to

Step 3: Apply the Intermediate Value Theorem

Obviously, . So we can apply the IVT.

Now that all IVT requirements are met, we can conclude that there is a value in [1, 4] such that .

Thus, must take on the value 7 at least once somewhere in the interval .

Remember, the IVT guarantees at least one solution. However, there may be more than one!

Example 3

Prove the equation has at least one solution on the interval .

Let's try this one without using a graph.

Step 1: Define f(x)

To define , we'll factor the initial equation.

So, we'll let

Step 2: Define a y-value for c

From our definition of in step 1, .

Step 3: Ensure f(x) meets the requirements of the IVT

From our knowledge of polynomial functions, we know that is continuous everywhere.

We will test our interval bounds, making . Remember, using the IVT, we need to confirm

Let :

Let b= 3:

Therefore, we have

Therefore, but the IVT, we can guarantee there is at least one solution to

on the interval .

Step 4: Apply the IVT

Now that all IVT requirements are met, we can conclude that there is a value in [0, 3] such that .

So, is solvable.

Proof of the Intermediate Value Theorem

To prove the Intermediate Value Theorem, grab a piece of paper and a pen. Let the left side of your paper represent the y -axis, and the bottom of your paper represent the x -axis. Then, draw two points. One point should be on the left side of the paper (a small x -value), and one point should be on the right side (a large x -value). Draw the points such that one point is closer to the top of the paper (a large y -value) and the other is closer to the bottom (a small y- value).

The Intermediate Value Theorem states that if a function is continuous and if endpoints and exist such that , then there is a point between the endpoints where the function takes on a function value between and . So, the IVT says that no matter how we draw the curve between the two points on our paper, it will go through some y-value between the two points.

Try to draw a line or curve between the two points (without lifting your pen to simulate a continuous function) on your paper that does not go through some point in the middle of the paper. It is impossible, right? No matter how you draw a curve, it will go through the middle of the paper at some point. So, the Intermediate Value Theorem holds.

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Intermediate Value Theorem - Key takeaways

• The Intermediate Value Theorem states that if a function f is continuous on the interval [a, b] and a function value N such that where , then there is at least one number in such that

  • Essentially, the IVT holds that a continuous function takes on all values between and

• IVT is used to guarantee a solution/solve equations and is a foundational theorem in Mathematics

• To prove that a function has a solution, follow the following procedure:

  • Step 1: Define the function

  • Step 2: Find the function value at

  • Step 3: Ensure that meets the requirements of IVT by checking that lies between the function value of the endpoints and

  • Step 4: Apply the IVT

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Source: https://www.studysmarter.co.uk/explanations/math/calculus/intermediate-value-theorem/

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